For example: I have an app on my phone called Countable.
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The word "wine" can be countable, but it's often uncountable, i.e.
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When deciding whether to use "less" or "fewer," ask yourself: Is this countable?
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Countable had already been a popular app among the civic-minded since 2013.
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Countable topped the list, accounting for more than 13,000 downloads on its own.
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An apeirogon is a polygon with an infinite yet countable number of sides.
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The email interface isn't as simple as Countable, but it gets the job done.
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"Fewer" should be used for numbered, countable things such as people or other plural nouns.
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"Fewer" should be used only for numbered, countable things, especially people or other plural nouns.
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Whenever they make a point about how human life is countable in units of time,
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Moreover, measuring both problems and solutions is difficult because beneficiaries may not be visible or countable.
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The controversy stems from whether or not data is to be considered a countable or uncountable noun.
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The most popular political app today, Countable, was downloaded over 200,000 times between November 2016 and January 48.83.
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But the pool of products Amazon now draws from is, in most categories, large and not readily countable.
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The cash and stock deal will make Brigade investors shareholders in Countable, and Mahan is taking an advisory role.
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Countable, VoteSpotter and Congress are all focused on helping users find their reps, track their votes and get in touch.
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I still discarded the correct entry here because it didn't fit as a singular, but as a countable plural it did.
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Countable also allows users to comment on issues, sending emails to the appropriate representatives, or video messages, which deliver links to lawmakers.
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Sure, a cross-country flight to play a nationally televised game on a Tuesday night may now register as countable athletic time.
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Now, they organize on social media, and download apps, like Countable, that allow them to track lawmakers' votes and to contact them.
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Countable users even get to see how they voted on important pieces of legislation so they can — wait for it — hold them accountable.
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App Annie says the following apps all saw above 5,000 downloads from November through January: Countable, Voter, We the People, VoteSpotter and Congress.
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According to a NCAA survey, 62 percent of Football Bowl Subdivision coaches want to increase the number of allowable countable hours for athletes.
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To further their contribution to the democracy innovation community, Countable agreed to let Brigade open-source its voter matching software before the sale.
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No conversation may include what the N.C.A.A. deems a "countable athletically related activity" — for example, no talk of X's and O's, much less actual practices.
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" A spokesman for Macomb County Clerk Carmella Sabaugh said it's "not unusual in a recount of this size to have some precincts that aren't countable.
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Often these descriptions employ topological properties: simple, countable characteristics that don't depend on size or shape, such as how often strings in a knot cross.
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To qualify for Element Care, a nonprofit health care program for older adults that brought him Sox, a patient's countable assets must not be greater than $2,0003.
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Included on the list are The Washington Post; theSkimm, which digests big issues into bite-sized content; and Countable, an app that surfaces political news relevant to a users' district.
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Though ghost imaging was predicted in the 1990s, arguments still rage about whether entanglement is playing a role or whether it works simply because light comes in discrete, countable photons.
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Countable is part of an emerging industry of political start-ups riding the wave of civic engagement that has spread across the country and over both sides of the aisle.
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The five most popular political apps — Countable, Voter, We the People, Votespotter, and Congress — have collectively gone from around 29,000 downloads between August and October to 300,000 downloads between November and January.
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According to Google, the Latin word for rhythm is "modum" and "rhythmus" is an Greek word that means "rhyme" or "countable," but "rithimus" is not a combination of letters that has ever meant anything.
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What this means is that part of the current strength of Chinese steel production is a statistical illusion, official countable output rising to fill the gap left by the closure of unofficial, uncounted output.
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After a packed day today and yesterday, I take the last hour of the day to clear my Gmail inbox; my Atlas Obscura, Refinery303, Countable, JSTOR Daily, and alumni emails have piled up since last week.
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No matter how many activities are deemed "countable," it's clear that coaches are still going to want to have as much practice as possible—which is totally understandable, since coaches are paid serious money in order to win.
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"So far, the algorithm described only works for problems where there are a countable number of actions you can take, so it would need modification before it could be used for continuous control problems like locomotion [for instance]," Hynes told Gizmodo.
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While most of the lawsuits are centered around the recount, some Democrats see the effort as a chance to change some of the standards governing how ballots are assessed in the Sunshine State, potentially widening the pool of countable votes.
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Because the messages are all sent from within the app, you lose the feeling of calling or emailing your congressperson directly, but apparently that hasn't deterred users: Countable says it has delivered over 1,500,000 messages to Capitol Hill since this past Election Day.
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In February, we caught wind of Brigade selling off its high-grade engineering team to Pinterest in an acqui-hire while it sought a home for its IP. Today, Brigade announces its technology and data have been acquired by politician-tracking service Countable.
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An apeirogon is a shape with an infinite but countable number of sides, and over the course of the book's 1,001 chapters — a nod to "One Thousand and One Nights" — McCann delves into the two men's lives, sometimes writing in their voices.
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However, this is definitely not as handy as some of the independent websites that provide this information to you with just a few clicks, including the newer resistance-focused services like Call to Action or 5 Calls, for instance, or even other mobile apps, like Countable.
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What the proposed welfare reform is really about is the fact that more than half of the people who receive benefits through this government gimmick have $20,85033 or more in countable assets, like cash or recreational vehicles, according to data from the United States Department of Agriculture.
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The Zeppelin knot, formed from two loops laid on top of each other, gets its strength from countable topological properties, said Patil: lots of rope crossings that tend to twist each other in opposite directions, like a towel being wrung out, and circulate in opposite directions to create friction.
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The startup's San Francisco-based engineering team was too pricey for civic tech companies to afford, but those that could pay the steep price didn't need Brigade's IP. So after approaching a half-dozen potential acquirers, Mahan split the company, selling the team to Pinterest and the tech to Countable.
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Above: Countable, which saw 200,000 downloads over past three months To put these new numbers in perspective, during the three months prior to the time frame App Annie analyzed (August-October 2016), this group of apps saw roughly one-third of the downloads they saw from the month of the election through January.
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The increased prominence of media intimidation globally, as well as the relative disinterest in the Balkans, compared to the clear and countable abuses such as the jailing of journalists by Turkey's strongman Recep Tayyip Erdogan, have made it easier for leaders like Vucic to quietly quash criticism without attracting too much attention.
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Other messages that staffers tend to disregard include tweets and Facebook posts (less out of dismissiveness than because of the difficulty of determining if they come from constituents), online petitions (because they require so little effort that they aren't seen as meaningful), comments submitted through apps like Countable, and mass e-mails that originate from the Web sites of advocacy groups.
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White men who founded the country, measured black lives and found them to be countable as only three-fifths of a person are still the face of our country's money Underneath some of the criticism about normalizing racists from some white people is an unspoken fear that the hater among us isn't just the Nazi sympathizer who takes his bride to Applebee's for date night.
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From an event space in Newton, where a hand-countable crowd whispered anxieties about Joseph R. Biden Jr., to a union hall in Ottumwa, where the filmmaker Michael Moore filled in for a Washington-bound Bernie Sanders with talk of democratic socialism and Icelandic gender parity, the restless final Iowa days of this endless pre-primary campaign have less resembled a resistance fantasy than a kind of rolling low-grade panic attack for Democrats.
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Any countable product of a first-countable space is first-countable, although uncountable products need not be.
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In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is in fact well- founded and can be expressed in the language of first-order set theory. A set is hereditarily countable if and only if it is countable, and every element of its transitive closure is countable. If the axiom of countable choice holds, then a set is hereditarily countable if and only if its transitive closure is countable.
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However any uncountable discrete space is first-countable but not second-countable. Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implications do not hold.
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Furthermore, a successor aleph need not be regular. For instance, the union of a countable set of countable sets need not be countable. It is consistent with ZF that \omega_1 be the limit of a countable sequence of countable ordinals as well as the set of real numbers be a countable union of countable sets. Furthermore, it is consistent with ZF that every aleph bigger than \aleph_0 is singular (a result proved by Moti Gitik).
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In a similar way if I=[a,b),\ b\leq+\infty or if I=(a,b)\ -\infty\leq a. In any interval I_n we have at most countable many points of discontinuity, and since a countable union of at most countable sets is at most countable, it follows that the set of all discontinuities is at most countable.
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Every second-countable manifold is separable and paracompact. Moreover, if a manifold is separable and paracompact then it is also second- countable. Every compact manifold is second-countable and paracompact.
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However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω1). Every first-countable space is compactly generated. Every subspace of a first-countable space is first-countable.
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Second-countability is a stronger notion than first- countability. A space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space.
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For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent. Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second- countable if and only if it has a countable number of connected components. In particular, a connected manifold is paracompact if and only if it is second- countable.
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The set of all subsets of X that are either countable or cocountable forms a σ-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the countable- cocountable algebra on X. It is the smallest σ-algebra containing every singleton set.
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This sequence has order type \omega, so \omega+\omega is the limit of a sequence of type less than \omega+\omega whose elements are ordinals less than \omega+\omega; therefore it is singular. \aleph_1 is the next cardinal number greater than \aleph_0, so the cardinals less than \aleph_1 are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So \aleph_1 cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.
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A 0-manifold is just a discrete space. A discrete space is second-countable if and only if it is countable.
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To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.
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There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent. The countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound. This is called the "countable chain condition" rather than the more logical term "countable antichain condition" for historical reasons related to certain chains of open sets in topological spaces and chains in complete Boolean algebras, where chain conditions sometimes happen to be equivalent to antichain conditions.
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SSI allows a single applicant to own no more than $2,000 in countable assets and a married applicant to own no more than $3,000 in countable assets. Certain assets, such as the home in which one is living, are specifically exempted and are not countable.
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In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete. The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Boolean algebra on a countable number of generators. Up to isomorphism, this is the only nontrivial Boolean algebra that is both countable and atomless.
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It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line. The real line also satisfies the countable chain condition: every collection of mutually disjoint, nonempty open intervals in is countable. In order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order- isomorphic to .
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Every countable subset of the real numbers that (i.e. finite or countably infinite) is null. For example, the set of natural numbers is countable, having cardinality \aleph_0 (aleph-zero or aleph-null), is null. Another example is the set of rational numbers, which is also countable, and hence null.
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In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable.
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In mathematical complex analysis, Radó's theorem, proved by , states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface. The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable.
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Hamkins proved that any two countable models of set theory are comparable by embeddability, and in particular that every countable model of set theory embeds into its own constructible universe.
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In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theorem, second-countable then implies metrizable. Conversely, a compact metric space is second-countable.
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For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.Willard, theorem 16.11, p. 112 Therefore, the lower limit topology on the real line is not metrizable.
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The set of definable numbers is broader, but still only countable.
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In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number. Some authors use countable set to mean countably infinite alone.
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In mathematics, a topological space X is called countably generated if the topology of X is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) by the convergent sequences. The countable generated spaces are precisely the spaces having countable tightness - therefore the name countably tight is used as well.
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But the semantics of second-order ZFC are quite different from those of MK. For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models.
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One of the most important properties of first-countable spaces is that given a subset A, a point x lies in the closure of A if and only if there exists a sequence {xn} in A which converges to x. (In other words, every first-countable space is a Fréchet-Urysohn space.) This has consequences for limits and continuity. In particular, if f is a function on a first-countable space, then f has a limit L at the point x if and only if for every sequence xn → x, where xn ≠ x for all n, we have f(xn) → L. Also, if f is a function on a first-countable space, then f is continuous if and only if whenever xn → x, then f(xn) → f(x). In first-countable spaces, sequential compactness and countable compactness are equivalent properties.
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In a revisionist account, considers that Löwenheim's proof was complete. gave a (correct) proof using formulas in what would later be called Skolem normal form and relying on the axiom of choice: :Every countable theory which is satisfiable in a model M, is satisfiable in a countable substructure of M. also proved the following weaker version without the axiom of choice: : Every countable theory which is satisfiable in a model is also satisfiable in a countable model. simplified . Finally, Anatoly Ivanovich Maltsev (Анато́лий Ива́нович Ма́льцев, 1936) proved the Löwenheim–Skolem theorem in its full generality .
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In second-countable spaces--as in metric spaces--compactness, sequential compactness, and countable compactness are all equivalent properties. Urysohn's metrization theorem states that every second-countable, Hausdorff regular space is metrizable. It follows that every such space is completely normal as well as paracompact. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.
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The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC). Paul Cohen showed that ACω, is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice . ACω holds in the Solovay model. ZF suffices to prove that the union of countably many countable sets is countable.
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In 2019, Countable acquired Causes and Brigade, a company owned by Sean Parker, which is a similar platform to countable.us and was the first application on Facebook. In 2020, Countable.us merged with Causes, and placed a redirect from Countable.
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The group of rotations by rational angles is countable, but still not cyclic.
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If is TVS then the following are equivalent: 1. is locally convex and pseudometrizable. 2. has a countable neighborhood base at the origin consisting of convex sets. 3. The topology of is induced by a countable family of (continuous) seminorms. 4.
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Every first-countable space is sequential, hence each second-countable space, metric space, and discrete space is sequential. Every first-countable space is a Fréchet–Urysohn space and every Fréchet- Urysohn space is sequential. Thus every metrizable and pseudometrizable space is a sequential space and a Fréchet–Urysohn space. A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences.
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The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, they cannot both be countable.
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In the case of countable languages, all prime models are at most countably infinite.
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Convergence can be defined in terms of sequences in first-countable spaces. Nets are a generalization of sequences that are useful in spaces which are not first countable. Filters further generalize the concept of convergence. In metric spaces, one can define Cauchy sequences.
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A countable intersection of open sets in a topological space is called a Gδ set. Trivially, every open set is a Gδ set. Dually, a countable union of closed sets is called an Fσ set. Trivially, every closed set is an Fσ set.
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Countable nouns, however, can be singular or plural and can be modified by numerical markers.
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
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Any countable totally ordered set can be mapped injectively into the rational numbers in an order-preserving way. Any dense countable totally ordered set with no highest and no lowest element can be mapped bijectively onto the rational numbers in an order- preserving way.
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For example, the three- dimensional Euclidean space is not a countable union of its affine planes.
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In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.
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A topological space X is said to be limit point compact if every infinite subset of X has a limit point in X, and countably compact if every countable open cover has a finite subcover. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom of choice). In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.Engelking, General Topology, Theorem 3.10.
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The image of f is the countable set whose members are themselves infinite (and possibly uncountable) sets. By using the axiom of countable choice we may choose one member from each of these sets, and this member is itself a finite subset of X. More precisely, according to the axiom of countable choice, a (countable) set exists, so that for every natural number n, g(n) is a member of f(n) and is therefore a finite subset of X of size n. Now, we define U as the union of the members of G. U is an infinite countable subset of X, and a bijection from the natural numbers to U, , can be easily defined. We may now define a bijection that takes every member not in U to itself, and takes h(n) for every natural number to .
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The topological spaces ω1 and its successor ω1+1 are frequently used as text-book examples of non-countable topological spaces. For example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one. The space ω1 is first-countable, but not second- countable, and ω1+1 has neither of these two properties, despite being compact. It is also worthy of note that any continuous function from ω1 to R (the real line) is eventually constant: so the Stone–Čech compactification of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much larger than ω).
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The Spanish language has nouns that express concrete objects, groups and classes of objects, qualities, feelings and other abstractions. All nouns have a conventional grammatical gender. Countable nouns inflect for number (singular and plural). However, the division between uncountable and countable nouns is more ambiguous than in English.
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Vaught's work is primarily focused on model theory. In 1957, he and Tarski introduced elementary submodels and the Tarski–Vaught test characterizing them. In 1962, he and Michael D. Morley pioneered the concept of a saturated structure. His investigations on countable models of first- order theories led him to the Vaught conjecture stating that the number of countable models of a complete first-order theory (in a countable language) is always either finite, or countably infinite, or equinumerous with the real numbers.
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In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
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Most books describing large countable ordinals are on proof theory, and unfortunately tend to be out of print.
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With Urs Stammbach, he proved there exists a non-free parafree group with every countable subgroup being free.
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Alternatively there is a sharper form of the conjecture that states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel, in "On Vaught's conjecture". Cabal Seminar 76—77 (Proc. Caltech-UCLA Logic Sem., 1976—77), pp.
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Oxymel (, ) is a mixture of honey and vinegar, used as a medicine. Its name is often found in Renaissance (and later) pharmacopoeiae in Late Latin form as either a countable or uncountable noun. As a countable noun, it is spelled variously as (singular) oxymellus and oxymellis, and plural oxymeli and oxymelli.
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"Nominal Reference, Temporal Constitution and Quantification in Event Semantics". In R. Bartsch, J. van Benthem, P. von Emde Boas (eds.), Semantics and Contextual Expression, Dordrecht: Foris Publication. Many nouns have both countable and uncountable uses; for example, soda is countable in "give me three sodas", but uncountable in "he likes soda".
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Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.
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The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
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Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.
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He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
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In short, one who takes the view that real numbers are (individually) effectively computable interprets Cantor's result as showing that the real numbers (collectively) are not recursively enumerable. Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be trivially represented as a real number, therefore the real numbers are no less than countable. They are, therefore exactly countable.
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Laver's theorem, in order theory, states that order embedding of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of the sequence to a later member. This result was previously known as Fraïssé's conjecture, after Roland Fraïssé, who conjectured it in 1948; Richard Laver proved the conjecture in 1971. More generally, Laver proved the same result for order embeddings of countable unions of scattered orders.
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Every contractible space is simply connected. ;Coproduct topology: If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous. ;Cosmic space: A continuous image of some separable metric space. ;Countable chain condition: A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
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Some quantifiers are restricted to either countable or uncountable nouns, while others can be used in conjunction with both. The four quantifiers used in the Matis language are as follows: dadenpa: “many, in large quantity” (used with both countable and uncountable nouns) kimo: “much” (used only with uncountable nouns) dabɨtsɨk: “few” (used with countable nouns) papitsɨk: “little, small amount” (used with uncountable nouns) _Example 1:_ waka -n i dadenpa river -loc. ray intens.qtd “many rays in the river” _Example 2:_ ɨnbi waka -∅ papitsɨk -∅ ak- -nu 1sg.erg.
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Thus, limits in first- countable spaces can be described by sequences. However, if the space is not first-countable, nets or filters must be used. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. Filters can be thought of as sets built from multiple nets.
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If P is a countable support iteration of proper forcings, then P is proper. Crucially, all proper forcings preserve \aleph_1 .
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On the other hand, Tennenbaum's theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is , which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
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In 1922, Thoralf Skolem proved that if conventional axioms of set theory are consistent, then they have a countable model. Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox, and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the model. Thus the model considers its set of real numbers to be uncountable, or more precisely, the first-order sentence that says the set of real numbers is uncountable is true within the model.. In 1963, Paul Cohen used countable models to prove his independence theorems..
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There is also a resource requirement for SNAP, although eligibility requirements vary slightly from state to state. Generally speaking, households may have up to $2,250 in a bank account or other countable sources. If at least one person is age 60 or older and/or has disabilities, households may have $3,500 in countable resources.
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The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2. Morley showed that the number of countable models is finite or ℵ0 or ℵ1 or 2, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture.
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In particular, when the branching at each node is done on a finite subset of an arbitrary set not assumed to be countable, the form of Kőnig's lemma that says "Every infinite finitely branching tree has an infinite path" is equivalent to the principle that every countable set of finite sets has a choice function, that is to say, the axiom of countable choice for finite sets., p. 273; compare , Exercise IX.2.18. This form of the axiom of choice (and hence of Kőnig's lemma) is not provable in ZF set theory.
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A similar extension is possible for countable powers and to products of powers of Cantor space and powers of Baire space.
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For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).
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A topological space X is called countably generated if V is closed in X whenever for each countable subspace U of X the set V \cap U is closed in U. Equivalently, X is countably generated if and only if the closure of any A \subset X equals the union of closures of all countable subsets of A.
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Basic electric equipment for these trains was manufactured by Electric Machinery plant of Riga. The majority of electric trainsets was released in 10 car versions (five countable sections), part in 8-car edition (four countable sections). The separate section edition was accepted too. (?) Electric trainsets can be used in 12-car, 6-car, as well as 4-car compositions.
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A P-space in the sense of Gillman–Henriksen is a topological space in which every countable intersection of open sets is open. An equivalent condition is that countable unions of closed sets are closed. In other words, Gδ sets are open and Fσ sets are closed. The letter P stands for both pseudo-discrete and prime.
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Every finite topological space is compact since any open cover must already be finite. Indeed, compact spaces are often thought of as a generalization of finite spaces since they share many of the same properties. Every finite topological space is also second-countable (there are only finitely many open sets) and separable (since the space itself is countable).
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In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
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The disjoint union of a countable family of n-manifolds is a n-manifold (the pieces must all have the same dimension).
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Not every infinite graph has a normal spanning tree. For instance, a complete graph on an uncountable set of vertices does not have one: a normal spanning tree in a complete graph can only be a path, but a path has only a countable number of vertices. However, every graph on a countable set of vertices does have a normal spanning tree.. See in particular Theorem 3, p. 193. Even in countable graphs, a depth-first search might not succeed in eventually exploring the entire graph, and not every normal spanning tree can be generated by a depth-first search: to be a depth-first search tree, a countable normal spanning tree must have only one infinite path or one node with infinitely many children (and not both).
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Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a nonzero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ... The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.
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Maharam's theorem states that every complete measure space is decomposable into a measure on the continuum, and a finite or countable counting measure.
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As to cardinality, almost all elements of the Cantor set are not endpoints of intervals, and the whole Cantor set is not countable.
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He also constructed one-to-one correspondences to prove that the n-dimensional spaces Rn (where R is the set of real numbers) and the set of irrational numbers have the same cardinality as R.. In 1883, Cantor extended the positive integers with his infinite ordinals. This extension was necessary for his work on the Cantor–Bendixson theorem. Cantor discovered other uses for the ordinals—for example, he used sets of ordinals to produce an infinity of sets having different infinite cardinalities.. His work on infinite sets together with Dedekind's set-theoretical work created set theory.. The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets.. In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions.. Building on the work of Borel and Baire, Henri Lebesgue created his theories of measure and integration, which were published from 1899 to 1901.. Countable models are used in set theory.
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A world manifold is a four-dimensional orientable real smooth manifold. It is assumed to be a Hausdorff and second countable topological space. Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space, a paracompact and completely regular space. Being paracompact, a world manifold admits a partition of unity by smooth functions.
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By definition, a Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to . In the past, Hilbert spaces were often required to be separable as part of the definition.
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The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable. The set of all periods is countable, and all periods are computable , and in particular definable.
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In recent years Henik pointed out the importance of non-countable dimensions (e.g., which object is larger in size, how much water is in the glass) to numerical cognition. In recent publications, Henik and colleagues suggested the existence of a magnitude sense rather than a number sense, with the former based on the ability to perceive and evaluate non- countable dimensions (e.g., size).
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This lemma shows that for a complex number a, the fiber f−1(a) is a discrete (and therefore countable) set, unless f ≡ a.
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Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of semi- norms.
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The back-and-forth method can be used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.
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In mathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
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In detail, a function f: X → Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous.
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If p\colon X \to Y is a perfect map and if X is locally compact, then Y is locally compact. 4\. If p\colon X \to Y is a perfect map and if X is second countable, then Y is second countable. 5\. Every injective perfect map is a homeomorphism. This follows from the fact that a bijective closed map has a continuous inverse. 6\.
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A discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems, which may also be called analog systems. A final discrete system is often modeled with a directed graph and is analyzed for correctness and complexity according to computational theory. Because discrete systems have a countable number of states, they may be described in precise mathematical models.
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In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. While the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says Y is cofinite.
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He also proved that any element in an at most countable conical refinement monoid is measured by a unique (up to isomorphism) V-measure on a unique at most countable Boolean algebra. He raised there the problem whether any conical refinement monoid is measurable. This was answered in the negative by Friedrich Wehrung in 1998. The counterexamples can have any cardinality greater than or equal to ℵ2.
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Burushaski is a double-marking language and word order is generally subject–object–verb. Nouns in Burushaski are divided into four genders: human masculine, human feminine, countable objects, and uncountable ones (similar to mass nouns). The assignment of a noun to a particular gender is largely predictable. Some words can belong both to the countable and to the uncountable class, producing differences in meaning.
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Meristics is an area of ichthyology which relates to counting quantitative features of fish, such as the number of fins or scales. A meristic (countable trait) can be used to describe a particular species of fish, or used to identify an unknown species. Meristic traits are often described in a shorthand notation called a meristic formula. Meristic characters are the countable structures occurring in series (e.g.
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This situation is impossible in finite dimensions. The tangent cone to the cube at the zero vector is the whole space. Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.
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A collection in a space is countably locally finite (or σ-locally finite) if it is the union of a countable family of locally finite collections of subsets of X. Countable local finiteness is a key hypothesis in the Nagata–Smirnov metrization theorem, which states that a topological space is metrizable if and only if it is regular and has a countably locally finite basis.
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In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence N1, N2, … of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with Ni contained in N. Since every neighborhood of any point contains an open neighborhood of that point the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.
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The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X. Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T1, as all singletons are closed. If X is an uncountable set, any two open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique.
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The second Prüfer theorem provides a straightforward extension of the fundamental theorem of finitely generated abelian groups to countable abelian p-groups without elements of infinite height: each such group is isomorphic to a direct sum of cyclic groups whose orders are powers of p. Moreover, the cardinality of the set of summands of order pn is uniquely determined by the group and each sequence of at most countable cardinalities is realized. Helmut Ulm (1933) found an extension of this classification theory to general countable p-groups: their isomorphism class is determined by the isomorphism classes of the Ulm factors and the p-divisible part. : Ulm's theorem.
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The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains the open sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closed under countable intersections as well. A short proof that the Borel algebra is well defined proceeds by showing that the entire powerset of the space is closed under complements and countable unions, and thus the Borel algebra is the intersection of all families of subsets of the space that have these closure properties. This proof does not give a simple procedure for determining whether a set is Borel.
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Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.
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Tennenbaum's theorem states that no countable nonstandard model of PA is recursive. Moreover, neither the addition nor the multiplication of such a model can be recursive.
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The word "beauty" is often used as a countable noun to describe a beautiful woman, an excellent example of something, or a pleasing feature of something.
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In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level Π sets of the Borel hierarchy.
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The proof consists in extending a lifting to ever larger sub-σ-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process.
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Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1 or \Omega.
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However, if the random variable has an infinite but countable probability space (i.e., corresponding to a die with infinite many faces) the 1965 paper demonstrates that for a dense subset of priors the Bernstein-von Mises theorem is not applicable. In this case there is almost surely no asymptotic convergence. Later in the 1980s and 1990s Freedman and Persi Diaconis continued to work on the case of infinite countable probability spaces.
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It can be shown that Wα is self-dual if and only if α is either 0, an even successor ordinal, or a limit ordinal of countable cofinality.
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In this case, one speaks of an ω-continuous poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is ω-algebraic.
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Pips are small but easily countable items, such as the dots on dominoes and dice, or the symbols on a playing card that denote its suit and value.
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Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from within the model, but are countable to someone outside the model. The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory. In the other direction, model theory itself can be formalized within ZFC set theory.
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In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who published this theorem for countable linear orders in 1940. More strongly, they showed that in the countable case there exists an order embedding into a proper subset of the given order; however, they provided examples showing that this strengthening does not always hold for uncountable orders. In reverse mathematics, the Dushnik–Miller theorem for countable linear orders has the same strength as the arithmetical comprehension axiom (ACA0), one of the "big five" subsystems of second-order arithmetic.
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Choose a collection of 2ℵ0 measure 0 subsets of R such that every measure 0 subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α. For each countable ordinal β choose a real number xβ that is not in any of the sets Sα for α < β, which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set X of all these real numbers xβ has only a countable number of elements in each set Sα, so is a Sierpiński set. It is possible for a Sierpiński set to be a subgroup under addition.
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A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved - the σ-algebra produced by this process is known as the Borel algebra on the real line, and can also be conceived as the smallest (i.e. "coarsest") σ-algebra containing all the open sets, or equivalently containing all the closed sets. It is foundational to measure theory, and therefore modern probability theory, and a related construction known as the Borel hierarchy is of relevance to descriptive set theory.
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We can imagine even larger ordinals that are still countable. For example, if ZFC has a transitive model (a hypothesis stronger than the mere hypothesis of consistency, and implied by the existence of an inaccessible cardinal), then there exists a countable \alpha such that L_\alpha is a model of ZFC. Such ordinals are beyond the strength of ZFC in the sense that it cannot (by construction) prove their existence. Even larger countable ordinals, called the stable ordinals, can be defined by indescribability conditions or as those \alpha such that L_\alpha is a 1-elementary submodel of L; the existence of these ordinals can be proved in ZFC,Barwise (1976), theorem 7.2.
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With this remark Froda's theorem takes the stronger form: Let f be a monotone function defined on an interval I. Then the set of discontinuities is at most countable.
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In mathematics, in the realm of topology, a paranormal space is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion.
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For any metric space , the following are equivalent (assuming countable choice): # is compact. # is complete and totally bounded (this is also equivalent to compactness for uniform spaces). # is sequentially compact; that is, every sequence in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable uniform spaces). # is limit point compact (also called countably compact); that is, every infinite subset of has at least one limit point in .
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2, p. 50. In logic, a theory that has only one model (up to isomorphism) with a given infinite cardinality is called -categorical. The fact that the Rado graph is the unique countable graph with the extension property implies that it is also the unique countable model for its theory. This uniqueness property of the Rado graph can be expressed by saying that the theory of the Rado graph is ω-categorical.
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In the 1976-1984 period the Railcar, Manufacturing Plant of Riga continued the release of ER2 electric trainsets (Manufacturing Sign 62-61) They were designed for 3000 V DC. Their construction began in 1962. The basic equipment for these trains were produced by Electric Machinery Plant of Riga. The majority of trainsets were released in the 10-car edition (five countable sections). Some were in the 12-car edition (six countable sections).
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The Borel sets of Aω are the smallest class of subsets of Aω that includes the open sets and is closed under complement and countable union. That is, the Borel sets are the smallest σ-algebra of subsets of Aω containing all the open sets. The Borel sets are classified in the Borel hierarchy based on how many times the operations of complement and countable union are required to produce them from open sets.
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Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.Stack Exchange, Hausdorff locally compact and second countable is sigma-compact A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds. These are just the connected components of M, which are open sets since manifolds are locally- connected. Being locally path connected, a manifold is path-connected if and only if it is connected.
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If is a TVS then the following are equivalent: 1. is metrizable. 2. is Hausdorff and pseudometrizable. 3. is Hausdorff and has a countable neighborhood base at the origin. 4.
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His book in English language, the Lake Manchar is his countable contribution. He was awarded with presidential national award of pride of performance for literary contribution on 14 August 1990.
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It can be shown that the set of prices is coherent when they satisfy the probability axioms and related results such as the inclusion–exclusion principle (but not necessarily countable additivity).
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George Mackey and Irving Kaplansky generalized Ulm's theorem to certain modules over a complete discrete valuation ring. They introduced invariants of abelian groups that lead to a direct statement of the classification of countable periodic abelian groups: given an abelian group A, a prime p, and an ordinal α, the corresponding αth Ulm invariant is the dimension of the quotient : pαA[p]/pα+1A[p], where B[p] denotes the p-torsion of an abelian group B, i.e. the subgroup of elements of order p, viewed as a vector space over the finite field with p elements. : A countable periodic reduced abelian group is determined uniquely up to isomorphism by its Ulm invariants for all prime numbers p and countable ordinals α.
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Here an algebra means a model for a language with a countable number of function symbols, in other words a set with a countable number of functions from finite products of the set to itself. A cardinal is a Jónsson cardinal if and only if there are no Jónsson algebras of that cardinality. The existence of Jónsson functions shows that if algebras are allowed to have infinitary operations, then there are no analogues of Jónsson cardinals.
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Measure words play a similar role to classifiers, except that they denote a particular quantity of something (a drop, a cupful, a pint, etc.), rather than the inherent countable units associated with a count noun. Classifiers are used with count nouns; measure words can be used with mass nouns (e.g. "two pints of mud"), and can also be used when a count noun's quantity is not described in terms of its inherent countable units (e.g. "two pints of acorns").
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In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions. The definition implies that it also includes the empty subset and that it is closed under countable intersections. The pair (X, Σ) is called a measurable space or Borel space. A σ-algebra is a type of algebra of sets.
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The Hahn–Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves: :A non- empty Hausdorff topological space is a continuous image of the unit interval if and only if it is a compact, connected, locally connected, second-countable space. Spaces that are the continuous image of a unit interval are sometimes called Peano spaces. In many formulations of the Hahn–Mazurkiewicz theorem, second-countable is replaced by metrizable. These two formulations are equivalent.
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In addition, some states pay additional SSI funds. As of January 2018, over 8 million people receive SSI. For some claimants, this program is harder to receive than funds from RSDI. To warrant a processing time of anything more than a day and an immediate denial, certain specific criteria must be met, including citizenship status, having less than $2,000.00 in countable financial resources, or having countable income of less than $718.00 per month from any source.
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In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.
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A topological space X is a 3-manifold if it is a second-countable Hausdorff space and if every point in X has a neighbourhood that is homeomorphic to Euclidean 3-space.
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If we drop the requirement for the space to be locally convex, we obtain F-spaces: vector spaces with complete translation-invariant metrics. LF-spaces are countable inductive limits of Fréchet spaces.
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Therefore, if X is a metrizable space with a countable basis, one implication of Bing's metrization theorem holds. In fact, Bing's metrization theorem is almost a corollary of the Nagata-Smirnov theorem.
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Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no.
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In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.. The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete..
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The Borel graph theorem states: An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. :Definition: A topological space is called a if it is the countable intersection of countable unions of compact sets. :Definition: A Hausdorff topological space is called K-analytic if it is the continuous image of a space (that is, if there is a space and a continuous map of onto ). Every compact set is K-analytic so that there are non-separable K-analytic spaces.
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In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general), a space is cosmic if and only if it has a countable network; namely a countable collection of subsets of the space such that any open set is the union of a subcollection of these sets. Cosmic spaces have several interesting properties. There are a number of unsolved problems about them.
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Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω1; their supremum is called Church–Kleene ω1 or ω1CK (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω1CK are the recursive ordinals (see below). Countable ordinals larger than this may still be defined, but do not have notations. Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted.
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In 1962 Spector extended Gödel's Dialectica interpretation of arithmetic to full mathematical analysis, by showing how the schema of countable choice can be given a Dialectica interpretation by extending system T with bar recursion.
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In both cases, we find that the limitation on the weakened \psi function comes not so much from the operations allowed on the countable ordinals as on the uncountable ordinals we allow ourselves to denote.
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In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size...
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In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
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For σ-compact spaces this is equivalent to Halmos's definition. For spaces that are not σ-compact the Baire sets under this definition are those under Halmos's definition together with their complements. However, in this case it is no longer true that a finite Baire measure is necessarily regular: for example the Baire probability measure that assigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co- countable subset is a Baire probability measure that is not regular.
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In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".
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In reverse mathematics, the version of the theorem for countable orders is denoted FRA (for Fraïssé) and the version for countable unions of scattered orders is denoted LAV (for Laver). In terms of the "big five" systems of second-order arithmetic, FRA is known to fall in strength somewhere between the strongest two systems, \Pi_1^1-CA0 and ATR0, and to be weaker than \Pi_1^1-CA0. However, it remains open whether it is equivalent to ATR0 or strictly between these two systems in strength.
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In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963. The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generated by a countable number of elements. As the size of countably generated complete Boolean algebras is unbounded, this shows that there is no free complete Boolean algebra on a countable number of elements.
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Example: The family of all dense open sets of having finite Lebesgue measure is a proper -system and a free prefilter. The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of . Since is a Baire space, every countable intersection of sets in is dense in (and also comeagre and non-meager) so the set of all countable intersections of elements of is a prefilter and -system; it is also finer than, and not equivalent to, .
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There exist topological vector spaces that are sequential but not -sequential (and thus not -sequential). where recall that every metrizable space is first countable. There also exist topological vector spaces that are -sequential but not sequential.
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If B_k is the entire boundary of C_k then C_k is called spanning. For \Delta not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of \Delta having analogous properties.
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The inverse of this bijection is an injection into the natural numbers of the computable numbers, proving that they are countable. But, again, this subset is not computable, even though the computable reals are themselves ordered.
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Hence S is not metrizable (since any compact metric space is separable); however, it is first countable. Also, S is connected but not path connected, nor is it locally path connected. Its fundamental group is trivial.
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A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
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No uncountable cover of a Lindelöf space can be locally finite, by essentially the same argument as in the case of compact spaces. In particular, no uncountable cover of a second-countable space is locally finite.
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The set of points at which a function is continuous is always a set. The set of discontinuities is an set. The set of discontinuities of a monotonic function is at most countable. This is Froda's theorem.
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He wrote many articles about history of Sindh but his book 'Sindh Ji Tareekh Jo Jadeed Mutalio' is his countable contribution. Abdullah Magsi mostly wrote about the neglected and unexplored heritage of the Dadu District including Sindh.
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Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say , produces an example with a different order type.
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Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).
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Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is originally due to S. P. Franklin (aka Stan Franklin) in 1965, who was investigating the question of "what are the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences?" Franklin arrived at the definition above by noting that every first-countable space can be specified completely by the knowledge of its convergent sequences, and then he abstracted properties of first countable spaces that allowed this to be true.
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Every field of characteristic zero contains a unique subfield isomorphic to Q. Q is the field of fractions of the integers Z. The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the field of algebraic numbers. The set of all rational numbers is countable, while the set of all real numbers (as well as the set of irrational numbers) is uncountable. Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.
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The top seven players with the most points accumulated in ATP Challenger tournaments during the year plus one wild card entrant from the host country qualified for the 2013 ATP Challenger Tour Finals. Countable points include points earned in 2013 until 21 October, plus points earned at late-season 2012 Challenger tournaments. However, players were only eligible to qualify for the tournament if they played a minimum of eight ATP Challenger Tour tournaments during the season. Moreover, the accumulated year-to-date points were only countable to a maximum of ten best results.
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Given a complete n-type p one can ask if there is a model of the theory that omits p, in other words there is no n-tuple in the model that realizes p. If p is an isolated point in the Stone space, i.e. if {p} is an open set, it is easy to see that every model realizes p (at least if the theory is complete). The omitting types theorem says that conversely if p is not isolated then there is a countable model omitting p (provided that the language is countable).
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Note that by Rice's theorem on index sets, most domains I are not recursive. Indeed, no effective map between all counting numbers \omega and the infinite (non-finite) indexing set I is asserted here, merely the subset relation I\subseteq\omega. Being dominated by a constructively non- countable set of numbers I, the name subcountable thus conveys that the uncountable set X is no bigger than \omega. The demonstration that X is subcountable also implies that it is classically (non-constructively) formally countable, but this does not reflect any effective countability.
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In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".
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While selectionists could insist on interpreting Fresnel's diffraction integrals in terms of discrete, countable rays, they could not do the same with his theory of polarization. For a selectionist, the state of polarization of a beam concerned the distribution of orientations over the population of rays, and that distribution was presumed to be static. For Fresnel, the state of polarization of a beam concerned the variation of a displacement over time. That displacement might be constrained but was not static, and rays were geometric constructions, not countable objects.
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There are several weaker statements that are not equivalent to the axiom of choice, but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (ACω or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
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Based on "Chomsky Definition", a language is assumed to be a countable set of sentences, each of finite length, and constructed out of a countable set of symbols. A theory of syntax is assumed to introduce symbols, and rules to construct well-formed sentences. A language is called fully interpreted, if meanings are attached to its sentences so that they all are either true or false. A fully interpreted language L which does not have a truth predicate can be extended to a fully interpreted language Ľ that contains a truth predicate T, i.e.
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This problem was resolved by defining measure only on a sub-collection of all subsets; the so- called measurable subsets, which are required to form a -algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.Halmos, Paul (1950), Measure theory, Van Nostrand and Co. Indeed, their existence is a non- trivial consequence of the axiom of choice.
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Intuitively, the expectation of a random variable taking values in a countable set of outcomes is defined analogously as the weighted sum of the outcome values, where the weights correspond to the probabilities of realizing that value. However, convergence issues associated with the infinite sum necessitate a more careful definition. A rigorous definition first defines expectation of a non-negative random variable, and then adapts it to general random variables. Let X be a non- negative random variable with a countable set of outcomes x_1, x_2, \ldots, occurring with probabilities p_1, p_2, \ldots, respectively.
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Iterated forcing with finite supports was introduced by Solovay and Tennenbaum to show the consistency of Suslin's hypothesis. Easton introduced another type of iterated forcing to determine the possible values of the continuum function at regular cardinals. Iterated forcing with countable support was investigated by Laver in his proof of the consistency of Borel's conjecture, Baumgartner, who introduced Axiom A forcing, and Shelah, who introduced proper forcing. Revised countable support iteration was introduced by Shelah to handle semi-proper forcings, such as Prikry forcing, and generalizations, notably including Namba forcing.
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This page contains examples of Markov chains and Markov processes in action. All examples are in the countable state space. For an overview of Markov chains in general state space, see Markov chains on a measurable state space.
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In September, 2012, DecodeDC moved to the Mule Radio Podcast Syndicate. In November 2013, DecodeDC was acquired by the E.W. Scripps Corp. Currently, Seabrook is the Managing Editor of Countable, the premier civic technology app in the U.S.
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In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of barrelled spaces.
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While the set of complex numbers is uncountable, the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the complex numbers; in this sense, almost all complex numbers are transcendental.
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In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.
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In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist.
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When the basis for a vector space is no longer countable, then the appropriate axiomatic formalization for the vector space is that of a topological vector space. The tensor product is still defined, it is the topological tensor product.
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A theorem of Gábor Szegő states that if f is in H^1, the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f (certainly countable in number) satisfy the Blaschke condition.
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Skolem's paradox is the seeming contradiction that on the one hand, the set of real numbers is uncountable (and this is provable from ZFC, or even from a small finite subsystem ZFC' of ZFC), while on the other hand there are countable transitive models of ZFC' (this is provable in ZFC), and the set of real numbers in such a model will be a countable set. The paradox can be resolved by noting that countability is not absolute to submodels of a particular model of ZFC. It is possible that a set X is countable in a model of set theory but uncountable in a submodel containing X, because the submodel may contain no bijection between X and ω, while the definition of countability is the existence of such a bijection. The Löwenheim–Skolem theorem, when applied to ZFC, shows that this situation does occur.
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If T is a compact operator, or, more generally, an inessential operator, then it can be shown that the spectrum is countable, that zero is the only possible accumulation point, and that any nonzero λ in the spectrum is an eigenvalue.
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In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of quasibarrelled spaces.
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If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω. If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β<α such that xι is in U for all ι≥β. Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 (omega-one, the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.
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Kc (and hence K) is a fine-structural countably iterable extender model below long extenders. (It is not currently known how to deal with long extenders, which establish that a cardinal is superstrong.) Here countable iterability means ω1+1 iterability for all countable elementary substructures of initial segments, and it suffices to develop basic theory, including certain condensation properties. The theory of such models is canonical and well understood. They satisfy GCH, the diamond principle for all stationary subsets of regular cardinals, the square principle (except at subcompact cardinals), and other principles holding in L. Kc is maximal in several senses.
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The completeness theorem can also be understood in terms of consistency, as a consequence of Henkin's model existence theorem. We say that a theory T is syntactically consistent if there is no sentence s such that both s and its negation ¬s are provable from T in our deductive system. The model existence theorem says that for any first-order theory T with a well-orderable language, :if T is syntactically consistent, then T has a model. Another version, with connections to the Löwenheim–Skolem theorem, says: :Every syntactically consistent, countable first-order theory has a finite or countable model.
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The dihedral group of order 8 requires two generators, as represented by this cycle diagram. In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated.
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This account is based mainly on . To understand the early history of model theory one must distinguish between syntactical consistency (no contradiction can be derived using the deduction rules for first-order logic) and satisfiability (there is a model). Somewhat surprisingly, even before the completeness theorem made the distinction unnecessary, the term consistent was used sometimes in one sense and sometimes in the other. The first significant result in what later became model theory was Löwenheim's theorem in Leopold Löwenheim's publication "Über Möglichkeiten im Relativkalkül" (1915): :For every countable signature σ, every σ-sentence which is satisfiable is satisfiable in a countable model.
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In practical terms, this means that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces. Many principles that imply the axiom of choice in their general form (such as "Every vector space has a basis") become provable in weak subsystems of second-order arithmetic when they are restricted. For example, "every field has an algebraic closure" is not provable in ZF set theory, but the restricted form "every countable field has an algebraic closure" is provable in RCA0, the weakest system typically employed in reverse mathematics.
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Jech (2003) p.684 The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports. MM implies that the value of the continuum is \aleph_2Jech (2003) p.685 and that the ideal of nonstationary sets on ω1 is \aleph_2-saturated.Jech (2003) p.687 It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ≥ω2 and every element of S has countable cofinality, then there is an ordinal α<κ such that S∩α is stationary in α. In fact, S contains a closed subset of order type ω1.
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Any two distinct points in [-1,1] are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in [-1,1], making [-1,1] with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlapping interval topology is second countable, with a countable basis being given by the intervals [-1,s), (r,s) and (r,1] with r < 0 < s and r and s rational.
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A σ-algebra Σ is just a σ-ring that contains the universal set X. A σ-ring need not be a σ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtained by their countable union yet its measure is not finite.
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Since coverings are local homeomorphisms, a covering of a topological n-manifold is an n-manifold. (One can prove that the covering space is second-countable from the fact that the fundamental group of a manifold is always countable.) However a space covered by an n-manifold may be a non-Hausdorff manifold. An example is given by letting C be the plane with the origin deleted and X the quotient space obtained by identifying every point with . If is the quotient map then it is a covering since the action of Z on C generated by is properly discontinuous.
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Solovay constructed his model in two steps, starting with a model M of ZFC containing an inaccessible cardinal κ. The first step is to take a Levy collapse M[G] of M by adding a generic set G for the notion of forcing that collapses all cardinals less than κ to ω. Then M[G] is a model of ZFC with the property that every set of reals that is definable over a countable sequence of ordinals is Lebesgue measurable, and has the Baire and perfect set properties. (This includes all definable and projective sets of reals; however for reasons related to Tarski's undefinability theorem the notion of a definable set of reals cannot be defined in the language of set theory, while the notion of a set of reals definable over a countable sequence of ordinals can be.) The second step is to construct Solovay's model N as the class of all sets in M[G] that are hereditarily definable over a countable sequence of ordinals.
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The spot-breasted oriole ranges only on the Pacific side of Central America. An introduced breeding population also exists on the Atlantic coast of southern Florida. The population is considered to be established enough to be "countable" for birdwatchers by the ABA.
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The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact. # is closed and bounded (as a subset of any metric space whose restricted metric is ). The converse may fail for a non-Euclidean space; e.g.
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Not all real numbers are computable. The entire set of computable numbers is countable, so most reals are not computable. Specific examples of noncomputable real numbers include the limits of Specker sequences, and algorithmically random real numbers such as Chaitin's Ω numbers.
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Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.
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Examples include Laver forcing. The concept is named after Richard Laver. Shelah proved that when proper forcings with the Laver property are iterated using countable supports, the resulting forcing notion will have the Laver property as well.Shelah, S., Proper and Improper Forcing, Springer (1992)C.
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A quotient of countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore the countably generated spaces form a coreflective subcategory of the category of topological spaces. They are the coreflective hull of all countable spaces.
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The priority method is now the main technique for establishing results about r.e. sets. The idea of the priority method for constructing a r.e. set X is to list a countable sequence of requirements that X must satisfy. For example, to construct a r.e.
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A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to second-countability and also to the Lindelöf property.
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Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first- countable. ;Metrizable/Metrisable: A space is metrizable if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff).
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Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces. Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.
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For locally compact Hausdorff topological spaces that are not σ-compact the three definitions above need not be equivalent, A discrete topological space is locally compact and Hausdorff. Any function defined on a discrete space is continuous, and therefore, according to the first definition, all subsets of a discrete space are Baire. However, since the compact subspaces of a discrete space are precisely the finite subspaces, the Baire sets, according to the second definition, are precisely the at most countable sets, while according to the third definition the Baire sets are the at most countable sets and their complements. Thus, the three definitions are non-equivalent on an uncountable discrete space.
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Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable while S is uncountable. Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.
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In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by . The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of order type that is homogeneous for (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal is the smallest cardinal such that : Existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal , there is an -Erdős cardinal". In fact, for every indiscernible satisfies "for every ordinal , there is an -Erdős cardinal in (the Levy collapse to make countable)".
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For other statements equivalent to ACω, see and . A common misconception is that countable choice has an inductive nature and is therefore provable as a theorem (in ZF, or similar, or even weaker systems) by induction. However, this is not the case; this misconception is the result of confusing countable choice with finite choice for a finite set of size n (for arbitrary n), and it is this latter result (which is an elementary theorem in combinatorics) that is provable by induction. However, some countably infinite sets of nonempty sets can be proven to have a choice function in ZF without any form of the axiom of choice.
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In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is the countable union of measurable sets with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e.
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Pi () is a well known transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial with rational coefficients. The best known transcendental numbers are and e. Though only a few classes of transcendental numbers are known, in part because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers compose a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set.
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Early research often considered mappings between interior algebras which were homomorphisms of the underlying Boolean algebras but which did not necessarily preserve the interior or closure operator. Such mappings were called Boolean homomorphisms. (The terms closure homomorphism or topological homomorphism were used in the case where these were preserved, but this terminology is now redundant as the standard definition of a homomorphism in universal algebra requires that it preserves all operations.) Applications involving countably complete interior algebras (in which countable meets and joins always exist, also called σ-complete) typically made use of countably complete Boolean homomorphisms also called Boolean σ-homomorphisms - these preserve countable meets and joins.
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Smooth manifolds are sometimes defined as embedded submanifolds of real coordinate space Rn, for some n. This point of view is equivalent to the usual, abstract approach, because, by the Whitney embedding theorem, any second-countable smooth (abstract) m-manifold can be smoothly embedded in R2m.
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If is a topological vector space (TVS) (where note in particular that is assumed to be a vector topology) then the following are equivalent: 1. is pseudometrizable (i.e. the vector topology is induced by a pseudometric on ). 2. has a countable neighborhood base at the origin. 3.
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To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show classically that the computable numbers are countable. And yet Cantor's diagonal argument shows that real numbers have higher cardinality. Furthermore, the diagonal argument seems perfectly constructive.
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In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact spaceMore generally, on any compactly generated space; e.g., a first-countable space. is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic.
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For more information on ccc in the context of forcing, see . More generally, if κ is a cardinal then a poset is said to satisfy the κ-chain condition if every antichain has size less than κ. The countable chain condition is the ℵ1-chain condition.
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Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal. Every locally compact group which is second-countable is metrizable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete.
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Let A be a countable admissible set. Let L be an A-finite relational language. Suppose \Gamma is a set of L_A-sentences, where \Gamma is a \Sigma_1 set with parameters from A, and every A-finite subset of \Gamma is satisfiable. Then \Gamma is satisfiable.
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The Rado graph is, up to graph isomorphism, the only countable graph with the extension property. For example, let G and H be two countable graphs with the extension property, let Gi and Hi be isomorphic finite induced subgraphs of G and H respectively, and let gi and hi be the first vertices in an enumeration of the vertices of G and H respectively that do not belong to Gi and Hi. Then, by applying the extension property twice, one can find isomorphic induced subgraphs Gi + 1 and Hi + 1 that include gi and hi together with all the vertices of the previous subgraphs. By repeating this process, one may build up a sequence of isomorphisms between induced subgraphs that eventually includes every vertex in G and H. Thus, by the back- and-forth method, G and H must be isomorphic.. Because the graphs constructed by the random graph construction, binary number construction, and Paley graph construction are all countable graphs with the extension property, this argument shows that they are all isomorphic to each other., Fact 2.
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It is easy to see that the existence of a dense orbit implies in topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable, complete metric space, then topological transitivity implies the existence of a dense set of points in X that have dense orbits.
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The converse of Jensen's covering theorem is also true: if 0# exists then the countable set of all cardinals less than ℵω cannot be covered by a constructible set of cardinality less than ℵω. In his book Proper Forcing, Shelah proved a strong form of Jensen's covering lemma.
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Accessed February 16, 2010 It is quite often incorrectly called hypodactyly, but the Greek prefixes hypo- and hyper- are used for continuous scales (e.g. in hypoglycaemia and hyperthermia). This as opposed to discrete or countable scales, where oligo- and poly- should be used (e.g. in oligarchy and polygamy).
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The definition of Markov chains has evolved during the 20th century. In 1953 the term Markov chain was used for stochastic processes with discrete or continuous index set, living on a countable or finite state space, see Doob.Joseph L. Doob: Stochastic Processes. New York: John Wiley & Sons, 1953.
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Urysohn proved that an Urysohn universal space exists, and that any two Urysohn universal spaces are isometric. This can be seen as follows. Take (X,d),(X',d'), two Urysohn spaces. These are separable, so fix in the respective spaces countable dense subsets (x_n)_n, (x'_n)_n.
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In mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every cluster point is a subsequential limit, but not conversely. For example {-1,1,-1,1,...} has a subsequential limit -1, but -1 is not a cluster point. In first-countable spaces, the two concepts coincide.
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The following is a standard proof that a complete pseudometric space \scriptstyle X is a Baire space. Let be a countable collection of open dense subsets. We want to show that the intersection is dense. A subset is dense if and only if every nonempty open subset intersects it.
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For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from . However, Fred Richman proved a reformulated version of the theorem that does work.See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from .
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The main occupation of the people of Padur is agriculture — either as agriculturalist or as laborer. A few are working in government services. A countable number of people work in other states of India and abroad, in either public or private sectors. A few entrepreneurs also live here.
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For S a stationary subset of \omega_1 we set P equal to the set of closed countable sequences from S. In V[G], we have that \bigcup G is a closed unbounded subset of S and \aleph_1 is preserved, and if CH holds then all cardinals are preserved.
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In mathematics, the Bachmann–Howard ordinal (or Howard ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced by and .
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In computability theory, the term "Gödel numbering" is used in settings more general than the one described above. It can refer to: #Any assignment of the elements of a formal language to natural numbers in such a way that the numbers can be manipulated by an algorithm to simulate manipulation of elements of the formal language. #More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation of the mathematical object. Also, the term Gödel numbering is sometimes used when the assigned "numbers" are actually strings, which is necessary when considering models of computation such as Turing machines that manipulate strings rather than numbers.
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Beckman and Quarles observe that the theorem is not true for the real line (one- dimensional Euclidean space). For, the function that returns if is an integer and returns otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry. Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. This example involves the composition of two discontinuous functions: one that maps every point of the Hilbert space onto a nearby point in a countable dense subspace, and a second that maps this dense set into a countable unit simplex (an infinite set of points all at unit distance from each other).
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The Löwenheim–Skolem theorem shows that if a first-order theory of cardinality λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in model theory, it implies that it is not possible to characterize countability or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ(x) such that an arbitrary structure M satisfies φ if and only if the domain of discourse of M is countable (or, in the second case, uncountable). The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic.
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Skolem was a pioneer model theorist. In 1920, he greatly simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model, then it has a countable model. His 1920 proof employed the axiom of choice, but he later (1922 and 1928) gave proofs using Kőnig's lemma in place of that axiom. It is notable that Skolem, like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists Charles Sanders Peirce and Ernst Schröder, including ∏, ∑ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, and Principles of Mathematical Logic.
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A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of S may be mapped into a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example. A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set).
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Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem showed in 1922 that every consistent theory of first-order predicate calculus, such as set theory, has an at most countable model. However, Cantor's theorem proves that there are uncountable sets. The root of this seeming paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in which the cardinality is measured. It is possible for a set to be uncountable in one model of set theory but countable in a larger model (because the bijections that establish countability are in the larger model but not the smaller one).
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The second Prüfer theorem states that a countable periodic abelian group whose elements have finite height is isomorphic to a direct sum of cyclic groups. Examples show that the assumption that the group be countable cannot be removed. The two Prüfer theorems follow from a general criterion of decomposability of an abelian group into a direct sum of cyclic subgroups due to L. Ya. Kulikov: > An abelian p-group A is isomorphic to a direct sum of cyclic groups if and > only if it is a union of a sequence {Ai} of subgroups with the property that > the heights of all elements of Ai are bounded by a constant (possibly > depending on i).
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Let A and B be countable abelian p-groups such that for every ordinal σ their Ulm factors are isomorphic, Uσ(A) ≅ Uσ(B) and the p-divisible parts of A and B are isomorphic, U∞(A) ≅ U∞(B). Then A and B are isomorphic. There is a complement to this theorem, first stated by Leo Zippin (1935) and proved in Kurosh (1960), which addresses the existence of an abelian p-group with given Ulm factors. : Let τ be an ordinal and {Aσ} be a family of countable abelian p-groups indexed by the ordinals σ < τ such that the p-heights of elements of each Aσ are finite and, except possibly for the last one, are unbounded.
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Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits a basis, but not orthonormal base Linear Functional Analysis Authors: Rynne, Bryan, Youngson, M.A. page 79; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice).
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It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra. He presented his solution, that amounts in modern terms to a combination of the Durand–Kerner method with the homotopy continuation principle, in 1891. Another proof of this kind was obtained by Hellmuth Kneser in 1940 and simplified by his son Martin Kneser in 1981. Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers (which are not constructively equivalent to the Cauchy real numbers without countable choice).
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Woodin cardinals are important in descriptive set theory. By a resultA Proof of Projective Determinacy of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that \Theta _0 is Woodin in the class of hereditarily ordinal-definable sets.
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In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University of Illinois at Urbana–Champaign... Let F be a family of distinct analytic functions on a given domain with the property that, for each x in the domain, the functions in F map x to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that F is necessarily itself countable.. As cited by . Paul Erdős in turn learned about the problem at the University of Michigan, likely via Lee Albert Rubel.
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Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is \aleph_0. Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.
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Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric.
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In 1924, Karl Menger introduced the following basis property for metric spaces: Every basis of the topology contains a countable family of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz observed that Menger's basis property can be reformulated to the above form using sequences of open covers.
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CRC press, 2010. Furthermore, a realization of a point process can be considered as a counting measure, so points processes are types of random measures known as random counting measures. In this context, the Poisson and other point processes has been studied on a locally compact second countable Hausdorff space.O. Kallenberg.
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The main occupation of this village is agriculture, especially mango, paddy, sugar cane, banana, cotton, and coconut. The majority of this village's people are depending upon agricultural sector. Very few countable people are government employees. Venkatachalam is the person who is the head of Thamaraikulam, Madurai District, now Theni District.
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J. H. C. Whitehead, motivated by the second Cousin problem, first posed the problem in the 1950s. answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problem was considered an important one in algebra for some years. Shelah's result was completely unexpected.
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Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.
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The space of all countable ordinals with the topology generated by "open intervals", is a locally compact Hausdorff space. The measure ("Dieudonné measure") that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.
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A discrete measure is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if its support is at most a countable set.
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Component Based Software 4.2.Multiple Input / Output interfaces For example, software development to change the field sizes for data in a data table does not represent changes in data processing capacity. However, this development requires work effort. Data Formatting is considered non-functional, and is countable under SNAP subcategory 1.3.
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Similar statements hold more generally, when X is not necessarily finite, not even countable. In that case, μ has to be a finite measure, and the lattice condition has to be defined using cylinder events; see, e.g., Section 2.2 of . For proofs, see the original or the Ahlswede–Daykin inequality (1978).
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Similarly, the question whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable .
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NCAA Bylaw 15.5.6.1 limits FBS football programs to a total number of scholarships to 85 "counters" annually including 25 scholarships for "initial counters." Counters (NCAA Bylaw 15.02.3) are individuals who are receiving institutional financial aid that is countable against the aid limitations in a sport, initial counters (NCAA Bylaw 15.02.
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In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = \aleph_0 = ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical. The notion is most important for countable first-order theories.
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The indefinite article of English takes the two forms a and an. Semantically, they can be regarded as meaning "one", usually without emphasis. They can be used only with singular countable nouns; for the possible use of some (or any) as an equivalent with plural and uncountable nouns, see Use of some below.
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According to the annotation of this photo by , it depicts Otto Toeplitz, E. Hagemann, D. Vieth, H. Ulm, Gottfried Köthe in 1930. Helmut Ulm (born June 21, 1908 in Gelsenkirchen; died June 13, 1975) was a German mathematician who established the classification of countable periodic abelian groups by means of their Ulm invariants.
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Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.
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Ebbinghaus, p. 184. He also objected strongly to the philosophical implications of countable models of set theory, which followed from Skolem's first-order axiomatization. According to the biography of Zermelo by Heinz-Dieter Ebbinghaus, Zermelo's disapproval of Skolem's approach marked the end of Zermelo's influence on the developments of set theory and logic.
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A real data type is a data type used in a computer program to represent an approximation of a real number. Because the real numbers are not countable, computers cannot represent them exactly using a finite amount of information. Most often, a computer will use a rational approximation to a real number.
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In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin. The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslin trees or Suslin lines.
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The family 𝓐(M) is closed under taking countable unions or intersections, but is not in general closed under taking complements. If M is the family of closed subsets of a topological space, then the elements of 𝓐(M) are called Suslin sets, or analytic sets if the space is a Polish space.
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In mathematics, specifically order theory, a partially ordered set is chain- complete if every chain in it has a least upper bound. It is ω-complete when every increasing sequence of elements (a type of countable chain) has a least upper bound; the same notion can be extended to other cardinalities of chains..
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Every distributive lattice with zero satisfies Schmidt's Condition; thus it is representable. This result has been improved further as follows, via a very long and technical proof, using forcing and Boolean-valued models. Theorem (Wehrung 2003). Every direct limit of a countable sequence of distributive lattices with zero and (∨,0)-homomorphisms is representable.
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In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
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It then proves that transcendental numbers exist: If there were no transcendental numbers, all the reals would be algebraic and hence countable, which contradicts what was just proved. This contradiction proves that transcendental numbers exist without constructing any. Oskar Perron, c. 1948 The correspondence containing Cantor's non-constructive reasoning was published in 1937.
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In particular, is a filter subbase if is countable (e.g. , , the primes), a meager set in , a set of finite measure, or a bounded subset of . If is a singleton set then is a subbasis for the Fréchet filter on . ;Characterizations of fixed ultra prefilters If a family of sets is fixed (i.e.
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The top eight players (or teams) with the most countable points accumulated in Grand Slam, ATP World Tour, and Davis Cup tournaments during the year qualify for the 2014 ATP World Tour Finals. Countable points include points earned in 2014, plus points earned at the 2013 Davis Cup final and the late-season 2013 Challengers played after the 2013 ATP World Tour Finals. To qualify, a player who finished in the 2013 year-end top 30 must compete in four Grand Slam tournaments and eight ATP World Tour Masters 1000 tournaments during 2014. They can count their best six results from ATP World Tour 500, ATP World Tour 250 and other events (Challengers, Futures, Davis Cup, Olympics) toward their ranking.
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A "Qualified Income Trust" (QIT) or "Miller Trust" can be used to qualify an applicant for Medicaid when that applicant has high long term medical expenses that consume their actual income, but still have countable income limits in excess of the Medicaid eligibility limit (which may vary in different states). The difference between the actual and countable income amounts is referred to as the "gap" from which this type of trust takes one of its several names. The QIT is most often used when nursing home (SNF) or adult living facility (HRF/ALF) costs, are sought from Medicaid. The Miller trust can be named as recipient of the individual's income from a pension plan, Social Security, or other source, effectively impoverishing them for this purpose.
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The top eight players (or teams) with the most countable points accumulated in Grand Slam, ATP World Tour and Davis Cup tournaments during the year qualify for the 2010 Barclays ATP World Tour Finals. Countable points include points earned in 2010, plus points earned at the 2009 Davis Cup final and the late-season 2009 Challengers played after the 2009 Barclays ATP World Tour Finals. To qualify, a player who finished in the 2009 year-end Top 30 must compete in four Grand Slam tournaments and eight ATP World Tour Masters 1000 tournaments during 2010. In addition, his best 4 ATP World Tour 500 events in 2010 and his best 2 ATP World Tour 250 events in 2010 will count towards his ranking.
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In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set (a countable intersection of dense open sets), with the dual concept being a closed nowhere dense set, or more generally a meagre set (a countable union of nowhere dense closed sets). However, density alone is not sufficient to characterize a generic property. This can be seen even in the real numbers, where both the rational numbers and their complement, the irrational numbers, are dense. Since it does not make sense to say that both a set and its complement exhibit typical behavior, both the rationals and irrationals cannot be examples of sets large enough to be typical.
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The top eight players (or teams) with the most countable points accumulated in Grand Slam, ATP World Tour, and Davis Cup tournaments during the year qualified for the 2011 Barclays ATP World Tour Finals. Countable points included points earned in 2011, plus points earned at the 2010 Davis Cup final and the late-season 2010 Challengers played after the 2010 Barclays ATP World Tour Finals. To qualify, a player who finished in the 2010 year-end top 30 must have competed in four Grand Slam tournaments and eight ATP World Tour Masters 1000 tournaments during 2011. The best four ATP World Tour 500 events in 2011 and best two ATP World Tour 250 events in 2011 counted towards the ranking.
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For a family (fi) of pseudometrics on X, the uniform structure defined by the family is the least upper bound of the uniform structures defined by the individual pseudometrics fi. A fundamental system of entourages of this uniformity is provided by the set of finite intersections of entourages of the uniformities defined by the individual pseudometrics fi. If the family of pseudometrics is finite, it can be seen that the same uniform structure is defined by a single pseudometric, namely the upper envelope sup fi of the family. Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric.
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Every finite planar graph can be colored with four colors, by the four-color theorem. The De Bruijn–Erdős theorem then shows that every graph that can be drawn without crossings in the plane, finite or infinite, can be colored with four colors. More generally, every infinite graph for which all finite subgraphs are planar can again be four-colored.. states the same result for the five-color theorem for countable planar graphs, as the four-color theorem had not yet been proven when he published his survey, and as the proof of the De Bruijn–Erdős theorem that he gives only applies to countable graphs. For the generalization to graphs in which every finite subgraph is planar (proved directly via Gödel's compactness theorem), see .
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If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite.
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The path taken along the graph forms the word. It is a finite graph because there are a countable number of nodes and edges, and only one path connects two distinct nodes. Gauss codes, created by Carl Friedrich Gauss in 1838, are developed from graphs. Specifically, a closed curve on a plane is needed.
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In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
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An E-chain is a countable family of nested graphs, each of which has an efficient dominating set. The Hamming codes in the n-cubes provide a classical example of E-chains. Dejter and SerraDejter I. J.; Serra O. "Efficient dominating sets in Cayley graphs", Discrete Appl. Math., 129 (2003), no. 2-3, 319-328.
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Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A Ck manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A C∞ or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.
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In mathematics, Ψ0(Ωω) is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem \Pi_1^1-CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).
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Metric spaces are first countable since one can use balls with rational radius as a neighborhood base. The metric topology on a metric space M is the coarsest topology on M relative to which the metric d is a continuous map from the product of M with itself to the non-negative real numbers.
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Earlier games in the series contained "secrets" which were a countable list of hidden extras. The games contained "beans", based on "Bertie Botts every flavour beans", used as currency, and Famous Witch or Wizard cards, used as collectables. However, in later games (specifically the final two entries), entries employ first person shooter and stealth sections.
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In set theory, a discipline within mathematics, an admissible set is a transitive set A\, such that \langle A,\in \rangle is a model of Kripke–Platek set theory (Barwise 1975). The smallest example of an admissible set is the set of hereditarily finite sets. Another example is the set of hereditarily countable sets.
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If y ≤ d, then g would not reach its maximum on [c,d] at z. Thus, y ∈ (d,b], and g(d) ≤ g(z) < g(y). This means that d ∈ S, which is a contradiction, thus establishing the lemma. The set E is open, so it is composed of a countable union of disjoint intervals (ak,bk).
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See Dual number: Slavic languages for a discussion of number phrases in Russian and other Slavic languages. The numeral "one" also has a plural form, used with pluralia tantum: одни джинсы/одни часы "one pair of jeans, one clock". The same form is used with countable nouns in meaning "only": Кругом одни идиоты "There are only idiots around".
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In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points.
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Trusts may be created to protect an individual's welfare or other state benefits. These are typically called "special needs trusts." Typically, an individual has Medicaid and Social Security Supplemental Security Income (SSI) coming in. For such individual to then be given access to funds in excess of, usually, $2,000 ("countable" assets), risks immediate termination of his government benefits.
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There is evidence of countable units of precious metal being used for exchange from the Vedic period onwards. A term Nishka appears in this sense in the Rigveda. Later texts speak of cows given as gifts being adorned with pādas of gold. A pāda, literally a quarter, would have been a quarter of some standard weight.
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Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on Borel sets.
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This is a generalization of the notion of a probability measure, where the probability axiom of countable additivity is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "expected value" of an outcome given a certain capacity can be found by taking the Choquet integral over the capacity.
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The annual pension is calculated by adding all of the person's countable income. Any deductions are then subtracted from that total. The remaining total is deducted from the maximum pension limitschedule (taking into account the number of dependents, spouse, etc.). This final number is the yearly pension; dividing it by 12 results in the monthly pension.
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Frequently this means that the ground field is uncountable and that the property is true except on a countable union of proper Zariski-closed subsets (i.e., the property holds on a dense Gδ set). For instance, this notion of very generic occurs when considering rational connectedness. However, other definitions of very generic can and do occur in other contexts.
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Since compact sets in X are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom. The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.
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92 A number that is disjunctive to every base is called absolutely disjunctive or is said to be a lexicon. Every string in every alphabet occurs within a lexicon. A set is called "comeager" or "residual" if it contains the intersection of a countable family of open dense sets. The set of absolutely disjunctive reals is residual.
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The proportion of graphs on n vertices with nontrivial automorphism tends to zero as n grows, which is informally expressed as "almost all finite graphs are asymmetric". In contrast, again informally, "almost all infinite graphs are symmetric." More specifically, countable infinite random graphs in the Erdős–Rényi model are, with probability 1, isomorphic to the highly symmetric Rado graph..
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In probability theory, Foster's theorem, named after Gordon Foster, is used to draw conclusions about the positive recurrence of Markov chains with countable state spaces. It uses the fact that positive recurrent Markov chains exhibit a notion of "Lyapunov stability" in terms of returning to any state while starting from it within a finite time interval.
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Lattice of varieties of regular bands. When partially ordered by inclusion, varieties of bands naturally form a lattice, in which the meet of two varieties is their intersection and the join of two varieties is the smallest variety that contains both of them. The complete structure of this lattice is known; in particular, it is countable, complete, and distributive.; ; ; .
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In mathematics, additivity (specifically finite additivity) and sigma additivity (also called countable additivity) of a function (often a measure) defined on subsets of a given set are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity, and σ-additivity implies additivity.
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Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure, also known as a content. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first.
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The set of natural numbers is an infinite set. By definition, this kind of infinity is called countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-naught ().
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Obviously, f(Kc) is countable, since it contains one point per component of Kc. Hence f(Kc) has measure zero, so f(K) has measure one. We need a strictly monotonic function, so consider g(x) = f(x) + x. Since g(x) is strictly monotonic and continuous, it is a homeomorphism. Furthermore, g(K) has measure one.
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Cambridge, MA: MIT Press. where it has been widely adopted and competence is the only level of language that is studied. According to Chomsky, competence is the ideal language system that enables speakers to produce and understand an infinite numberIn his use of "infinite number", Chomsky assumed no upper bound for the length of a sentence. See countable infinity.
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A suffix -j following the noun or adjective suffixes -o or -a makes a word plural. Without this suffix, a countable noun is understood to be singular. Direct objects take an accusative case suffix -n, which goes after any plural suffix. (The resulting sequence -ojn rhymes with English coin, and -ajn rhymes with fine.) Adjectives agree with nouns.
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Froda's major contribution was in the field of mathematical analysis. His first important result was concerned with the set of discontinuities of a real-valued function of a real variable. In this theorem Froda proves that the set of simple discontinuities of a real-valued function of a real variable is at most countable. In a paperA.
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The space }, along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. In terms of compactness, the space }, along with the double origin topology fails to be either compact, paracompact or locally compact, however, X is second countable. Finally, it is an example of an arc connected space.
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"garbage" (but countable), offensive. Etymology uncertain, theories suggested include the acronym MUS for "Moscow Criminal Investigation [Office]" (Московский Уголовный Сыск) in Tzarist Russia and Hebrew for "informer." Also, in Belarus, acronym MUS stands for Ministry for Home Affairs (Belarusian: Міністэрства ўнутраных спраў, МУС), and is embroidered on policeman uniform. ; Mountie(s): Canada, colloquial, Royal Canadian Mounted Police.
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Let M be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of M is the group of all Cr diffeomorphisms of M to itself, denoted by Diffr(M) or, when r is understood, Diff(M). This is a "large" group, in the sense that—provided M is not zero-dimensional—it is not locally compact.
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A topological characterization of Cantor spaces is given by Brouwer's theorem:. The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as: This theorem is also equivalent (via Stone's representation theorem for Boolean algebras) to the fact that any two countable atomless Boolean algebras are isomorphic.
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Indeed, a Hausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular.Topospaces subwiki, Locally compact Hausdorff implies completely regular Assume such a space X is σ-compact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable.
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Let X be a topological space. A collection {Ga} of subsets of X is said to be locally discrete, if each point of the space has a neighbourhood intersecting at most one element of the collection. A collection of subsets of X is said to be countably locally discrete, if it is the countable union of locally discrete collections.
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Let X be a topological space. For a subset S of X let S denote the closure of S. Then a point x is called a Pytkeev point if for every set A with , there is a countable \pi -net of infinite subsets of A. A Pytkeev space is a space in which every point is a Pytkeev point.
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The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal). The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic). Many larger epsilon numbers can be defined using the Veblen function.
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Martin-Löf's idea was to limit the definition to measure 0 sets that are effectively describable; the definition of an effective null cover determines a countable collection of effectively describable measure 0 sets and defines a sequence to be random if it does not lie in any of these particular measure 0 sets. Since the union of a countable collection of measure 0 sets has measure 0, this definition immediately leads to the theorem that there is a measure 1 set of random sequences. Note that if we identify the Cantor space of binary sequences with the interval [0,1] of real numbers, the measure on Cantor space agrees with Lebesgue measure. The martingale characterization conveys the intuition that no effective procedure should be able to make money betting against a random sequence.
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Much of Larson's research is in infinitary combinatorics, studying versions of Ramsey's theorem for infinite sets. Her doctoral dissertation, On Some Arrow Relations, was in this subject. She has been called a "prominent figure in the field of partition relations", particularly for her "expertise in relations for countable ordinals". Five of her publications are with Paul Erdős, who became her most frequent collaborator.
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By translating the charts obtained from the countable neighborhood basis used above one obtains slice charts around every point in . This shows that is an embedded submanifold of . Moreover, multiplication , and inversion in are analytic since these operations are analytic in and restriction to a submanifold (embedded or immersed) with the relative topology again yield analytic operations and . Proposition 8.22.
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They divorced in 1915, the same year as their only son Barclay Raunkiær died. Raunkiær later married the botanist Agnete Seidelin (1874–1956). Raunkiær's research axiom was that everything countable in nature should be subjected to numerical analysis, e.g. the number of male and female catkins in monecious plants and the number of male and female individuals in dioecious plants.
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In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. showed that they exist if the continuum hypothesis is true. On the other hand, they do not exist if Martin's axiom for ℵ1 is true.
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If there is a supercompact cardinal, then there is a model of set theory in which PFA holds. The proof uses the fact that proper forcings are preserved under countable support iteration, and the fact that if \kappa is supercompact, then there exists a Laver function for \kappa . It is not yet known how much large cardinal strength comes from PFA.
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It is worth noting that "the number of sales" is fundamentally countable and therefore discrete. A continuous simulation of sales implies the possibility of fractional sales e.g. 1/3 of a sale. For that reason, a continuous simulation of sales does not model reality but nevertheless may make useful predictions that match a discrete simulation's predictions for whole numbers of sales.
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Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ2. The Haar system is an example of a basis for Lp([0, 1]), when 1 ≤ p < ∞. When , another example is the trigonometric system defined below. The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the supremum norm, admits a Schauder basis.
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A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
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For some purposes, we also need this ideal to be a sigma- ideal, so that countable unions of negligible sets are also negligible. If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets. The opposite of a negligible set is a generic property, which has various forms.
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One can check that is indeed a measure. It is not -finite, as not every Borel set is at most a countable union of finite sets. Let be the usual Lebesgue measure on this Borel algebra. Then, is absolutely continuous with respect to , since for a set one has only if is the empty set, and then is also zero.
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Frank Cruz (born 1959) is an American college baseball coach, who most recently served as the head coach of the USC Trojans baseball team. He held the position from 2011 through 2012. Cruz was relieved of his duties for "knowingly violating NCAA Countable Athletically-Related Activities limitations" just two days prior to the beginning of the 2013 NCAA Division I baseball season.
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In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.
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However, the use of the adjective alone is fairly common in the case of superlatives such as biggest, ordinal numbers such as first, second, etc., and other related words such as next and last. Many adjectives, though, have undergone conversion so that they can be used regularly as countable nouns; examples include Catholic, Protestant, red (with various meanings), green, etc.
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The founders began departing the company in August 2015 with Jason Putorti, then Michael Capone in April 2016, and James Windon in March 2017. Following the 2018 midterms, Brigade's assets were acquired by Countable (app) and the employees were acqui-hired by Pinterest. The app no longer exists on the App Store (iOS) and the web site is listed as for sale.
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This is the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. The real line can also be given the lower limit topology.
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In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras. An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open).
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In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily Tychonoff spaces. Adding the Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of σ-compactness and second-countability are the same.
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A Hausdorff space cannot have a locally discrete basis unless it is itself discrete. The same property holds for a T1 space. 4\. The following is known as Bing's metrization theorem: A space X is metrizable iff it is regular and has a basis that is countably locally discrete. 5\. A countable collection of sets is necessarily countably locally discrete.
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The paintings are located between and above the river's water level. The main area contains about 1900 discrete countable images arranged in about 110 groups. The paintings have a red color and were executed using a mixture of red ochre (hematite), animal glue, and blood. They depict human figures as well as animals along with bronze drums, knives, swords, bells, and ships.
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In the countable sense, a verse is formally a single metrical line in a poetic composition. However, verse has come to represent any division or grouping of words in a poetic composition, with groupings traditionally having been referred to as stanzas. In the uncountable (mass noun) sense verse refers to "poetry" as contrasted to prose.Wiktionary, "Verse" (accessed 8 August 2014).
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Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875,The Cantor Set Before Cantor Mathematical Association of America is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers. Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem.
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This rule can be seen in the examples "there is less flour in this canister" and "there are fewer cups (grains, pounds, bags, etc.) of flour in this canister", which are based on the reasoning that flour is uncountable whereas the unit used to measure the flour (cup, etc.) is countable. Nevertheless, even most prescriptivists accept the most common usage "there are less cups of flour in this canister" and prescribe the rule addition that "less" should be used with units of measurement (other examples: "less than 10 pounds/dollars"). Prescriptivists would, however, consider "fewer cups of coffee" to be correct in a sentence such as "there are fewer cups of coffee on the table now", where the cups are countable separate objects. In addition, "less" is recommended in front of counting nouns that denote distance, amount, or time.
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For example, some people like football, while others prefer rugby, or I've got some money, but not enough to lend you any. It can also be used as an indefinite pronoun, not qualifying a noun at all (Give me some!) or followed by a prepositional phrase (I want some of your vodka); the same applies to any. Some can also be used with singular countable nouns, as in There is some person on the porch, which implies that the identity of the person is unknown to the speaker (which is not necessarily the case when a(n) is used). This usage is fairly informal, although singular countable some can also be found in formal contexts: We seek some value of x such that... When some is used just as an indefinite article, it is normally pronounced weakly, as .
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The notion of saturated model is dual to the notion of prime model in the following way: let T be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and let P be a prime model of T. Then P admits an elementary embedding into any other model of T. The equivalent notion for saturated models is that any "reasonably small" model of T is elementarily embedded in a saturated model, where "reasonably small" means cardinality no larger than that of the model in which it is to be embedded. Any saturated model is also homogeneous. However, while for countable theories there is a unique prime model, saturated models are necessarily specific to a particular cardinality. Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories.
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The purpose of the concept of a net, first introduced by E. H. Moore and Herman L. Smith in 1922, is to generalize the notion of a sequence so as to confirm the equivalence of the conditions (with "sequence" being replaced by "net" in condition 2). In particular, rather than being defined on a countable linearly ordered set, a net is defined on an arbitrary directed set. In particular, this allows theorems similar to that asserting the equivalence of condition 1 and condition 2, to hold in the context of topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behaviour.
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A random variable has a probability distribution, which specifies the probability of Borel subsets of its range. Random variables can be discrete, that is, taking any of a specified finite or countable list of values (having a countable range), endowed with a probability mass function that is characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals (having an uncountable range), via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both. Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
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In his paper on the problem, Erdős credited an anonymous mathematician with the observation that, when each x is mapped to a finite set of values, F is necessarily finite. However, as Erdős showed, the situation for countable sets is more complicated: the answer to Wetzel's question is yes if and only if the continuum hypothesis is false.. That is, the existence of an uncountable set of functions that maps each argument x to a countable set of values is equivalent to the nonexistence of an uncountable set of real numbers whose cardinality is less than the cardinality of the set of all real numbers. One direction of this equivalence was also proven independently, but not published, by another UIUC mathematician, Robert Dan Dixon. It follows from the independence of the continuum hypothesis, proved in 1963 by Paul Cohen,.
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Every CW-complex is sequential, as it can be considered as a quotient of a metric space. The prime spectrum of a commutative Noetherian ring with the Zariski topology is sequential. ;Sequential spaces that are not first countable Take the real line and identify the set of integers to a point. It is a sequential space since it is a quotient of a metric space.
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Then the payoff to A is 0 if , 1 if and if . Thus each player seeks to choose the larger number, but there is a penalty of for choosing too large a number. A large number of variants have been studied, where the set may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.
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The result was the truth predicate is well arithmetically, it is even \Delta^0_2. So far down in the arithmetic hierarchy, and that goes for any recursively axiomatized (countable, consistent) theories. Even if you are true in all the natural numbers \Pi^0_1 formulas to the axioms. This classic proof is a very early, original application of the arithmetic hierarchy theory to a general-logical problem.
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We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet space or an F-space . If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC) (, ). A complete LMC algebra is called an Arens- Michael algebra .
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His first construction shows how to write the real algebraic numbersThe real algebraic numbers are the real roots of polynomial equations with integer coefficients. as a sequence a1, a2, a3, .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence.
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The set of reals R = T ∪ {an} = T0 ∪ {tn} ∪ {an} where an is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T0 and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T0, f(t2n-1) = tn, and f(t2n) = an.
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1; Dauben 1977, p. 89 15n. Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".
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To which is subjoined a short account of the natives of Madagascar, with suggestions as to their civilizations by J. Hatchard, L.B. Seeley and T. Hamilton, London, 1820. but has become much more common in the later 20th century, sometimes just meaning culture (itself in origin an uncountable noun, made countable in the context of ethnography)."Civilization" (1974), Encyclopædia Britannica 15th ed. Vol. II, Encyclopædia Britannica, Inc.
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In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.
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In other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, . In which case, the cardinality of the set is denoted \aleph_0 (aleph-null)—the first in the series of aleph numbers. This terminology is not universal. Some authors use countable to mean what is here called countably infinite, and do not include finite sets.
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For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does \omega_\omega or an uncountable cofinality. The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1.
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The element Σ Zntn is a group-like element of the Hopf algebra of formal power series over NSymm, so over the rationals its logarithm is primitive. The coefficients of its logarithm generate the free Lie algebra on a countable set of generators over the rationals. Over the rationals this identifies the Hopf algebra NSYmm with the universal enveloping algebra of the free Lie algebra.
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In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte. It is sometimes said to be the first impredicative ordinal,Kurt Schütte, Proof theory, Grundlehren der Mathematischen Wissenschaften, Band 225, Springer-Verlag, Berlin, Heidelberg, New York, 1977, xii + 302 pp.
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The iconography follows the late Middle Ages depiction. Christ was nailed to the cross on the ground. Since the holes are made on the cross beforehand and are found to be too far apart when Christ is laid on it, his body is stretched to reach the hole. As a result of the stretch, the bones of the body are clearly shown and countable.
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Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.
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The surrounding space matters: a set may be nowhere dense when considered as a subset of a topological space , but not when considered as a subset of another topological space . Notably, a set is always dense in its own subspace topology. A countable union of nowhere dense sets is called a meagre set. Meager sets play an important role in the formulation of the Baire category theorem.
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The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
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This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.
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Thus an enduring physical object, like an Aristotelian substance, undergoes changes and adventures during the course of its existence. In some contexts, especially in the theory of relativity in physics, the word 'event' refers to a single point in Minkowski or in Riemannian space-time. A point event is not a process in the sense of Whitehead's metaphysics. Neither is a countable sequence or array of points.
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The axioms of Zermelo–Fraenkel set theory without the axiom of choice (ZF) are not strong enough to prove that every infinite set is Dedekind-infinite, but the axioms of Zermelo–Fraenkel set theory with the axiom of countable choice () are strong enough. Other definitions of finiteness and infiniteness of sets than that given by Dedekind do not require the axiom of choice for this, see .
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Then S can be partitioned into \kappa many disjoint stationary sets. This result is due to Solovay. If \kappa is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix. H. Friedman has shown that for every countable successor ordinal \beta, every stationary subset of \omega_1 contains a closed subset of order type \beta.
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The empty set is always a null set. More generally, any countable union of null sets is null. Any measurable subset of a null set is itself a null set. Together, these facts show that the m-null sets of X form a sigma-ideal on X. Similarly, the measurable m-null sets form a sigma-ideal of the sigma-algebra of measurable sets.
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It is true, however, that condition 1 implies condition 2. The difficulty encountered when attempting to prove that condition 2 implies condition 1 lies in the fact that topological spaces are, in general, not first-countable. If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent. In particular, the two conditions are equivalent for metric spaces.
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In linguistics, a count noun (also countable noun) is a noun that can be modified by a numeral and that occurs in both singular and plural forms, and that co-occurs with quantificational determiners like every, each, several, etc. A mass noun has none of these properties, because it cannot be modified by a numeral, cannot occur in plural, and cannot co-occur with quantificational determiners.
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Theorem (Wehrung 1999). Let R be a von Neumann regular ring. Then the (∨,0)-semilattices Idc R and Conc L(R) are both isomorphic to the maximal semilattice quotient of V(R). Bergman proves in a well-known unpublished note from 1986 that any at most countable distributive (∨,0)-semilattice is isomorphic to Idc R, for some locally matricial ring R (over any given field).
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The following is a topological version of this theorem: If X is a second-countable Hausdorff space and \Sigma contains the Borel sigma-algebra, then the set of recurrent points of f has full measure. That is, almost every point is recurrent. For a proof, see the cited reference. More generally, the theorem applies to conservative systems, and not just to measure-preserving dynamical systems.
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In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X may be written uniquely as the disjoint union of a perfect set and a countable set.
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In Step 3, the sphere was partitioned into orbits of our group H. To streamline the proof, the discussion of points that are fixed by some rotation was omitted; since the paradoxical decomposition of F2 relies on shifting certain subsets, the fact that some points are fixed might cause some trouble. Since any rotation of S2 (other than the null rotation) has exactly two fixed points, and since H, which is isomorphic to F2, is countable, there are countably many points of S2 that are fixed by some rotation in H. Denote this set of fixed points as D. Step 3 proves that S2 − D admits a paradoxical decomposition. What remains to be shown is the Claim: S2 − D is equidecomposable with S2. Proof. Let λ be some line through the origin that does not intersect any point in D. This is possible since D is countable.
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For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like Joseph Doob and Kai Lai Chung. Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.
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For some transitive models M of ZFC, every sequence of integers in M is definable relative to M; for others, only some integer sequences are (Hamkins et al. 2013). There is no systematic way to define in M itself the set of sequences definable relative to M and that set may not even exist in some such M. Similarly, the map from the set of formulas that define integer sequences in M to the integer sequences they define is not definable in M and may not exist in M. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model (Hamkins et al. 2013). If M contains all integer sequences, then the set of integer sequences definable in M will exist in M and be countable and countable in M.
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The top eight players (or teams) with the most countable points accumulated in Grand Slam, ATP World Tour, and Davis Cup tournaments during the year qualify for the 2013 ATP World Tour Finals. Countable points include points earned in 2013, plus points earned at the 2012 Davis Cup final and the late-season 2012 Challengers played after the 2012 ATP World Tour Finals. To qualify, a player who finished in the 2012 year-end top 30 must compete in four Grand Slam tournaments and eight ATP World Tour Masters 1000 tournaments during 2013. They can count their best six (6) results from ATP World Tour 500, ATP World Tour 250 and other events (Challengers, Futures, Davis Cup, Olympics) toward their ranking. To count their best six (6), players must have fulfilled their commitment to 500 events – 4 total per year (at least 1 after the US Open).
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The top eight players (or teams) with the most countable points accumulated in Grand Slam, ATP World Tour, and Davis Cup tournaments during the year qualify for the 2012 ATP World Tour Finals. Countable points include points earned in 2012, plus points earned at the 2011 Davis Cup final and the late-season 2011 Challengers played after the 2011 ATP World Tour Finals. To qualify, a player who finished in the 2011 year-end top 30 must compete in four Grand Slam tournaments and eight ATP World Tour Masters 1000 tournaments during 2012. They can count their best six (6) results from ATP World Tour 500, ATP World Tour 250 and other events (Challengers, Futures, Davis Cup, Olympics) toward their ranking. To count their best six (6), players must have fulfilled their commitment to 500 events – 4 total per year (at least 1 after the US Open).
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A sequential space may fail to be a -sequential space and also a -sequential space may fail to be a sequential space. In particular, it should not be assumed that a sequential space has the properties described in the next definitions. As with -sequential spaces, it should not be assumed that a sequential space has the properties described in the next definition. Every first-countable space is -sequential.
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On November 8, Utah defeated the Delta Devils 143–49 to set an NCAA record for largest margin of victory (94 points) over a Division I opponent. Two players recently received "Delta Devil of the Day" awards. The recipients were: Richard "Big Tuna" Rivers, a 6' 11" center from Pennsylvania and Caleb Hunter, a 5' 11" guard from Michigan. The game against North American University is a non-countable game.
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In mathematics, S-space is a regular topological space that is hereditarily separable but is not a Lindelöf space. L-space is a regular topological space that is hereditarily Lindelöf but not separable. A space is separable if it has a countable dense set and hereditarily separable if every subspace is separable. It had been believed for a long time that S-space problem and L-space problem are dual, i.e.
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A space is called ω-bounded if the closure of every countable set is compact. For example, the long line and the closed long ray are ω-bounded but not compact. When restricted to a metric space ω-boundedness is equivalent to compactness. The bagpipe theorem states that every ω-bounded connected surface is the connected sum of a compact connected surface and a finite number of long pipes.
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In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one in which every subspace with property P has compact closure. Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact.
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The rational numbers form a metric space by using the absolute difference metric and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected.
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Accordingly, Medicaid coverage does vary from state to state in certain aspects, but there are also mandatory Federal law provisions. To qualify for Medicaid and its long-term medical and nursing care benefits, the applicant must be "impoverished." There is a strict limit to the countable assets which a Medicaid recipient can own. To qualify for Medicaid, an applicant must meet the asset guidelines for Supplemental Security Income (SSI).
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A topological space with a countable dense subset is called separable. A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable for a cardinal κ if it contains κ pairwise disjoint dense sets.
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Let J be the set of angles, α, such that for some natural number n, and some P in D, r(nα)P is also in D, where r(nα) is a rotation about λ of nα. Then J is countable. So there exists an angle θ not in J. Let ρ be the rotation about λ by θ. Then ρ acts on S2 with no fixed points in D, i.e.
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In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities.Cantor 1878, p. 242. In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.
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Stanley Tennenbaum (April 11, 1927 – May 4, 2005) was an American mathematician who contributed to the field of logic. In 1959, he published Tennenbaum's theorem, which states that no countable nonstandard model of Peano arithmetic (PA) can be recursive, i.e. the operations + and × of a nonstandard model of PA are not recursively definable in the + and × operations of the standard model. He was a Professor at Yeshiva University in the 1960s.
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The collective form is therefore similar in many respects to an English mass noun like "rice", which in fact refers to a collection of items which are logically countable. However, English has no productive process of forming singulative nouns (just phrases such as "a grain of rice"). Therefore, English cannot be said to have a singulative number. In other languages, singulatives can be regularly formed from collective nouns; e.g.
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Nouns in Nalik are categorized as being an uncountable noun, or a countable noun. Nouns can be part of a noun phrase or can be an independent subject referenced in a verbal complex. When used as subjects, some uncountable nouns are co-referential with plural subject markers however those are the exceptions and are usually marked with singular subject markers. With uncountable nouns, numerical markers cannot be used.
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The image of a regular space under an injective, continuous open map is always regular. #Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces. #Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable. #The Moore plane (also known as the Niemytski space) is an example of a non- metrizable Moore space.
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For non-Hausdorff spaces the definitions of Baire sets in terms of continuous functions need not be equivalent to definitions involving Gδ compact sets. For example, if X is an infinite countable set whose closed sets are the finite sets and the whole space, then the only continuous real functions on X are constant, but all subsets of X are in the σ-algebra generated by compact closed Gδ sets.
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Every algebra of sets is closed under finite unions and finite intersections. However, an algebra of sets is not required to be closed under countable (let alone arbitrary) unions or intersections. Assuming the nullary union convention (i.e. that the union of sets is the empty set ), a family of sets is an algebra of sets if and only if it is closed under complementation and closed under finite unions.
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Such a poset is called algebraic. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an uncountable set. In some cases, however, the base for a poset is countable.
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On the topic of infinite chess, Hamkins, Brumleve and Schlicht proved that the mate-in-n problem of infinite chess is decidable. Hamkins and Evans investigated transfinite game values in infinite chess, proving that every countable ordinal arises as the game value of a position in infinite three-dimensional chess.C. D. A. Evans and J. D. Hamkins, "Transfinite game values in infinite chess," Integers, volume 14, Paper No. G2, 36, 2014.
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However, existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for (using ordinal parameters), then existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to . And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel. If κ is -Erdős, then it is -Erdős in every transitive model satisfying " is countable".
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For example, when countable, báalt means 'apple' but when uncountable, it means 'apple tree' (Grune 1998). Noun morphology consists of the noun stem, a possessive prefix (mandatory for some nouns, and thus an example of inherent possession), and number and case suffixes. Distinctions in number are singular, plural, indefinite, and grouped. Cases include absolutive, ergative/oblique, genitive, and several locatives; the latter indicate both location and direction and may be compounded.
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Suppose that X is a subset of the reals, and each pair of elements of X is colored either black or white, with set of white pairs being open. The open coloring axiom states that either X has an uncountable subset such that any pair from this subset is white, or X can be partitioned into a countable number of subsets such that any pair from the same subset is black.
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By definition, a set containing an infinite structure falls outside the area that FMT deals with. Note that infinite structures can never be discriminated in FO, because of the Löwenheim–Skolem theorem, which implies that no first-order theory with an infinite model can have a unique model up to isomorphism. The most famous example is probably Skolem's theorem, that there is a countable non-standard model of arithmetic.
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Thus, regular spaces are usually studied to find properties and theorems, such as the ones below, that are actually applied to completely regular spaces, typically in analysis. There exist Hausdorff spaces that are not regular. An example is the set R with the topology generated by sets of the form U — C, where U is an open set in the usual sense, and C is any countable subset of U.
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For this reason, one considers instead a smaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets; that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.
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The grammar in Malaysian English may become simplified in the mesolectal and basilectal varieties. For example, articles and past-tense markers may sometimes be omitted, question structures may be simplified, and the distinction between countable and mass nouns may be blurred. In the basilectal variety, omission of the object pronoun or the subject pronoun is common. The modal auxiliary system is also often reduced, and sometimes, a verb may be absent.
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Scott took up a post as Assistant Professor of Mathematics, back at the University of California, Berkeley, and involved himself with classical issues in mathematical logic, especially set theory and Tarskian model theory. During this period he started supervising Ph.D. students, such as James Halpern (Contributions to the Study of the Independence of the Axiom of Choice) and Edgar Lopez-Escobar (Infinitely Long Formulas with Countable Quantifier Degrees).
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Some of the difficulties and paradoxes of presentism can be resolved by changing the normal view of time as a container or thing unto itself and seeing time as a measure of changing spatial relationships among objects. Thus, observers need not be extended in time to exist and to be aware, but they rather exist and the changes in internal relationships within the observer can be measured by stable countable events.
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Suppose that X is the first uncountable ordinal, with the finite measure where the measurable sets are either countable (with measure 0) or the sets of countable complement (with measure 1). The (non-measurable) subset E of X×X given by pairs (x,y) with x The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory. The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while showed that it is consistent with ZFC that a strong Fubini-type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.
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Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem. The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics. When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA0 (a second- order variant of Peano arithmetic restricted to induction over Σ01 formulas). Weak König's lemma is provable in ZF, the system of Zermelo–Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF. However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.
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In a more concise (although more obscure) way: :\psi(\alpha) is the smallest ordinal which cannot be expressed from 0, 1, \omega and \Omega using sums, products, exponentials, and the \psi function itself (to previously constructed ordinals less than \alpha). Here is an attempt to explain the motivation for the definition of \psi in intuitive terms: since the usual operations of addition, multiplication and exponentiation are not sufficient to designate ordinals very far, we attempt to systematically create new names for ordinals by taking the first one which does not have a name yet, and whenever we run out of names, rather than invent them in an ad hoc fashion or using diagonal schemes, we seek them in the ordinals far beyond the ones we are constructing (beyond \Omega, that is); so we give names to uncountable ordinals and, since in the end the list of names is necessarily countable, \psi will “collapse” them to countable ordinals.
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Valentina Harizanov is a Serbian-American mathematician and professor of mathematics at The George Washington University. Her main research contributions are in computable structure theory (roughly at the intersection of computability theory and model theory), where she introduced the notion of degree spectra of relations on computable structures and obtained the first significant results concerning uncountable, countable, and finite Turing degree spectra. Her recent interests include algorithmic learning theory and spaces of orders on groups.
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Many other word classes exist in different languages, such as conjunctions like "and" that serve to join two sentences, articles that introduce a noun, interjections such as "wow!", or ideophones like "splash" that mimic the sound of some event. Some languages have positionals that describe the spatial position of an event or entity. Many languages have classifiers that identify countable nouns as belonging to a particular type or having a particular shape.
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Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number. Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.Cantor's construction starts with the set of transcendentals T and removes a countable subset {tn} (for example, tn = e / n). Call this set T0. Then T = T0 ∪ {tn} = T0 ∪ {t2n-1} ∪ {t2n}.
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These conditions are equivalent for metrizable spaces, but neither one implies the other in the class of all topological spaces. It is almost trivial to prove that the product of two sequentially compact spaces is sequentially compact—one passes to a subsequence for the first component and then a subsubsequence for the second component. An only slightly more elaborate "diagonalization" argument establishes the sequential compactness of a countable product of sequentially compact spaces.
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More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring. In algebraic geometry and commutative algebra, rings of formal power series are especially tractable topologically complete local rings, allowing calculus-like arguments within a purely algebraic framework. They are analogous in many ways to p-adic numbers. Formal power series can be created from Taylor polynomials using formal moduli.
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Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element. See the Topology and ordinals section of the "Order topology" article.
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In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with G for Gebiet (German: area, or neighbourhood) meaning open set in this case and δ for Durchschnitt (German: intersection). The term inner limiting set is also used. Gδ sets, and their dual, Fσ sets, are the second level of the Borel hierarchy.
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In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
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In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0\. The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral.
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In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.
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The proof runs as follows: since D is countable, one can enumerate the dense subsets of P as D1, D2, …. By assumption, there exists p ∈ P. Then by density, there exists p1 ≤ p with p1 ∈ D1. Repeating, one gets … ≤ p2 ≤ p1 ≤ p with pi ∈ Di. Then G = { q ∈ P: ∃ i, q ≥ pi} is a D-generic filter. The Rasiowa–Sikorski lemma can be viewed as an equivalent to a weaker form of Martin's axiom.
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In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem deals with the existence of probability measures for probabilistic events consisting of a countably infinite number of individual probabilistic events. In particular, the individual events may be independent or dependent with respect to each other. Thus, the statement goes beyond the mere existence of countable product measures. The theorem was proved by Cassius Ionescu-Tulcea in 1949.
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Every free abelian group is slender. The additive group of rational numbers Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, and as Q is injective this homomorphism extends over the whole of ZN. Therefore, a slender group must be reduced. Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.
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A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa. Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
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In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets in , smooth curves in the plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then it is "large", meaning that it is not a countable union of negligible subsets.
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However, these set models are non-standard. In particular, they do not use the normal membership relation and they are not well-founded. If there is no standard model then the minimal model cannot exist as a set. However in this case the class of all constructible sets plays the same role as the minimal model and has similar properties (though it is now a proper class rather than a countable set).
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If a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. The converse to this statement is the so- called von Neumann conjecture, which was disproved by Olshanskii in 1980 using his Tarski monsters. Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic, they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented.
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Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values {0, 1, 2, ...}. On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa.
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With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hypothesis), there are still highly nontrivial ZFC theorems about cardinal exponentiation. Shelah constructed a Jónsson group, an uncountable group for which every proper subgroup is countable. He showed that Whitehead's problem is independent of ZFC. He gave the first primitive recursive upper bound to van der Waerden's numbers V(C,N).
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Sapienza had a degree in economics and countable sciences from the Pontifical Catholic University of São Paulo. In 1962, he started working as an accountant for the State of São Paulo. During this period of time, he had an active role at the State of São Paulo accountant union, and held various posts such as President, Secretary and Council member. In 1986, Sapienza successfully ran for a spot at the Legislative Assembly of São Paulo.
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Peace Corps, Panama (unpublished). The regular noun phrase consists of the nucleus (head noun or pronoun) followed by the possible addition of a modifier, quantifier, or demonstrative. Articles are not used in Ngäbere. :Monso chi iti (A small child) :child small one :Mrö ñakare (No food) :Food none :Ñö mrene krubäte (Very salty water) :Water salty very :Mütü kri ye (That large pig) :Pig large that 'Plurality' All nouns are countable in Ngäbere.
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The symbols A and B are "stand-ins" for strings; this form of notation is called an "axiom schema" (i.e., there is a countable number of specific forms the notation could take). This can be read in a manner similar to IF-THEN but with a difference: given symbol string IF A and A implies B THEN B (and retain only B for further use). But the symbols have no "interpretation" (e.g.
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There is a probability of that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example. Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see . The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
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There are numerous characterizations that tell when a second-countable topological space is metrizable, such as Urysohn's metrization theorem. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology. There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game.
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When treating large degrees of freedom at the macroscopic level, statistical mechanics becomes useful. Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in thermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity.
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Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω1 is often written as [0,ω1), to emphasize that it is the space consisting of all ordinals smaller than ω1. If the axiom of countable choice holds, every increasing ω-sequence of elements of [0,ω1) converges to a limit in [0,ω1). The reason is that the union (i.e.
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The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existence of ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0# implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0#. Chang's conjecture implies the existence of 0#.
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This is true regardless of whether the Hilbert space is finite-dimensional or not. Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set. It follows from the spectral theorem for compact self-adjoint operators that every mixed state is a countable convex combination of pure states.
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When there is an uncountably infinite collection of formulas, the Axiom of Choice (or at least some weak form of it) is needed. Using the full AC, one can well-order the formulas, and prove the uncountable case with the same argument as the countable one, except with transfinite induction. Other approaches can be used to prove that the completeness theorem in this case is equivalent to the Boolean prime ideal theorem, a weak form of AC.
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In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism). There is a different Borel conjecture (named for Émile Borel) in set theory. It asserts that every strong measure zero set of reals is countable.
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As part of her thesis work, in 1952, Morel found two different countable ordinal numbers whose squares are equal. After Wacław Sierpiński simplified her construction, they published it jointly. In 1955, Morel published a converse to the Knaster–Tarski theorem, according to which every incomplete lattice has an increasing function with no fixed point. Her 1965 paper with Thomas Frayne and Dana Scott, "Reduced direct products", provides the main definitions of reduced products in model theory.
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A set is countable if there exists an injective function from to the natural numbers }.Since there is an obvious bijection between and }, it makes no difference whether one considers 0 a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which takes 0 as a natural number. If such an can be found that is also surjective (and therefore bijective), then is called countably infinite.
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This is an eponymous album as he used one of his stage names, Aleph-1. The concept of the album and its name, Aleph-1, derive from the theories of German mathematician Georg Cantor, who was a teacher in Halle, Saxony-Anhalt, Germany, a city, to which Alva Noto is deeply connected with through his family. In mathematical terms, \aleph_1 is the cardinality of the set of all countable ordinal numbers or a number of elements in endless successions.
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The liberties of groups are countable. Situations where mutually opposing groups must capture each other or die are called capturing races, or semeai. In a capturing race, the group with more liberties (and/or better "shape") will ultimately be able to capture the opponent's stones. Capturing races and the elements of life or death are the primary challenges of Go. A player may pass on determining that the game offers no further opportunities for profitable play.
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If is an absorbing disk in a vector space then the sequence defined by forms a string beginning with . This is called the natural string of Moreover, if a vector space has countable dimension then every string contains an absolutely convex string. Summative sequences of sets have the particularly nice property that they define non- negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces.
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A subset of Rn is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of Rn has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on Rn (or any metric Lipschitz equivalent to it).
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Robert Michael "Mike" Canjar"Dissertation: Model-Theoretic Properties of Countable Ultraproducts without the Continuum Hypothesis" by Robert Michael Kanjar, Ph.D., University of Michigan, 1982 (September 9, 1953 - May 7, 2012) was a Professor in the Department of Mathematics and Computer Science at University of Detroit Mercy (UDM). He started there in 1995, and served as department Chairman from 1995–2002. He was promoted to Full Professor in 2001. He previously taught at several universities, including the University of Baltimore.
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Asserting all laws of classical logic, the disjunctive property of I discussed above indeed does hold for all sets. Then, for nonempty X, the properties numerable (X injects into \omega), countable (\omega has X as its range), subcountable (a subset of \omega surjects into X) and also not \omega- productive (a countability property essentially defined in terms of subsets of X, formalized below) are all equivalent and express that a set is finite or countably infinite.
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Sanxi stream, also known as Dingxi stream, originates from the mountains at the junction of Changle and Fuqing and is known as the Changle No. 1 Stream. There are many bridges above Sanxi stream. Pingqiao bridge, Daqiao bridge, Xiaoqiao bridge, Xiashi bridge, and Dangqiao bridge, the five ancient stone bridges, which crossed the stream like a rainbow, were built in the Tang and Song Dynasties. The stream is crystal clear and the fish are clearly countable.
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In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is (\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega). The axiom of constructibility implies that Chang's conjecture fails.
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A translation and dilation of a set of uniqueness is a set of uniqueness. A union of a countable family of closed sets of uniqueness is a set of uniqueness. There exists an example of two sets of uniqueness whose union is not a set of uniqueness, but the sets in this example are not Borel. It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness.
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In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement. That is, given any open cover of the topological space, there is a refinement which is again an open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point finite open refinement.
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The existential determinative (or determiner) some is sometimes used as a functional equivalent of a(n) with plural and uncountable nouns (also called a partitive). For example, Give me some apples, Give me some water (equivalent to the singular countable forms an apple and a glass of water). Grammatically this some is not required; it is also possible to use zero article: Give me apples, Give me water. The use of some in such cases implies some limited quantity.
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The definition of separability can also be stated for other index sets and state spaces,, p. 22 such as in the case of random fields, where the index set as well as the state space can be n-dimensional Euclidean space. The concept of separability of a stochastic process was introduced by Joseph Doob,. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.
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The "universal sorter" describes one way in which mass nouns are understood when they are used in the plural. Harry Bunt suggested the universal sorter in his 1981 doctoral dissertation. When an ordinarily uncountable noun such as wine appears with plural form (several wines), it can be understood as referring to various abstract kinds (for example, varieties of wine). The "universal packager" likewise describes how mass nouns are understood when they are used as countable nouns.
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English nouns are inflected for grammatical number, meaning that if they are of the countable type, they generally have different forms for singular and plural. This article discusses the variety of ways in which English plural nouns are formed from the corresponding singular forms, as well as various issues concerning the usage of singulars and plurals in English. For plurals of pronouns, see English personal pronouns. Phonological transcriptions provided in this article are for Received Pronunciation and General American.
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The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if A ≤W B and B is a countable intersection of open sets, then so is A. The same works for all levels of the Borel hierarchy and the difference hierarchy. The Wadge hierarchy plays an important role in models of the axiom of determinacy.
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In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "a negligible amount of elements of X".
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Let N denote the natural numbers and R the reals. It follows from the theorem that the theory of (N, +, ×, 0, 1) (the theory of true first-order arithmetic) has uncountable models, and that the theory of (R, +, ×, 0, 1) (the theory of real closed fields) has a countable model. There are, of course, axiomatizations characterizing (N, +, ×, 0, 1) and (R, +, ×, 0, 1) up to isomorphism. The Löwenheim–Skolem theorem shows that these axiomatizations cannot be first- order.
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For example, in the theory of the real numbers, the completeness of a linear order used to characterize R as a complete ordered field, is a non- first-order property. Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable. This counterintuitive situation came to be known as Skolem's paradox; it shows that the notion of countability is not absolute.
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If there is a set in that is a standard model of ZF, and the ordinal is the set of ordinals that occur in , then is the of . If there is a set that is a standard model of ZF, then the smallest such set is such a . This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.
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In some occasions, it is possible for a set S and its proper subset to be equinumerous. For example, the set of even natural numbers is equinumerous to the set of all natural numbers. A set that is equinumerous to a proper subsets of itself is called Dedekind-infinite. The axiom of countable choice (ACω), a weak variant of the axiom of choice (AC), is needed to show that a set that is not Dedekind-infinite is actually finite.
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If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be stationary. See also: Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose \kappa has countable cofinality.
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In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory. One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.
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This is known as the Heine–Borel theorem. Note that compactness depends only on the topology, while boundedness depends on the metric. Lebesgue's number lemma states that for every open cover of a compact metric space M, there exists a "Lebesgue number" \delta such that every subset of M of diameter r<\delta is contained in some member of the cover. Every compact metric space is second countable, and is a continuous image of the Cantor set.
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The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension.
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In classical statistical mechanics, the number of microstates is actually uncountably infinite, since the properties of classical systems are continuous. For example, a microstate of a classical ideal gas is specified by the positions and momenta of all the atoms, which range continuously over the real numbers. If we want to define Ω, we have to come up with a method of grouping the microstates together to obtain a countable set. This procedure is known as coarse graining.
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Kechris has made contributions to the theory of Borel equivalence relations and the theory of automorphism groups of uncountable structures. His research interests cover foundations of mathematics, mathematical logic and set theory and their interactions with analysis and dynamical systems. Kechris earned his Ph.D. in 1972 under the direction of Yiannis N. Moschovakis, with a dissertation titled Projective Ordinals and Countable Analytic Sets. During his academic career he advised 23 PhD students and sponsored 20 postdoctoral researchers.
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Chapter nine discusses ways to weaken Ramsey's theorem, and the final chapter discusses stronger theorems in combinatorics including the Dushnik–Miller theorem on self-embedding of infinite linear orderings, Kruskal's tree theorem, Laver's theorem on order embedding of countable linear orders, and Hindman's theorem on IP sets. An appendix provides a proof of a theorem of Jiayi Liu, part of the collection of results showing that the graph Ramsey theorem does not fall into the big five subsystems.
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Underside the head is a radiodont-like mouthpart forming by multiple layers of plates and teeth-like structures. The trunk is wide and annulated, with a pair of well-developed lobopodous limbs on each body segment. Only 8 segment/limb pairs are countable in the incomplete fossil materials which lacking posterior region, so it may have had more (possibly up to 11 to 13) in nature. It also has pairs of digestive glands similar to those of basal arthropods.
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For any locally compact abelian (LCA) group A, the group of continuous homomorphisms :Hom(A, S1) from A to the circle group is again locally compact. Pontryagin duality asserts that this functor induces an equivalence of categories :LCAop -> LCA. This functor exchanges several properties of topological groups. For example, finite groups correspond to finite groups, compact groups correspond to discrete groups, and metrisable groups correspond to countable unions of compact groups (and vice versa in all statements).
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In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects. In this article, a Lie group refers to a real Lie group. For the complex and p-adic cases, see complex Lie group and p-adic Lie group. In this article, manifolds (in particular Lie groups) are assumed to be second countable; in particular, they have at most countably many connected components.
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Using functional principal component analysis and the Karhunen-Loève expansion, these processes can be equivalently expressed as a countable sequence of their functional principal component scores (FPCs) and eigenfunctions. In the FAM the responses (scalar or functional) conditional on the predictor functions are modeled as function of the functional principal component scores of the predictor function in an additive structure. This model can be categorized as a Frequency Additive Model since it is additive in the predictor FPC scores.
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A topological space X is called locally Euclidean if there is a non- negative integer n such that every point in X has a neighbourhood which is homeomorphic to real n-space Rn. A topological manifold is a locally Euclidean Hausdorff space. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact or second-countable. In the remainder of this article a manifold will mean a topological manifold.
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A DPO graph transformation system (or graph grammar) consists of a finite graph, which is the starting state, and a finite or countable set of labeled spans in the category of finite graphs and graph homomorphisms, which serve as derivation rules. The rule spans are generally taken to be composed of monomorphisms, but the details can vary."Double-pushout graph transformation revisited", Habel, Annegret and Müller, Jürgen and Plump, Detlef, Mathematical Structures in Computer Science, vol. 11, no. 05.
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For example, in Finnish, join vettä, "I drank (some) water", the word vesi, "water", is in the partitive case. The related sentence join veden, "I drank (the) water", using the accusative case instead, assumes that there was a specific countable portion of water that was completely drunk. The work of logicians like Godehard Link and Manfred Krifka established that the mass/count distinction can be given a precise, mathematical definition in terms of quantization and cumulativity.
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Leopold Löwenheim (1915) and Thoralf Skolem (1920) obtained the Löwenheim–Skolem theorem, which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model. This counterintuitive fact became known as Skolem's paradox. In his doctoral thesis, Kurt Gödel (1929) proved the completeness theorem, which establishes a correspondence between syntax and semantics in first-order logic.
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First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse. Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality.
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Vaught's "Never 2" theorem states that a complete first-order theory cannot have exactly two nonisomorphic countable models. He considered his best work was his paper "Invariant sets in topology and logic", introducing the Vaught transform. He is known for the Tarski–Vaught test for elementary substructures, the Feferman–Vaught theorem, the Łoś–Vaught test for completeness and decidability, the Vaught two-cardinal theorem, and his conjecture on the nonfinite axiomatizability of totally categorical theories (this work eventually led to geometric stability theory).
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In the late 1700s and early 1800s, during the French Revolution, "civilization" was used in the singular, never in the plural, and meant the progress of humanity as a whole. This is still the case in French. The use of "civilizations" as a countable noun was in occasional use in the 19th century,E.g. in the title A narrative of the loss of the Winterton East Indiaman wrecked on the coast of Madagascar in 1792; and of the sufferings connected with that event.
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It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. The α-th infinite initial ordinal is written \omega_\alpha, it is always a limit ordinal. Its cardinality is written \aleph_\alpha. For example, the cardinality of ω0 = ω is \aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals).
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One of the requirements to receive SSI is that the individual's income must be below certain limits.(SSA POMS SI 00810.001) These limits may vary based on the state in which the individual lives, living arrangements, the number of people living in the residence, and the type of income. Not all income is counted when determining an individual's "countable income" for SSI eligibility purposes. Certain payments such as: grants, scholarships, SNAP benefits, home energy assistance, and small infrequent payments are not included.
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There are other ordinal notations capable of capturing ordinals well past \varepsilon_0, but because there are only countably many strings over any finite alphabet, for any given ordinal notation there will be ordinals below \omega_1 (the first uncountable ordinal) that are not expressible. Such ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions, and more general primitive recursive ordinal functions can be used to describe larger ordinals.
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The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of the doublet are topologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable. An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together.
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Shallow potholes in a road surface. The Great Blue Hole near Ambergris Caye, Belize, is an underwater sinkhole. It has been noted that holes occupy an unusual ontological position in human psychology, as people tend to refer to them as tangible and countable objects, when in fact they are the absence of something in another object. An example of this reasoning can be found in the Beatles lyric from the song, "A Day in the Life", from their 1967 album Sgt.
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Kanoê is a polysynthetic language, where the more complex words are the verbs (Payne 1997). It is also primarily an agglutinative language, and many words are formed by simple roots, juxtaposition and suffixation. The gender can be expressed by suffixation or by a hyperonym, and while Kanoê does not make a distinction of number, it does make a distinction between uncountable and countable nouns, where the suffix {-te} is added . The syntax order of Kanoê follows SOV = subject + object + verb.
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The hyperfinite II1 factor R is the unique smallest infinite dimensional factor in the following sense: it is contained in any other infinite dimensional factor, and any infinite dimensional factor contained in R is isomorphic to R. The outer automorphism group of R is an infinite simple group with countable many conjugacy classes, indexed by pairs consisting of a positive integer p and a complex pth root of 1. The projections of the hyperfinite II1 factor form a continuous geometry.
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7 This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of : if every module has a unique maximal injective submodule, then the ring is hereditary. A complete classification of countable reduced periodic abelian groups is given by Ulm's theorem.
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A band in a Riesz space is defined to be an ideal with the extra property, that for any element in for which its absolute value is the supremum of an arbitrary subset of positive elements in , that is actually in . -Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'. Clearly every band is a -ideal, but the converse is not true in general. The intersection of an arbitrary family of bands is again a band.
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His address was published in the Conference Proceedings (The Theory of Models, North-Holland Publishing Co., 1965) as "On the denumerable models of theories with extra predicates", pp 376–389. In this paper he characterizes the countable ("denumerable") structures which can be made into models of a theory by adding interpretations of the extra predicates used in defining the theory. His characterization involves (infinite) expressions beginning with an infinite sequence of alternating quantifiers. Such expressions are now interpreted using infinite two-person games.
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Let G be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring measure, the Haar measure. (This is Borel regular measure when G is second-countable; there are both left and right measures when G is compact.) Consider the Banach space L∞(G) of essentially bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure). Definition 1.
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In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω2 × ω to {0,1} and if . This poset satisfies the countable chain condition. Forcing with this poset adds ω2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis. More generally, one can replace ω2 by any cardinal κ so construct a model where the continuum has size at least κ.
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The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets . It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra is sometimes called the Cohen algebra, though "Cohen algebra" usually refers to a different type of Boolean algebra.) The complete Cantor algebra was studied by von Neumann in 1935 (later published as ), who showed that it is not isomorphic to the random algebra of Borel subsets modulo measure zero sets.
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This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself. ;Limit point compact: See Weakly countably compact. ;Lindelöf: A space is Lindelöf if every open cover has a countable subcover. ;Local base: A set B of neighbourhoods of a point x of a space X is a local base (or local basis, neighbourhood base, neighbourhood basis) at x if every neighbourhood of x contains some member of B. ;Local basis: See Local base.
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Schematic representation of the Dirac measure by a line surmounted by an arrow. The Dirac measure is a discrete measure whose support is the point 0. The Dirac measure of any set containing 0 is 1, and the measure of any set not containing 0 is 0. In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set.
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Let f be a real-valued monotone function defined on an interval I. Then the set of discontinuities of the first kind is at most countable. One can proveWalter Rudin, Principles of Mathematical Analysis, McGraw–Hill 1964 (Corollary, p. 83)Miron Nicolescu, Nicolae Dinculeanu, Solomon Marcus, Mathematical Analysis (Bucharest 1971), Vol.1, p. 213, [in Romanian] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
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A differentiable manifold is a Hausdorff and second countable topological space , together with a maximal differentiable atlas on . Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of bump functions and partitions of unity, both of which are used ubiquitously. The notion of a manifold is identical to that of a topological manifold.
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Fresco of a Roman woman from Pompeii, c. 50 AD The word "beauty" is often used as a countable noun to describe a beautiful woman. The characterization of a person as “beautiful”, whether on an individual basis or by community consensus, is often based on some combination of inner beauty, which includes psychological factors such as personality, intelligence, grace, politeness, charisma, integrity, congruence and elegance, and outer beauty (i.e. physical attractiveness) which includes physical attributes which are valued on an aesthetic basis.
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Count nouns or countable nouns are common nouns that can take a plural, can combine with numerals or counting quantifiers (e.g., one, two, several, every, most), and can take an indefinite article such as a or an (in languages which have such articles). Examples of count nouns are chair, nose, and occasion. Mass nouns or uncountable (or non-count) nouns differ from count nouns in precisely that respect: they cannot take plurals or combine with number words or the above type of quantifiers.
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Kolmogorov’s probability theory is a typical mathematical discipline. In it, the subject matter is an abstract probability space and the scope of research is the mathematical relationships between its elements. The physical phenomenon of statistical stability of the relative frequency of events which constitutes the foundation of this discipline formally would not then appear to play any role. This phenomenon is taken into account in an idealized form by accepting the axiom of countable additivity, which is equivalent to acceptance of the hypothesis of perfect statistical stability.
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For example, Lω1ω permits countable conjunctions and disjunctions. The set of free variables in a formula of Lκω can have any cardinality strictly less than κ, yet only finitely many of them can be in the scope of any quantifier when a formula appears as a subformula of another.Some authors only admit formulas with finitely many free variables in Lκω, and more generally only formulas with < λ free variables in Lκλ. In other infinitary logics, a subformula may be in the scope of infinitely many quantifiers.
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In computer science, a list or sequence is an abstract data type that represents a countable number of ordered values, where the same value may occur more than once. An instance of a list is a computer representation of the mathematical concept of a tuple or finite sequence; the (potentially) infinite analog of a list is a stream. Lists are a basic example of containers, as they contain other values. If the same value occurs multiple times, each occurrence is considered a distinct item.
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The term comics refers to the comics medium when used as an uncountable noun and thus takes the singular: "comics is a medium" rather than "comics are a medium". When comic appears as a countable noun it refers to instances of the medium, such as individual comic strips or comic books: "Tom's comics are in the basement." Panels are individual images containing a segment of action, often surrounded by a border. Prime moments in a narrative are broken down into panels via a process called encapsulation.
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Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.For more details on Cantor's article, see Georg Cantor's first set theory article and .
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Thus, if one enlarges the group to allow arbitrary bijections of , then all sets with non- empty interior become congruent. Likewise, one ball can be made into a larger or smaller ball by stretching, or in other words, by applying similarity transformations. Hence, if the group is large enough, -equidecomposable sets may be found whose "size"s vary. Moreover, since a countable set can be made into two copies of itself, one might expect that using countably many pieces could somehow do the trick.
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Gödel also considered the case where there are a countably infinite collection of formulas. Using the same reductions as above, he was able to consider only those cases where each formula is of degree 1 and contains no uses of equality. For a countable collection of formulas \phi^i of degree 1, we may define B^i_k as above; then define D_k to be the closure of B^1_1...B^1_k, ..., B^k_1...B^k_k . The remainder of the proof then went through as before.
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The basis problem was posed by Stefan Banach in his book, Theory of Linear Operators. Banach asked whether every separable Banach space has a Schauder basis. A Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that for Hamel bases we use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.
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A few other related results are: #The Nielsen–Schreier theorem: Every subgroup of a free group is free. #A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks. #The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [am, bn] for non-zero m and n.
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Classrooms are the best and favorite places of all children, from Class I to XII. The school does not just have a teacher standing in the corner of a classroom and dictating stuff which makes children sleep and the act of education boring. All classrooms of the school are equipped with a Superior Digital Technology now common to only a countable number of schools in Kerala. It is the Plasma TV. Bhavan's Girinagar was the first school to introduce it in the Kochi City of Kerala.
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1) Tychonoff's 1930 proof used the concept of a complete accumulation point. 2) The theorem is a quick corollary of the Alexander subbase theorem. More modern proofs have been motivated by the following considerations: the approach to compactness via convergence of subsequences leads to a simple and transparent proof in the case of countable index sets. However, the approach to convergence in a topological space using sequences is sufficient when the space satisfies the first axiom of countability (as metrizable spaces do), but generally not otherwise.
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In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, α) for the number of models of T (up to isomorphism) of cardinality α. The spectrum problem is to describe the possible behaviors of I(T, α) as a function of α. It has been almost completely solved for the case of a countable theory T.
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Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space". defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with l2. The convention still persists in most rigorous treatments of quantum mechanics; see for instance . Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms.
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Silver proved the consistency of Chang's conjecture from the consistency of an ω1-Erdős cardinal. Hans-Dieter Donder showed the reverse implication: if CC holds, then ω2 is ω1-Erdős in K. More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ). The consistency of (\omega_3,\omega_2)\twoheadrightarrow(\omega_2,\omega_1) was shown by Laver from the consistency of a huge cardinal.
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Bendixson started out very much as a pure mathematician but later in his career he turned to also consider problems from applied mathematics. His first research work was on set theory and the foundations of mathematics, following the ideas which Georg Cantor had introduced. He contributed important results in point set topology. As a young student Bendixson made his name by proving that every uncountable closed set can be partitioned into a perfect set (the Bendixson derivative of the original set) and a countable set.
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The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory.
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Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context. ; almost all: A shorthand term for "all except for a set of measure zero", when there is a measure to speak of. For example, "almost all real numbers are transcendental" because the algebraic real numbers form a countable subset of the real numbers with measure zero.
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In mathematics, a universal graph is an infinite graph that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph family : that is, an infinite graph belonging to F that contains all finite graphs in . For instance, the Henson graphs are universal in this sense for the -clique-free graphs.
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In graph theory, the Henson graph is an undirected infinite graph, the unique countable homogeneous graph that does not contain an -vertex clique but that does contain all -free finite graphs as induced subgraphs. For instance, is a triangle-free graph that contains all finite triangle-free graphs. These graphs are named after C. Ward Henson, who published a construction for them (for all ) in 1971.. The first of these graphs, , is also called the homogeneous triangle-free graph or the universal triangle-free graph.
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In quantum mechanics, the discrete spectrum of an observable corresponds to the eigenvalues of the operator used to model that observable. According to the mathematical theory of such operators, its eigenvalues are a discrete set of isolated points, which may be either finite or countable. Discrete spectra are usually associated with systems that are bound in some sense (mathematically, confined to a compact space). The position and momentum operators have continuous spectra in an infinite domain, but a discrete (quantized) spectrum in a compact domainL.
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A superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties: :1. all axioms of intuitionistic logic belong to L; :2. if F and G are formulas such that F and F → G both belong to L, then G also belongs to L (closure under modus ponens); :3. if F(p1, p2, ..., pn) is a formula of L, and G1, G2, ..., Gn are any formulas, then F(G1, G2, ..., Gn) belongs to L (closure under substitution).
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An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0\. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable. Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.
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In his original definition, Baire defined a notion of category (unrelated to category theory) as follows. :Definition: A subset of a topological space is called nowhere dense or rare if its closure has empty interior. Note that a closed subset is nowhere dense if and only if its interior is empty. :Definition: A subset of a topological space is said to be meagre in , a meagre subset of , or of the first category in if it is a countable union of nowhere dense subsets of .
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In topology, the split interval is a space that results from splitting each interior point in a closed interval into two adjacent points. It may be defined as the lexicographic product [0, 1] × {0, 1} without the isolated edge points, (0,1) and (1,0), equipped with the order topology. It is also known as the Alexandrov double arrow space or two arrows space. The split interval is compact Hausdorff, and it is hereditarily Lindelöf and hereditarily separable, but it is not metrizable; its metrizable subspaces are all countable.
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Zermelo set theory (Z) is Zermelo–Fraenkel set theory without the axiom of replacement. It differs from ZF in that Z does not prove that the power set operation can be iterated uncountably many times beginning with an arbitrary set. In particular, Vω \+ ω, a particular countable level of the cumulative hierarchy, is a model of Zermelo set theory. The axiom of replacement, on the other hand, is only satisfied by Vκ for significantly larger values of κ, such as when κ is a strongly inaccessible cardinal.
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The CFU/plate is read from a plate in the linear range, and then the CFU/g (or CFU/mL) of the original is deduced mathematically, factoring in the amount plated and its dilution factor. serially diluted in order to obtain at least one plate with a countable number of bacteria. In this figure, the "x10" plate is suitable for counting. An advantage to this method is that different microbial species may give rise to colonies that are clearly different from each other, both microscopically and macroscopically.
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A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of all subsets of R of the form (n, n + 2) with integer n. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of R of the form (−n, n) with natural n. If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite.
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A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and a theory is categorical in some uncountable cardinal greater than or equal to then it is categorical in all cardinalities greater than .
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In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing functions fα: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity.
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In mathematical set theory, the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various models are all equally valid or true, though some may be more useful or attractive than others. The opposite view is the "universe" view of set theory in which all sets are contained in some single ultimate model. The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse".
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For a simplified proof of Läuchli's theorem by Mycielski, see . The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a theory of second-order arithmetic, to Kőnig's infinity lemma. For a counterexample to the theorem in models of set theory without choice, let be an infinite graph in which the vertices represent all possible real numbers. In , connect each two real numbers and by an edge whenever one of the values is a rational number.
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In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a countable open set. The axiom of choice implies the existence of sets of reals that do not have the perfect set property, such as Bernstein sets. However, in Solovay's model, which satisfies all axioms of ZF but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary. Every analytic set has the perfect set property.
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He generalized the Cantor–Bernstein theorem, which said the collection of countable order types has the cardinality of the continuum and showed that the collection of all graded types of an idempotent cardinality has a cardinality of 2. For the summer semester 1910 Hausdorff was appointed as professor to the University of Bonn. In Bonn, he began a lecture on set theory, which he repeated in the summer semester 1912, substantially revised and expanded. In the summer of 1912 he also began work on his magnum opus, the book Basics of set theory.
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In fact, if every countable subgroup of a locally finite group has only countably many maximal p-subgroups, then every maximal p-subgroup of the group is conjugate . The class of locally finite groups behaves somewhat similarly to the class of finite groups. Much of the 1960s theory of formations and Fitting classes, as well as the older 19th century and 1930s theory of Sylow subgroups has an analogue in the theory of locally finite groups . Similarly to the Burnside problem, mathematicians have wondered whether every infinite group contains an infinite abelian subgroup.
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Dem Deutschen Volke ("To the German People"), the dedication on the Reichstag building in Berlin. The German noun Volk () translates to people, both uncountable in the sense of people as in a crowd, and countable (plural Völker) in the sense of a people as in an ethnic group or nation (compare the English term folk). Within an English-language context, the German word is of interest primarily for its use in German philosophy, as in Volksseele ("national soul"), and in German nationalism – notably the derived adjective völkisch ("national, ethnic").
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A disabled beneficiary's own assets can form the corpus of a supplemental needs trust. Although an individual's assets are usually considered to be countable resources for purposes of qualification for Medicaid, the supplemental needs trust statute permits an individual to fund an SNT without being penalized. Generally, divestment of assets for purposes of Medicaid qualification will trigger a 36-to-60 month "look back" by Medicaid, in which all asset transfers of the would-be beneficiary are examined. If found to be made specifically to qualify for Medicaid the transfer will be disallowed.
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Income that is routed into a Miller Trust each month, as received, is no longer counted for Medicaid eligibility. The trust provides a specific manner in which funds in the trust will be spent each month. A Miller Trust does not provide any assistance with the "countable resources" requirement for Medicaid, and assets (other than monthly income) are not contributed to a Miller Trust. Upon the death of the beneficiary, the state Medicaid agency must be paid back for its medical assistance from any remaining assets in the Miller trust.
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The cofinality of an ordinal α is the smallest ordinal δ which is the order type of a cofinal subset of α. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set. Thus for a limit ordinal α, there exists a δ-indexed strictly increasing sequence with limit α. For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω.
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Given any model M of ZFC, the collection of hereditarily finite sets in M will satisfy the GST axioms. Therefore, GST cannot prove the existence of even a countable infinite set, that is, of a set whose cardinality is ℵ0. Even if GST did afford a countably infinite set, GST could not prove the existence of a set whose cardinality is \aleph_1, because GST lacks the axiom of power set. Hence GST cannot ground analysis and geometry, and is too weak to serve as a foundation for mathematics.
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If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X→M. Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family ZN of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f in ZN :f has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.
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All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real- valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. We also assume that always denotes a finite sequence of non-negative integers and we will use the notation: : and .
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If Ω is uncountable, still, it may happen that p(ω) ≠ 0 for some ω; such ω are called atoms. They are an at most countable (maybe empty) set, whose probability is the sum of probabilities of all atoms. If this sum is equal to 1 then all other points can safely be excluded from the sample space, returning us to the discrete case. Otherwise, if the sum of probabilities of all atoms is between 0 and 1, then the probability space decomposes into a discrete (atomic) part (maybe empty) and a non-atomic part.
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In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space (for example, a scale might be defined on a set of real numbers). Scales were originally isolated as a concept in the theory of uniformization,Kechris and Moschovakis 2008:28 but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of wellorderings of a given complexity, and showing (under certain assumptions) that there are largest countable sets of certain complexities.
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Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula). Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.
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An LF-space that is the inductive limit of a countable sequence of separable spaces is separable. LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck). If is the strict inductive limit of an increasing sequence of Fréchet space then a subset of is bounded in if and only if there exists some such that is a bounded subset of . A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.
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For this one modifies the construction above by choosing a real number xβ that is not in any of the countable number of sets of the form (Sα + X)/n for α < β, where n is a positive integer and X is an integral linear combination of the numbers xα for α < β. Then the group generated by these numbers is a Sierpiński set and a group under addition. More complicated variations of this construction produce examples of Sierpiński sets that are subfields or real-closed subfields of the real numbers.
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Discrete event simulation Continuous simulation Continuous simulation must be clearly differentiated from discrete and discrete event simulation. Discrete simulation relies upon countable phenomena like the number of individuals in a group, the number of darts thrown, or the number of nodes in a Directed graph. Discrete event simulation produces a system which changes its behaviour only in response to specific events and typically models changes to a system resulting from a finite number of events distributed over time. A continuous simulation applies a Continuous function using Real numbers to represent a continuously changing system.
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Helmut Ulm's father was an elementary school teacher in Elberfeld. After finishing high school in Wuppertal in 1926, he attended the universities of Göttingen (1926–1927), Jena (1927) and Bonn (1927–1930), where he studied mathematics and physics, attending the lectures of Richard Courant, Erich Bessel-Hagen, Felix Hausdorff, and the joint Hausdorff–Otto Toeplitz seminar. He graduated summa cum laude in 1930 with a thesis about countable periodic abelian groups (1933). In 1933–1935 he was an assistant in Göttingen and worked with Wilhelm Magnus and Olga Taussky-Todd editing David Hilbert's Collected Works.
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Thus e.g. a utility function defines a preference relation. In this context, weak orderings are also known as preferential arrangements.. If X is finite or countable, every weak order on X can be represented by a function in this way.. However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences.
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The event was Ohio State University's second annual "Big Wish Gala". Their chart-topping song, "My Wish" is also used on ESPN as the soundtrack for its series that follows the Make-a-Wish Foundation as they turn dreams into reality for children with life-threatening illnesses. Since then, they have also contributed countable hours of their time—and $4 million—to Monroe Carell Jr. Children's Hospital at Vanderbilt which is among the nation's leading pediatric facilities, where the Rascal Flatts Pediatric Surgery Center was named in recognition of the trio's long-standing involvement.
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The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology if and only if they are of the form U \ A where U is open in the Euclidean topology and A is countable. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff). There exist spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff.
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Todorčević's work involves mathematical logic, set theory, and their applications to pure mathematics. In Todorčević's 1978 master’s thesis, he constructed a model of MA + ¬wKH in a way to allow him to make the continuum any regular cardinal, and so derived a variety of topological consequences. Here MA is an abbreviation for Martin's axiom and wKH stands for the weak Kurepa Hypothesis. In 1980, Todorčević and Abraham proved the existence of rigid Aronszajn trees and the consistency of MA + the negation of the continuum hypothesis + there exists a first countable S-space.
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In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.
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As noted in the previous paragraph, second- order comprehension axioms easily generalize to the higher-order framework. However, theorems expressing the compactness of basic spaces behave quite differently in second- and higher-order arithmetic: on one hand, when restricted to countable covers/the language of second-order arithmetic, the compactness of the unit interval is provable in WKL0 from the next section. On the other hand, given uncountable covers/the language of higher-order arithmetic, the compactness of the unit interval is only provable from (full) second-order arithmetic (). Other covering lemmas (e.g.
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All the conditions in the statement of the theorem are necessary: 1\. We cannot eliminate the Hausdorff condition; a countable set (with at least two points) with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable. 2\. We cannot eliminate the compactness condition, as the set of rational numbers shows. 3\. We cannot eliminate the condition that one point sets cannot be open, as any finite space with the discrete topology shows. Corollary.
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It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.) The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point. The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.
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Kőnig's lemma may be considered to be a choice principle; the first proof above illustrates the relationship between the lemma and the axiom of dependent choice. At each step of the induction, a vertex with a particular property must be selected. Although it is proved that at least one appropriate vertex exists, if there is more than one suitable vertex there may be no canonical choice. In fact, the full strength of the axiom of dependent choice is not needed; as described below, the axiom of countable choice suffices.
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Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverse images. The complement of an analytic set need not be analytic. Suslin proved that if the complement of an analytic set is analytic then the set is Borel. (Conversely any Borel set is analytic and Borel sets are closed under complements.) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set containing one and disjoint from the other.
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Players may "pass" if they believe they cannot beat an earlier bid. Once two players have passed, the third player wins the auction, takes the talon adds it to his hand (it is mandatory to show the other players the said cards), selects 2 cards from his hand and places them face down in his pile of countable cards for scoring. The player who wins the "auction" is the one who decides the trump suit for that deal. When play is finished, the declarer counts his cards to see if he has achieved his bid.
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In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic, particularly when the statistic is a function or a random process and the notion of conditional density is not applicable. If one possible σ-algebra on X is where ∅ is the empty set. In general, a finite algebra is always a σ-algebra. If {A1, A2, A3, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
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In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ (an infinite cardinal number), if the Stone space of every model of T of size ≤ λ has itself size ≤ λ. T is called a stable theory if there is no upper bound for the cardinals κ such that T is stable in κ. The stability spectrum of T is the class of all cardinals κ such that T is stable in κ. For countable theories there are only four possible stability spectra.
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Navy's doctoral thesis, "Nonparacompactness in Para- Lindelöf Spaces", was important in the development of metrizability theory. The paper examines the properties of para-Lindelöf topological spaces, which are a generalization of both Lindelöf spaces and paracompact spaces. In a para-Lindelöf space, every open cover has a locally countable open refinement, that is, one such that each point of the space has a neighborhood that intersects only countably many elements of the refinement. The spaces constructed by Navy are counterexamples to the conjecture that all para- Lindelöf spaces are paracompact.
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However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assigns a measure of b − a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because it is countable. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using Carathéodory's extension theorem) does not make it obvious whether non- measurable sets exist. The answer to that question involves the axiom of choice.
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In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
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In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available.
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Recursive ordinals (or computable ordinals) are certain countable ordinals: loosely speaking those represented by a computable function. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer (Turing machine, say) can manipulate them (and, essentially, compare them). A different definition uses Kleene's system of ordinal notations.
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An ordinal that is both admissible and a limit of admissibles, or equivalently such that \alpha is the \alpha-th admissible ordinal, is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals. For example, we can define recursively Mahlo ordinals: these are the \alpha such that every \alpha-recursive closed unbounded subset of \alpha contains an admissible ordinal (a recursive analog of the definition of a Mahlo cardinal). But note that we are still talking about possibly countable ordinals here.
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The model N is an inner model of M[G] satisfying ZF + DC such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property. The proof of this uses the fact that every real in M[G] is definable over a countable sequence of ordinals, and hence N and M[G] have the same reals. Instead of using Solovay's model N, one can also use the smaller inner model L(R) of M[G], consisting of the constructible closure of the real numbers, which has similar properties.
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A set of unary relations Pi for i in some set I is called independent if for every two disjoint finite subsets A and B of I there is some element x such that Pi(x) is true for i in A and false for i in B. Independence can be expressed by a set of first-order statements. The theory of a countable number of independent unary relations is complete, but has no atomic models. It is also an example of a theory that is superstable but not totally transcendental.
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The number density (symbol: n or ρN) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one- dimensional linear number density. Population density is an example of areal number density. The term number concentration (symbol: lowercase n, or C, to avoid confusion with amount of substance indicated by uppercase N) is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.
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In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner. An axiom introduced by states that any two ℵ1-dense subsets of the real line are order-isomorphic. Todorcevic showed that this Baumgartner's Axiom is a consequence of the Proper Forcing Axiom. Another axiom introduced by states that Martin's axiom for partially ordered sets MAP(κ) is true for all partially ordered sets P that are countable closed, well met and ℵ1-linked and all cardinals κ less than 2ℵ1.
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One of Ajtai's results states that the length of proofs in propositional logic of the pigeonhole principle for n items grows faster than any polynomial in n. He also proved that the statement "any two countable structures that are second- order equivalent are also isomorphic" is both consistent with and independent of ZFC. Ajtai and Szemerédi proved the corners theorem, an important step toward higher-dimensional generalizations of the Szemerédi theorem. With Komlós and Szemerédi he proved the ct2/log t upper bound for the Ramsey number R(3,t).
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It is possible to accommodate countably infinitely many coachloads of countably infinite passengers each, by several different methods. Most methods depend on the seats in the coaches being already numbered (or use the axiom of countable choice). In general any pairing function can be used to solve this problem. For each of these methods, consider a passenger's seat number on a coach to be n, and their coach number to be c, and the numbers n and c are then fed into the two arguments of the pairing function.
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This relation is dual to in sense that if and only if . The relation is closely related to the _downward closure_ of a set, which is defined by }, in a manner similar to how is related to the upward closure. ;Other topological examples Example: The set of all dense open subsets of a topological space is a proper -system and a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a -system and a prefilter that is finer than .
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N is an elementary substructure of M if N and M are structures of the same signature σ such that for all first- order σ-formulas φ(x1, …, xn) with free variables x1, …, xn, and all elements a1, …, an of N, φ(a1, …, an) holds in N if and only if it holds in M: :N \models φ(a1, …, an) iff M \models φ(a1, …, an). It follows that N is a substructure of M. If N is a substructure of M, then both N and M can be interpreted as structures in the signature σN consisting of σ together with a new constant symbol for every element of N. Then N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σN-structures. If N is an elementary substructure of M, one writes N \preceq M and says that M is an elementary extension of N: M \succeq N. The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
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The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski (1948) established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. (He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic.) A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem, proved by Michael D. Morley (1965), states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e.
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This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous – see Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs from diagonal argument that he gave in 1891.
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The continuum hypothesis has been proven independent of the ZF axioms of set theory, so within that system, the proposition can neither be proven true nor proven false. A formalist would therefore say that the continuum hypothesis is neither true nor false, unless you further refine the context of the question. A platonist, however, would assert that there either does or does not exist a transfinite set with a cardinality less than the continuum but greater than any countable set. So, regardless of whether it has been proven unprovable, the platonist would argue that an answer nonetheless does exist.
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For each element g of G introduce a countable set of variables gi for i>0. Define exp(gt) to be the formal power series in t :\exp(gt) = 1+g_1t+g_2t^2+g_3t^3+\cdots. The exp ring of G is the commutative ring generated by all the elements gi with the relations :\exp((g+h)t) = \exp(gt)\exp(ht) for all g, h in G; in other words the coefficients of any power of t on both sides are identified. The ring Exp(G) can be made into a commutative and cocommutative Hopf algebra as follows.
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Although the Rado graph is universal for induced subgraphs, it is not universal for isometric embeddings of graphs, where an isometric embedding is a graph isomorphism which preserves distance. The Rado graph has diameter two, and so any graph with larger diameter does not embed isometrically into it. has described a family of universal graphs for isometric embedding, one for each possible finite graph diameter; the graph in his family with diameter two is the Rado graph. The Henson graphs are countable graphs (one for each positive integer ) that do not contain an -vertex clique, and are universal for -clique-free graphs.
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176-179 Democratic representation in the Washington legislature would during this period at times be countable on one hand,Schattschneider, Elmer Eric; The Semisovereign People: A Realist's View of Democracy in America, pp. 76-84 and no Democrat other than Woodrow Wilson in 1916 would henceforth carry even one county in the state before Catholic Al Smith carried German-settled Ferry County in 1928. Republican primaries would take over as the chief mode of political competition when introduced later in the decade.Murray, Keith; ‘Issues and Personalities of Pacific Northwest Politics, 1889-1950’, The Pacific Northwest Quarterly, vol.
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Khinchin's theorem on the factorization of distributions says that every probability distribution P admits (in the convolution semi-group of probability distributions) a factorization :P = P_1 \otimes P_2 where P1 is a probability distribution without any indecomposable factor and P2 is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions. The factorization is not unique, in general. The theorem was proved by A. Ya. Khinchin for distributions on the line, and later it became clear that it is valid for distributions on considerably more general groups. A broad class (seeD.
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A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω if and only if the corresponding projection-valued measure π on X is homogeneous of multiplicity n. In fact, X breaks up into a countable disjoint family {Xn} 1 ≤ n ≤ ω of Borel sets such that π is homogeneous of multiplicity n on Xn. It is also easy to show Xn is G invariant. Lemma. Any system of imprimitivity is an orthogonal direct sum of homogeneous ones. It can be shown that if the action of G on X is transitive, then any system of imprimitivity on X is homogeneous.
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This result is of particular interest when the action of H on X is such that every ergodic quasi-invariant measure on X is transitive. In that case, each such measure is the image of (a totally finite version) of Haar measure on X by the map : g \mapsto g \cdot x_0. A necessary condition for this to be the case is that there is a countable set of H invariant Borel sets which separate the orbits of H. This is the case for instance for the action of the Lorentz group on the character space of R4.
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Skolem (1934) pioneered the construction of non-standard models of arithmetic and set theory. Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.
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Fewer versus less is the debate revolving around grammatically using the use words "fewer" and "less" correctly. According to prescriptive grammar, "fewer" should be used (instead of "less") with nouns for countable objects and concepts (discretely quantifiable nouns, or count nouns). According to this rule, "less" should be used only with a grammatically singular noun (including mass nouns). However, descriptive grammarians (who describe language as actually used) point out that this rule does not correctly describe the most common usage of today or the past and in fact arose as an incorrect generalization of a personal preference expressed by a grammarian in 1770.
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In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).
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According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points. Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set (Rademacher's theorem). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well- behaved.
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Let (X, d) be a complete metric space and let E be a subset of X. Let B(x, r) denote the closed ball in (X, d) with centre x ∈ X and radius r > 0\. E is said to be porous if there exist constants 0 < α < 1 and r0 > 0 such that, for every 0 < r ≤ r0 and every x ∈ X, there is some point y ∈ X with :B(y, \alpha r) \subseteq B(x, r) \setminus E. A subset of X is called σ-porous if it is a countable union of porous subsets of X.
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The reals are uncountable; that is: there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of N. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
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In the Internet Engineering Task Force (IETF), decisions are assumed to be taken by rough consensus. The IETF has studiously refrained from defining a mechanical method for verifying such consensus, apparently in the belief that any such codification leads to attempts to "game the system." Instead, a working group (WG) chair or BoF chair is supposed to articulate the "sense of the group." One tradition in support of rough consensus is the tradition of humming rather than (countable) hand-raising; this allows a group to quickly discern the prevalence of dissent, without making it easy to slip into majority rule.
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Beth's earlier definability theorem is a consequence of Svenonius' Theorem. The other two papers include a characterization of theories having only one countable model, obtained also by the Polish logician Czesław Ryll-Nardzewski, and results on prime models, obtained also by Robert Vaught at Berkeley. All of these results are classics of modern model theory. Presumably as a result of these papers he was named a Visiting Associate Professor at The University of California, Berkeley, for 1962-1963, and gave an Invited Address at the International Symposium on the Theory of Models held there in 1963.
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A metric space M is compact if every sequence in M has a subsequence that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers. Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaces with finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact. A metric space is compact if and only if it is complete and totally bounded.
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Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics.
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The space Qp of p-adic numbers is complete for any prime number . This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric. If is an arbitrary set, then the set of all sequences in becomes a complete metric space if we define the distance between the sequences and to be , where is the smallest index for which is distinct from , or if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discrete space .
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In loop quantum gravity (LQG), a spin network represents a "quantum state" of the gravitational field on a 3-dimensional hypersurface. The set of all possible spin networks (or, more accurately, "s-knots" – that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG Hilbert space. In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral (functional integration) description of quantum gravity. It is closely related to loop quantum gravity.
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At the extreme, the piling on of such countable interventions amounts to interventionism, a flawed model of care lacking holistic circumspection—merely treating discrete problems (in billable increments) rather than maintaining health. Therapy and treatment, in the middle of the semantic field, can connote either the holism of care or the discreteness of intervention, with context conveying the intent in each use. Accordingly, they can be used in both noncount and count senses (for example, therapy for chronic kidney disease can involve several dialysis treatments per week). The words aceology and iamatology are obscure and obsolete synonyms referring to the study of therapies.
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If the graph is countable, the vertices are well-ordered and one can canonically choose the smallest suitable vertex. In this case, Kőnig's lemma is provable in second-order arithmetic with arithmetical comprehension, and, a fortiori, in ZF set theory (without choice). Kőnig's lemma is essentially the restriction of the axiom of dependent choice to entire relations R such that for each x there are only finitely many z such that xRz. Although the axiom of choice is, in general, stronger than the principle of dependent choice, this restriction of dependent choice is equivalent to a restriction of the axiom of choice.
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Similarly with the higher axioms of infinity. Now \aleph_1 is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set C [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach C. Thus C is greater than \aleph_n, \aleph_\omega, \aleph_a, where a = \aleph_\omega, etc.
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For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice. In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space.
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In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers... Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers.. Cantor's construction builds a one-to-one correspondence between the set of transcendental numbers and the set of real numbers. In this article, Cantor only applies his construction to the set of irrational numbers. Cantor's work established the ubiquity of transcendental numbers.
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Andrea Seabrook (born 1974) is an American journalist reporting in various formats: radio, print, podcast & digital. She is known for her coverage of politics, Congress and the White House, and for her work hosting NPR's signature news programs, All Things Considered, Weekend Edition, Talk of the Nation, and others. Seabrook was among the first on-air public radio personalities to leave NPR and start a successful, independent podcast, DecodeDC, which was later acquired by the E.W. Scripps Corp. Seabrook went on to serve as DC Bureau Chief of Marketplace with Kai Ryssdal, and then Managing Editor of the civic-tech app, Countable.
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In English, such words are almost always mass nouns. Some uncountable nouns can be alternatively used as count nouns when meaning "a type of", and the plural means "more than one type of". For example, strength is uncountable in Strength is power, but it can be used as a countable noun to mean an instance of [a kind of] strength, as in My strengths are in physics and chemistry. Some words, especially proper nouns such as the name of an individual, are nearly always in the singular form because there is only one example of what that noun means.
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A subset X of the real line is a strong measure zero set if to every sequence (εn) of positive reals there exists a sequence of intervals (In) which covers X and such that In has length at most εn. Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC. A subset X of the real line is \aleph_1-dense if every open interval contains \aleph_1-many elements of X. Whether all \aleph_1-dense sets are order-isomorphic is independent of ZFC.Baumgartner, J., All \aleph_1-dense sets of reals can be isomorphic, Fund. Math.
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Aristotle responded to these paradoxes by developing the notion of a potential countable infinity, as well as the infinitely divisible continuum. Unlike the eternal and unchanging cycles of time, he believed that the world is bounded by the celestial spheres and that cumulative stellar magnitude is only finitely multiplicative. The Indian philosopher Kanada, founder of the Vaisheshika school, developed a notion of atomism and proposed that light and heat were varieties of the same substance.Will Durant, Our Oriental Heritage: In the 5th century AD, the Buddhist atomist philosopher Dignāga proposed atoms to be point-sized, durationless, and made of energy.
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In the XVIII century, in addition to the listed expeditions, the Collegium of Commerce included a countable expedition and several commissions on commerce. The counting expedition was established temporarily ("before the state was approved") by decree on March 31 (April 11) 1732 and destroyed by decree on June 21 (July 2) 1743. It had the character of an audit institution and carried out the case, which was normally distributed among the office of the college and the audit college. The Commission on Commerce was established in 1727, "thanks to the merchants, seeing it in poor condition - for correction and consideration thereof".
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In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function , where denotes Euclidean space and denotes the complex numbers, the lemma gives a precise way of partitioning into two sets: one where is essentially small; the other a countable collection of cubes where is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of , wherein is written as the sum of "good" and "bad" functions, using the above sets.
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An example is the set of natural numbers, In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers is larger than , because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable".
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Note that here the model parameters p consist of a function and that the response of a model also consists of a function denoted by d(x). This equation is an extension to infinite dimension of the matrix equation d=Fp given in the case of discrete problems. For sufficiently smooth K the operator defined above is compact on reasonable Banach spaces such as the L^2. F. Riesz theory states that the set of singular values of such an operator contains zero (hence the existence of a null-space), is finite or at most countable, and, in the latter case, they constitute a sequence that goes to zero.
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An ω-model of T is a model of T whose domain includes the natural numbers and whose specified names and symbol N are standardly interpreted, respectively as those numbers and the predicate having just those numbers as its domain (whence there are no nonstandard numbers). If N is absent from the language then what would have been the domain of N is required to be that of the model, i.e. the model contains only the natural numbers. (Other models of T may interpret these symbols nonstandardly; the domain of N need not even be countable, for example.) These requirements make the ω-rule sound in every ω-model.
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A trust is a legal arrangement in which legal title to assets is held by a trustee under certain defined restrictions written within the governing instrument (usually a will or a written trust agreement) for the benefit of another party known as the beneficiary. Trusts can be used as a vehicle to make assets available to a beneficiary but still significantly restrict them. Such Trusts are called spendthrift trusts. A beneficiary does not necessarily have to be disabled to benefit from a spendthrift trust, but most spendthrift trusts would not suffice to qualify their beneficiary for Medicaid as the assets held within them would be countable.
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The Rado graph was first constructed by in two ways, with vertices either the hereditarily finite sets or the natural numbers. (Strictly speaking Ackermann described a directed graph, and the Rado graph is the corresponding undirected graph given by forgetting the directions on the edges.) constructed the Rado graph as the random graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. rediscovered the Rado graph as a universal graph, and gave an explicit construction of it with vertex set the natural numbers.
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In one of Ackermann's original 1937 constructions, the vertices of the Rado graph are indexed by the hereditarily finite sets, and there is an edge between two vertices exactly when one of the corresponding finite sets is a member of the other. A similar construction can be based on Skolem's paradox, the fact that there exists a countable model for the first-order theory of sets. One can construct the Rado graph from such a model by creating a vertex for each set, with an edge connecting each pair of sets where one set in the pair is a member of the other., Theorem 2.
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Chads from punched cards. Each chad is about 1/8 inch (3 mm) long. Votomatic voting machines of the type used in the 2000 election in Florida The chip (chad) receiver from a UNIVAC key punch Pouring chads from a jar at the Computer History Museum Asymmetrical chad produced by a railroad ticket punch Chad refers to fragments sometimes created when holes are made in a paper, card or similar synthetic materials, such as computer punched tape or punched cards. The word "chad" has been used both as a mass noun (as in "a pile of chad") and as a countable noun (pluralizing as in "many chads").
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A continuous-time Markov chain (Xt)t ≥ 0 is defined by a finite or countable state space S, a transition rate matrix Q with dimensions equal to that of the state space and initial probability distribution defined on the state space. For i ≠ j, the elements qij are non-negative and describe the rate of the process transitions from state i to state j. The elements qii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one. There are three equivalent definitions of the process.
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262 (1961), pp. 455-475. On the other hand, it is an easy theorem that every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (up to isomorphism) exactly the finitely generated subgroups of finitely presented groups. Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups. As a corollary, there is a universal finitely presented group that contains all finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism).
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On the other hand, a set may have topological dimension less than n and have positive n-dimensional Lebesgue measure. An example of this is the Smith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure. In order to show that a given set A is Lebesgue-measurable, one usually tries to find a "nicer" set B which differs from A only by a null set (in the sense that the symmetric difference (A − B) \cup(B − A) is a null set) and then show that B can be generated using countable unions and intersections from open or closed sets.
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On December 21, 2017, an unofficial report was published on Reddit claiming that the Seminoles were not bowl eligible due to an NCAA rule stating that for an FCS opponent to be countable towards bowl eligibility, the FCS program must have awarded 90% of the FCS scholarship limit. Delaware State, an FCS team that lost to FSU earlier in the season, did not meet the 90% threshold set by the NCAA. Without this win, FSU stood at 5–6 on the season. However, on December 22, 2017, Florida State addressed the issue and stated that Delaware State verified its scholarship situation as eclipsing the 90-percent threshold.
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Sometimes cross-stitch is done on designs printed on the fabric (stamped cross-stitch); the stitcher simply stitches over the printed pattern. Cross- stitch is often executed on easily countable fabric called aida cloth whose weave creates a plainly visible grid of squares with holes for the needle at each corner. Fabrics used in cross-stitch include linen, aida, and mixed- content fabrics called 'evenweave' such as jobelan. All cross-stitch fabrics are technically "evenweave" as the term refers to the fact that the fabric is woven to make sure that there are the same number of threads per inch in both the warp and the weft (i.e.
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The topological Vaught conjecture is the statement that whenever a Polish group acts continuously on a Polish space, there are either countably many orbits or continuum many orbits. The topological Vaught conjecture is more general than the original Vaught conjecture: Given a countable language we can form the space of all structures on the natural numbers for that language. If we equip this with the topology generated by first-order formulas, then it is known from A. Gregorczyk, A. Mostowski, C. Ryll- Nardzewski, "Definability of sets of models of axiomatic theories" (Bulletin of the Polish Academy of Sciences (series Mathematics, Astronomy, Physics), vol. 9 (1961), pp.
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Normal spanning trees are also closely related to the ends of an infinite graph, equivalence classes of infinite paths that, intuitively, go to infinity in the same direction. If a graph has a normal spanning tree, this tree must have exactly one infinite path for each of the graph's ends. An infinite graph can be used to form a topological space by viewing the graph itself as a simplicial complex and adding a point at infinity for each end of the graph. With this topology, a graph has a normal spanning tree if and only if its set of vertices can be decomposed into a countable union of closed sets.
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The Prüfer -group with presentation , illustrated as a subgroup of the unit circle in the complex plane In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.
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In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system (X_n, i_{nm}) of Fréchet spaces. This means that X is a direct limit of a direct system (X_n, i_{nm}) in the category of locally convex topological vector spaces and each X_n is a Fréchet space. If each of the bonding maps i_{nm} is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on by is identical to the original topology on .
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Any finite or countable -clique- free graph can be found as an induced subgraph of by building it one vertex at a time, at each step adding a vertex whose earlier neighbors in match the set of earlier neighbors of the corresponding vertex in . That is, is a universal graph for the family of -clique-free graphs. Because there exist -clique-free graphs of arbitrarily large chromatic number, the Henson graphs have infinite chromatic number. More strongly, if a Henson graph is partitioned into any finite number of induced subgraphs, then at least one of these subgraphs includes all -clique-free finite graphs as induced subgraphs.
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One approach for avoiding mathematical construction issues of stochastic processes, proposed by Joseph Doob, is to assume that the stochastic process is separable. Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set. Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied. Another approach is possible, originally developed by Anatoliy Skorokhod and Andrei Kolmogorov, for a continuous-time stochastic process with any metric space as its state space.
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While a finitely additive measure is sufficient for most intuition of area, and is analogous to Riemann integration, it is considered insufficient for probability, because conventional modern treatments of sequences of events or random variables demand countable additivity. In this respect, the plane is similar to the line; there is a finitely additive measure, extending Lebesgue measure, which is invariant under all isometries. When you increase in dimension the picture gets worse. The Hausdorff paradox and Banach–Tarski paradox show that you can take a three-dimensional ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1.
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A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous.
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In probability theory, a standard probability space, also called Lebesgue- Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms. The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be effectively substituted for many of these in probability theory.
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On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every real- closed field satisfies the same first-order sentences in the signature \langle +,\cdot,\le\rangle as the real numbers.) In second-order logic, it is possible to write formal sentences which say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective.
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Extremely complex sets of natural numbers can be defined by simple formulas in the language of set theory (which can quantify over arbitrary sets). In the context of second-order arithmetic, results such as Post's theorem establish a close link between the complexity of a formula and the (non)computability of the set it defines. Another effect of using second-order arithmetic is the need to restrict general mathematical theorems to forms that can be expressed within arithmetic. For example, second-order arithmetic can express the principle "Every countable vector space has a basis" but it cannot express the principle "Every vector space has a basis".
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Mathematical Games, September 1976 Scientific American Volume 235, Issue 3 The book is roughly divided into two sections: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the first section, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind cut. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo–Fraenkel axioms. The section also covers what Conway (adopting Knuth's nomenclature) termed the "surreal numbers".
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Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup. Examples of groups with property FA include SL3(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL2(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C4 and C6 along C2.
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2-adic integers, with selected corresponding characters on their Pontryagin dual group In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual. The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being second-countable and either compact or discrete.
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The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s of the minimal model can be named; in other words there is a first-order sentence φ(x) such that s is the unique element of the minimal model for which φ(s) is true. gave another construction of the minimal model as the strongly constructible sets, using a modified form of Gödel's constructible universe. Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZFC (assuming ZFC is consistent).
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The set of rotational symmetries of a polygon forms a finite cyclic group.. If there are n different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to Z/nZ. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but which are not all rotations around an axis, but instead rotoreflections. The group of all rotations of a circle S1 (the circle group, also denoted S1) is not cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C∞ is countable, while S1 is not.
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In 1976, Michael Atiyah introduced l^2 -cohomology of manifolds with a free co-compact action of a discrete countable group (e.g. the universal cover of a compact manifold together with the action of the fundamental group by deck transformations.) Atiyah defined also l^2-Betti numbers as von Neumann dimensions of the resulting l^2-cohomology groups, and computed several examples, which all turned out to be rational numbers. He therefore asked if it is possible for l^2-Betti numbers to be irrational. Since then, various researchers asked more refined questions about possible values of l^2-Betti numbers, all of which are customarily referred to as "Atiyah conjecture".
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In general, Desano words follow a CVCV structure in terms of consonants and vowels , which is similar to that of Japanese. Desano nouns generally have a masculine-feminine distinction, as demonstrated in its pronoun inventory. Furthermore, its verbs distinguish between ‘animate’ and ‘inanimate’ entities, which is of close relation to the nature. On top of that, it is evident that Desano presents a clear cut between human beings and non human beings in regards to its lexicon, as illustrated by its strict structure of verb class. Desano’s verb class also details in singular or plural, high class or low class animates, and countable and uncountables .
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The proof of the "razor" is based on the known mathematical properties of a probability distribution over a countable set. These properties are relevant because the infinite set of all programs is a denumerable set. The sum S of the probabilities of all programs must be exactly equal to one (as per the definition of probability) thus the probabilities must roughly decrease as we enumerate the infinite set of all programs, otherwise S will be strictly greater than one. To be more precise, for every \epsilon > 0, there is some length l such that the probability of all programs longer than l is at most \epsilon.
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Georg Cantor, 1870 Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874.
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A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true or false though we have not yet been able to decide which. In the multiverse view it is meaningless to ask whether the continuum hypothesis is true or false before selecting a model of set theory. Another difference is that the statement "For every transitive model of ZFC there is a larger model of ZFC in which it is countable" is true in some versions of the multiverse view of mathematics but is false in the universe view.
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In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992. A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ, there is a nontrivial elementary embedding of M into M with critical point below κ. Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice. In fact, the existence of Berkeley cardinals is inconsistent with the axiom of countable choice.
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An identifier is a name that identifies (that is, labels the identity of) either a unique object or a unique class of objects, where the "object" or class may be an idea, physical [countable] object (or class thereof), or physical [noncountable] substance (or class thereof). The abbreviation ID often refers to identity, identification (the process of identifying), or an identifier (that is, an instance of identification). An identifier may be a word, number, letter, symbol, or any combination of those. The words, numbers, letters, or symbols may follow an encoding system (wherein letters, digits, words, or symbols stand for (represent) ideas or longer names) or they may simply be arbitrary.
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Let X0 be the countable subset given by the finitely many Fn-orbits of the fixed points hi ±∞, the fixed points of the hi and all their conjugates. Since X is uncountable, there is an element of g with fixed points outside X0 and a point w outside X0 different from these fixed points. Then for some subsequence (gm) of (gn) :gm = h1n(m,1) ··· hkn(m,k), with each n(m,i) constant or strictly monotone. On the one hand, by successive use of the rules for computing limits of the form hn·wn, the limit of the right hand side applied to x is necessarily a fixed point of one of the conjugates of the hi's.
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In complex analysis, a branch of mathematics, the identity theorem for holomorphic functions states: given functions f and g holomorphic on a domain D (open and connected subset), if f = g on some S \subseteq D, where S has an accumulation point, then f = g on D. Thus a holomorphic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D (provided this contains a converging sequence). This is not true for real-differentiable functions. In comparison, holomorphy, or complex-differentiability, is a much more rigid notion. Informally, one sometimes summarizes the theorem by saying holomorphic functions are "hard" (as opposed to, say, continuous functions which are "soft").
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If each of the bonding maps f_{i}^{j} is an embedding of TVSs onto proper vector subspaces and if the system is directed by with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each is a vector subspace of and that the subspace topology induced on by is identical to the original topology on . In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces can be described by specifying that an absolutely convex subset is a neighborhood of if and only if is an absolutely convex neighborhood of in for every .
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An emission theory of light was one that regarded the propagation of light as the transport of some kind of matter. While the corpuscular theory was obviously an emission theory, the converse did not follow: in principle, one could be an emissionist without being a corpuscularist. This was convenient because, beyond the ordinary laws of reflection and refraction, emissionists never managed to make testable quantitative predictions from a theory of forces acting on corpuscles of light. But they did make quantitative predictions from the premises that rays were countable objects, which were conserved in their interactions with matter (except absorbent media), and which had particular orientations with respect to their directions of propagation.
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Thus the first-order theory of real numbers and sets of real numbers has many models, some of which are countable. The second-order theory of the real numbers has only one model, however. This follows from the classical theorem that there is only one Archimedean complete ordered field, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic. This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models.
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Whorf's study of Hopi time has been the most widely discussed and criticized example of linguistic relativity. In his analysis he argues that there is a relation between how the Hopi people conceptualize time, how they speak of temporal relations, and the grammar of the Hopi language. Whorf's most elaborate argument for the existence of linguistic relativity was based on what he saw as a fundamental difference in the understanding of time as a conceptual category among the Hopi. He argued that the Hopi language, in contrast to English and other SAE languages, does not treat the flow of time as a sequence of distinct countable instances, like "three days" or "five years", but rather as a single process.
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A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the axiom of dependent choice). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which would allow us to apply the axiom of dependent choice to construct a countable chain. As a result, any partially ordered set with exclusively finite chains must have a maximal element. More generally, strengthening the axiom of dependent choice to higher ordinals allows us to generalize the statement in the previous paragraph to higher cardinalities.
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In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori).
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In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth. More generally, a Swiss cheese may be all or part of Euclidean space Rn - or of an even more complicated manifold - with "holes" in it.
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The other two hands (usually the right) are in the abhaya (fearlessness) and varada (blessing) mudras, which means her initiated devotees (or anyone worshipping her with a true heart) will be saved as she will guide them here and in the hereafter.White (2000), p. 477. She has a garland consisting of human heads, variously enumerated at 108 (an auspicious number in Hinduism and the number of countable beads on a japa mala or rosary for repetition of mantras) or 51, which represents Varnamala or the Garland of letters of the Sanskrit alphabet, Devanagari. Hindus believe Sanskrit is a language of dynamism, and each of these letters represents a form of energy, or a form of Kali.
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In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula.
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Baumgartner's axiom A is an axiom for partially ordered sets introduced in . A partial order (P, ≤) is said to satisfy axiom A if there is a family ≤n of partial orderings on P for n = 0, 1, 2, ... such that # ≤0 is the same as ≤ #If p ≤n+1q then p ≤nq #If there is a sequence pn with pn+1 ≤n pn then there is a q with q ≤n pn for all n. #If I is a pairwise incompatible subset of P then for all p and for all natural numbers n there is a q such that q ≤n p and the number of elements of I compatible with q is countable.
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In a metrizable space (or more generally first-countable spaces, or Fréchet–Urysohn space), sequences usually suffices to characterize, or "describe," most topological properties (such as the closures of sets or continuity of functions). But there are many spaces where sequences can _not_ be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets and filters, which never fail to characterize topological properties. ;Generalizing sequence convergence Nets directly generalize the notion of a sequence since nets are, by definition, maps from an arbitrary directed set into the space , so that a sequence is just a net whose domain is .
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Technically, definitions are always "if and only if" statements; some texts — such as Kelley's General Topology — follow the strict demands of logic, and use "if and only if" or iff in definitions of new terms.For instance, from General Topology, p. 25: "A set is countable iff it is finite or countably infinite." [boldface in original] However, this logically correct usage of "if and only if" is relatively uncommon, as the majority of textbooks, research papers and articles (including English Wikipedia articles) follow the special convention to interpret "if" as "if and only if", whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover").
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"A History of Montezuma Well". 1990. The earliest of the ruins located on the property (with the exception of the irrigation canal), a "pithouse" in the traditional Hohokam style, dates to about 1050 CE. More than 50 countable "rooms" are found inside the park boundaries; it is likely that some were used for purposes other than living space, including food storage and religious ceremonies. The Sinagua people, and possibly earlier cultures, intensively farmed the land surrounding the Well using its constant outflow as a reliable source of irrigation. Beginning about 700 CE, the Well's natural drainage into the immediately adjacent Wet Beaver Creek was diverted into a man-made canal running parallel to the creek, segments of which still conduct the outflow today.
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The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.
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However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904).
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What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set. However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers. A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers (i.e.
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In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them. While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable. Here, for any given computable number x, the well ordering principle provides that there is a minimal element in S which corresponds to x, and therefore there exists a subset consisting of the minimal elements, on which the map is a bijection.
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So ω can be identified with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since, for example, \aleph_0^2 = \aleph_0 whereas \omega^2 > \omega). Also, \omega_1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and \omega_1 is the order type of that set), \omega_2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and \omega_\omega is the limit of the \omega_n for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_n).
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In axiomatic set theory, the Rasiowa–Sikorski lemma (named after Helena Rasiowa and Roman Sikorski) is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a poset (P, ≤) is called dense in P if for any p ∈ P there is e ∈ E with e ≤ p. If D is a family of dense subsets of P, then a filter F in P is called D-generic if :F ∩ E ≠ ∅ for all E ∈ D. Now we can state the Rasiowa–Sikorski lemma: :Let (P, ≤) be a poset and p ∈ P. If D is a countable family of dense subsets of P then there exists a D-generic filter F in P such that p ∈ F.
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In point set topology, a set A is closed if it contains all its boundary points. The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets.
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An impression (in the context of online advertising) is when an ad is fetched from its source, and is countable. Whether the ad is clicked is not taken into account.Yahoo Search Marketing Glossary Each time an ad is fetched, it is counted as one impression.Google AdWords Help: Impression Because of the possibility of click fraud, robotic activity is usually filtered and excluded, and a more technical definition is given for accounting purposed by the IAB, a standards and watchdog industry group: "Impression" is a measurement of responses from a Web server to a page request from the user browser, which is filtered from robotic activity and error codes, and is recorded at a point as close as possible to opportunity to see the page by the user.
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To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the compactness theorem and the upward Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic. Certain fragments of second order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic which allow non-linear ordering of quantifier dependencies, like first-order logic extended with Henkin quantifiers, Hintikka and Sandu's independence- friendly logic, and Väänänen's dependence logic.
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For the theory associated to the infinite unitary group, U, the space BU is the classifying space for stable complex vector bundles (a Grassmannian in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space, Ω2BU of BU. Here, Ω is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially BU again; more precisely, :\Omega^2BU\simeq \Z\times BU is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is :\Omega^2U\simeq U . Either of these has the immediate effect of showing why (complex) topological K-theory is a 2-fold periodic theory.
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A series of interactions between two Michael Portillo obsessed, gay Daleks, travelling in a space/time machine called "The Turdis" (a play on the TARDIS and "turd", a countable noun for a piece of faeces), during the sketches, the pair treat their relationship with an argumentative, tense approach which often is ridden with their addressing relationship problems, crude bickering, and finally becoming so turned on by their arguing, that they begin to orgasm, leading to a variation of the classic Dalek catchphrase "Exterminate", now "Exsperminate". In the pilot episode, dubbed clips from the 1979 serial Destiny of the Daleks were used. A later attempt at reviving the Gay Daleks as an animated series was blocked by the Terry Nation Estate.
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Kc computes the successors of measurable and many singular cardinals correctly. Also, it is expected that under an appropriate weakening of countable certifiability, Kc would correctly compute the successors of all weakly compact and singular strong limit cardinals correctly. If V is closed under a mouse operator (an inner model operator), then so is Kc. Kc has no sharp: There is no natural non-trivial elementary embedding of Kc into itself. (However, unlike K, Kc may be elementarily self-embeddable.) If in addition there are also no Woodin cardinals in this model (except in certain specific cases, it is not known how the core model should be defined if Kc has Woodin cardinals), we can extract the actual core model K. K is also its own core model.
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The concept of theories of arithmetic whose integers are the true mathematical integers is captured by ω-logic.J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. Let T be a theory in a countable language which includes a unary predicate symbol N intended to hold just of the natural numbers, as well as specified names 0, 1, 2, ..., one for each (standard) natural number (which may be separate constants, or constant terms such as 0, 1, 1+1, 1+1+1, ..., etc.). Note that T itself could be referring to more general objects, such as real numbers or sets; thus in a model of T the objects satisfying N(x) are those that T interprets as natural numbers, not all of which need be named by one of the specified names.
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This is the solution favored by mathematicians Allis and Koetsier. It is the juxtaposition of this argument that the vase is empty at noon, together with the more intuitive answer that the vase should have infinitely many balls, that has warranted this problem to be named the Ross–Littlewood paradox. Ross's probabilistic version of the problem extended the removal method to the case where whenever a ball is to be withdrawn that ball is uniformly randomly selected from among those present in the vase at that time. He showed in this case that the probability that any particular ball remained in the vase at noon was 0 and therefore, by using Boole's inequality and taking a countable sum over the balls, that the probability the vase would be empty at noon was 1.
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So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure E. The quotient algebra T/E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all possible terms involving m, i, e and the variables; so for example, m(i(x), m(x,m(y,e))) would be an element of the term algebra.
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After optical heterodyne became an established technique, consideration was given to the conceptual basis for operation at such low signal light levels that "only a few, or even fractions of, photons enter the receiver in a characteristic time interval". It was concluded that even when photons of different energies are absorbed at a countable rate by a detector at different (random) times, the detector can still produce a difference frequency. Hence light seems to have wave-like properties not only as it propagates through space, but also when it interacts with matter. Progress with photon counting was such that by 2008 it was proposed that, even with larger signal strengths available, it could be advantageous to employ local oscillator power low enough to allow detection of the beat signal by photon counting.
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If an infinite graph G has a normal spanning tree, so does every connected graph minor of G. It follows from this that the graphs that have normal spanning trees have a characterization by forbidden minors. One of the two classes of forbidden minors consists of bipartite graphs in which one side of the bipartition is countable, the other side is uncountable, and every vertex has infinite degree. The other class of forbidden minors consists of certain graphs derived from Aronszajn trees.. The details of this characterization depend on the choice of set-theoretic axiomatization used to formalize mathematics. In particular, in models of set theory for which Martin's axiom is true and the continuum hypothesis is false, the class of bipartite graphs in this characterization can be replaced by a single forbidden minor.
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Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property: > Given any n + m elements a_1,\ldots, a_n,b_1,\ldots, b_m \in V, there is a > vertex c in V that is adjacent to each of a_1,\ldots, a_n and is not > adjacent to any of b_1,\ldots, b_m. It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.
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When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first- order formalization of set theory. It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as Skolem in 1933 provided an explicit construction of such a nonstandard model.
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In set theory, Hamkins has investigated the indestructibility phenomenon of large cardinals, proving that small forcing necessarily ruins the indestructibility of supercompact and other large cardinals and introducing the lottery preparation as a general method of forcing indestructibility. Hamkins introduced the modal logic of forcing and proved with Benedikt Löwe that if ZFC is consistent, then the ZFC-provably valid principles of forcing are exactly those in the modal theory known as S4.2. Hamkins, Linetsky and Reitz proved that every countable model of Gödel-Bernays set theory has a class forcing extension to a pointwise definable model, in which every set and class is definable without parameters. Hamkins and Reitz introduced the ground axiom, which asserts that the set- theoretic universe is not a forcing extension of any inner model by set forcing.
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This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form } for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection. During the latter half of the 19th century, most mathematicians simply assumed that a set is infinite if and only if it is Dedekind-infinite. However, this equivalence cannot be proved with the axioms of Zermelo–Fraenkel set theory without the axiom of choice (AC) (usually denoted "ZF"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is strictly weaker than the axiom of countable choice (CC).
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That every Dedekind-infinite set is infinite can be easily proven in ZF: every finite set has by definition a bijection with some finite ordinal n, and one can prove by induction on n that this is not Dedekind-infinite. By using the axiom of countable choice (denotation: axiom CC) one can prove the converse, namely that every infinite set X is Dedekind-infinite, as follows: First, define a function over the natural numbers (that is, over the finite ordinals) , so that for every natural number n, f(n) is the set of finite subsets of X of size n (i.e. that have a bijection with the finite ordinal n). f(n) is never empty, or otherwise X would be finite (as can be proven by induction on n).
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Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every set in F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing F. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated by F. Then σ(F) consists of all the subsets of X that can be made from elements of F by a countable number of complement, union and intersection operations. If F is empty, then σ(F) = }, since an empty union and intersection produce the empty set and universal set, respectively. For a simple example, consider the set X = {1, 2, 3}.
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In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures. It gives an "if and only if" characterisation of when two such measures are equivalent, and hence it is extremely useful when trying to establish change-of-measure formulae for measures on function spaces. The result is due to the Japanese mathematician Shizuo Kakutani. Kakutani's theorem can be used, for example, to determine whether a translate of a Gaussian measure \mu is equivalent to \mu (only when the translation vector lies in the Cameron–Martin space of \mu), or whether a dilation of \mu is equivalent to \mu (only when the absolute value of the dilation factor is 1, which is part of the Feldman–Hájek theorem).
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137, where the De Bruijn–Erdős theorem is first announced (but not proven), with a hint that Kőnig's lemma can be used for countable graphs. A seven-coloring of the plane, and the four-chromatic Moser spindle drawn as a unit distance graph in the plane, providing upper and lower bounds for the Hadwiger–Nelson problem. Another application of the De Bruijn–Erdős theorem is to the Hadwiger–Nelson problem, which asks how many colors are needed to color the points of the Euclidean plane so that every two points that are a unit distance apart have different colors. This is a graph coloring problem for an infinite graph that has a vertex for every point of the plane and an edge for every two points whose Euclidean distance is exactly one.
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In the same year the French mathematician Jules Richard used a variant of Cantor's diagonal method to obtain another contradiction in naive set theory. Consider the set A of all finite agglomerations of words. The set E of all finite definitions of real numbers is a subset of A. As A is countable, so is E. Let p be the nth decimal of the nth real number defined by the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal if p is not equal either to 8 or 9, and unity if p is equal to 8 or 9. This number N is not defined by the set E because it differs from any finitely defined real number, namely from the nth number by the nth digit.
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Mahakali is most often depicted in blue/black complexion in popular Indian art. Her most common four armed iconographic image shows each hand carrying variously a sword, a trishul (trident), a severed head of a demon and a bowl or skull-cup (kapala) catching the blood of the severed head. Her eyes are described as red with intoxication and in absolute rage, Her hair is shown disheveled, small fangs sometimes protrude out of Her mouth and Her tongue is lolling. She is adorned with a garland consisting of the heads of demons she has slaughtered, variously enumerated at 108 (an auspicious number in Hinduism and the number of countable beads on a Japa Mala, similar to a rosary, for repetition of Mantras) or 50, which represents the letters of the Sanskrit alphabet, Devanagari, and wears a skirt made of demon arms.
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It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology. The analogue of the Plancherel theorem is abstractly given by identifying a measure on the unitary dual, the Plancherel measure, with respect to which the direct integral is taken. (For Pontryagin duality the Plancherel measure is some Haar measure on the dual group to G, the only issue therefore being its normalization.) For general locally compact groups, or even countable discrete groups, the von Neumann group algebra need not be of type I and the regular representation of G cannot be written in terms of irreducible representations, even though it is unitary and completely reducible. An example where this happens is the infinite symmetric group, where the von Neumann group algebra is the hyperfinite type II1 factor.
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The Scots textbooks of the divine right of kings were written in 1597–1598 by James VI of Scotland despite Scotland never having believed in the theory and where the monarch was regarded as the "first among equals" on a par with his people. His Basilikon Doron, a manual on the powers of a king, was written to edify his four-year-old son Henry Frederick that a king "acknowledgeth himself ordained for his people, having received from the god a burden of government, whereof he must be countable". He based his theories in part on his understanding of the Bible, as noted by the following quote from a speech to parliament delivered in 1610 as James I of England: James's reference to "God's lieutenants" is apparently a reference to the text in Romans 13 where Paul refers to "God's ministers".
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The 1908 United States presidential election in Washington took place on November 3, 1908. All contemporary 46 states were part of the 1908 United States presidential election. Voters chose five electors to the Electoral College, which selected the president and vice president. Washington had been established earlier in the 1900s as a one-party Republican bastion, which it would remain at a Presidential level apart from the 1910s GOP split until Franklin D. Roosevelt rose to power in 1932, and more or less continuously at state level during this era.Burnham, Walter Dean; ‘The System of 1896’, in Kleppner, Paul (editor), The Evolution of American Electoral Systems, pp. 176-179 Democratic representation in the Washington legislature would during this period at times be countable on one hand,Schattschneider, Elmer Eric; The Semisovereign People: A Realist's View of Democracy in America, pp.
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The problem of constructing practical algorithms to determine whether sentences with large numbers of propositional variables are tautologies is an area of contemporary research in the area of automated theorem proving. The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. This method for verifying tautologies is an effective procedure, which means that given unlimited computational resources it can always be used to mechanistically determine whether a sentence is a tautology. This means, in particular, the set of tautologies over a fixed finite or countable alphabet is a decidable set.
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A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such that A \bigtriangleup U is meager (where \bigtriangleup denotes the symmetric difference).. Further, A has the Baire property in the restricted sense if for every subset E of X the intersection A\cap E has the Baire property relative to E . . The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set is almost open, and any countable union or intersection of almost open sets is again almost open. Since every open set is almost open (the empty set is meager), it follows that every Borel set is almost open.
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In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logicIn the sense of Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors, Model-theoretic logics, 1985 page 43 (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.A companion to philosophical logic by Dale Jacquette 2005 page 329 Lindström's theorem is perhaps the best known result of what later became known as abstract model theory, the basic notion of which is an abstract logic; the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category-theoretical one. Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers.
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In many cases, statistical physics uses probability measures, but not all measures it uses are probability measures.A course in mathematics for students of physics, Volume 2 by Paul Bamberg, Shlomo Sternberg 1991 page 802The concept of probability in statistical physics by Yair M. Guttmann 1999 page 149 In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.An introduction to measure-theoretic probability by George G. Roussas 2004 page 47 The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g.
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In Pattern theory and computational vision in Medical imaging, jump-diffusion processes were first introduced by Grenander and Miller as a form of random sampling algorithm which mixes "focus" like motions, the diffusion processes, with "saccade" like motions, via jump processes. The approach modelled sciences of electron- micrographs as containing multiple shapes, each having some fixed dimensional representation, with the collection of micrographs filling out the sample space corresponding to the unions of multiple finite-dimensional spaces. Using techniques from Pattern theory, a posterior probability model was constructed over the countable union of sample space; this is therefore a hybrid system model, containing the discrete notions of object number along with the continuum notions of shape. The jump-diffusion process was constructed to have ergodic properties so that after initially flowing away from its initial condition it would generate samples from the posterior probability model.
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The continuum of the spatial dimension is a three dimensional axes composed of distinct ordered points. Suppose absolute succession of points along a dimension corresponds to direct contact of parts. According to a moderate formulation of connection, composition is instantiated by two objects separated by a countable number of discrete points (x), where (x) need not be one, but cannot be unbounded. Unfortunately, even the more moderate formulation is untenable. Criticizing the possibility of bounding degree, Sider (2001) takes as given these premises: (1) On a continuum of discrete points, if there are both instances of both composition and not, then the series of points instantiating composition (e.g. (1, 2, 3, 4)) is continuous with any series not (e.g. (5, 6, 7)). (2) There is no principled way determine a cutoff for composition along such continuums (no non-arbitrary way to determine between (1, 2, 3) and (1, 2, 3, 4)).
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In trying to formalize the argument for the reflection principle of the previous section in ZF set theory, it turns out to be necessary to add some conditions about the collection of properties A (for example, A might be finite). Doing this produces several closely related "reflection theorems" of ZFC all of which state that we can find a set that is almost a model of ZFC. One form of the reflection principle in ZFC says that for any finite set of axioms of ZFC we can find a countable transitive model satisfying these axioms. (In particular this proves that, unless inconsistent, ZFC is not finitely axiomatizable because if it were it would prove the existence of a model of itself, and hence prove its own consistency, contradicting Gödel's second incompleteness theorem.) This version of the reflection theorem is closely related to the Löwenheim–Skolem theorem.
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In the opening game against Alabama, quarterback Deondre Francois suffered a season ending knee injury which resulted in true freshman James Blackman being named the starter for the remainder of the season, leading to the program's worst start since 1976 although the Seminoles went on to become bowl eligible for the 36th consecutive year. Following the game against Florida, Jimbo Fisher resigned as coach; associate coach Odell Haggins was named interim head coach for the remainder of the season. On December 21, 2017, an unofficial report was published on Reddit claiming that the Seminoles were not bowl eligible due to an NCAA rule stating that for an FCS opponent to be countable towards bowl eligibility, the FCS program must have awarded 90% of the FCS scholarship limit. Delaware State, an FCS team that lost to FSU earlier in the season, did not meet the 90% threshold set by the NCAA.
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In the meantime, Weber, who kept operating from Los Angeles, had become an important player in the German movie industry. End of 2005 he and his partner Helge Sasse together with a group of German-American investors acquired a 50.1 per cent stake in the insolvent long-established distribution company Senator Entertainment AG. Thanks to a target-oriented acquisitions and production strategy Weber was able to give the company a new profile.Ed Meza, "Senator comes back from the dead", Variety, March 29, 2007 In 2007 he showed countable success with the Academy Award-nominated Pan's Labyrinth (287,905 admissions – source: German Federal Film Board/GFFB), 1408 (537,334 admissions – source: GFFB) and Quentin Tarantino’s Death Proof (572,906 admissions – source: GFFB). With the newly founded label Autobahn he tried to propagate innovative genre films Katja Hofmann, "Germans mine indie gems", Variety, July 10, 2006 – a strategy that worked for low-budget movies like Hard Candy (77,794 admissions – source: GFFB) starring Ellen Page.
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In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure. The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a -paradoxical decomposition of , the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets and a countable set such that, on the one hand, are pairwise congruent, and on the other hand, is congruent with the union of and .
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Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is uniquely defined, among countable graphs, by an extension property that guarantees the correctness of this algorithm: no matter which vertices have already been chosen to form part of the induced subgraph, and no matter what pattern of adjacencies is needed to extend the subgraph by one more vertex, there will always exist another vertex with that pattern of adjacencies that the greedy algorithm can choose. The Rado graph is highly symmetric: any isomorphism of its induced subgraphs can be extended to a symmetry of the whole graph. The first-order logic sentences that are true of the Rado graph are also true of almost all random finite graphs, and the sentences that are false for the Rado graph are also false for almost all finite graphs.
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For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of \varepsilon_\cdot ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded. A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality).
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Formally, if \kappa is a limit ordinal, then a set C\subseteq\kappa is closed in \kappa if and only if for every \alpha<\kappa, if \sup(C\cap \alpha)=\alpha e0, then \alpha\in C. Thus, if the limit of some sequence from C is less than \kappa, then the limit is also in C. If \kappa is a limit ordinal and C\subseteq\kappa then C is unbounded in \kappa if for any \alpha<\kappa, there is some \beta\in C such that \alpha<\beta. If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
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The words care, therapy, treatment, and intervention overlap in a semantic field, and thus they can be synonymous depending on context. Moving rightward through that order, the connotative level of holism decreases and the level of specificity (to concrete instances) increases. Thus, in health care contexts (where its senses are always noncount), the word care tends to imply a broad idea of everything done to protect or improve someone's health (for example, as in the terms preventive care and primary care, which connote ongoing action), although it sometimes implies a narrower idea (for example, in the simplest cases of wound care or postanesthesia care, a few particular steps are sufficient, and the patient's interaction with that provider is soon finished). In contrast, the word intervention tends to be specific and concrete, and thus the word is often countable; for example, one instance of cardiac catheterization is one intervention performed, and coronary care (noncount) can require a series of interventions (count).
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In mathematics, specifically in order theory and functional analysis, a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A (i.e. is contained in an interval [a, b] := { z ∈ X : a ≤ z and z ≤ b } for some a and b belonging to A), the supremum sup S and the infimum inf S both exist and are elements of A. An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.
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If a base vertex is chosen in each connected component of G, then each end of G contains a unique ray starting from one of the base vertices, so the ends may be placed in one-to-one correspondence with these canonical rays. Every countable graph G has a spanning forest with the same set of ends as G.More precisely, in the original formulation of this result by in which ends are defined as equivalence classes of rays, every equivalence class of rays of G contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order ℵ0 between G and its spanning tree T for which \beta_T(X)\subset \beta_G(X) for every finite set X and every corresponding pair of havens βT and βG. However, there exist uncountably infinite graphs with only one end in which every spanning tree has infinitely many ends.
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There is a relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least a reasonable fragment of Peano arithmetic). Certain computable ordinals are so large that while they can be given by a certain ordinal notation o, a given formal system might not be sufficiently powerful to show that o is, indeed, an ordinal notation: the system does not show transfinite induction for such large ordinals. For example, the usual first-order Peano axioms do not prove transfinite induction for (or beyond) ε0: while the ordinal ε0 can easily be arithmetically described (it is countable), the Peano axioms are not strong enough to show that it is indeed an ordinal; in fact, transfinite induction on ε0 proves the consistency of Peano's axioms (a theorem by Gentzen), so by Gödel's second incompleteness theorem, Peano's axioms cannot formalize that reasoning. (This is at the basis of the Kirby–Paris theorem on Goodstein sequences.) We say that ε0 measures the proof-theoretic strength of Peano's axioms.
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In response to the ongoing debate in Congress concerning the admission of Missouri as a state and its effect on the existing balance of slave and free states, Tallmadge, an opponent of slavery, sought to impose conditions on Missouri's statehood that would provide for the eventual termination of legal slavery and the emancipation of current slaves: There were two Senators from each state regardless of the population of the state. The number of seats in the House of Representatives, however, was based on the population of the state, and to further complicate matters, slave states were allowed to count three-fifths of their slave population to increase their number of representatives. The population of the North had grown more rapidly than that of the South, and the South also had a large percentage of slaves, which resulted in a lower countable populace. Thus, the proposed Tallmadge Amendment was seen as a way to further restrict the weight of the slaveholding South in Congress.
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Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces. Every subset A of a topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The function f is called open if the image of every open set in X is open in Y. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.
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If X and Y are measure spaces with measures, there are several natural ways to define a product measure on their product. The product X×Y of measure spaces (in the sense of category theory) has as its measurable sets the σ-algebra generated by the products A×B of measurable subsets of X and Y. A measure μ on X×Y is called a product measure if μ(A×B)=μ1(A)μ2(B) for measurable subsets A⊂X and B⊂Y and measures µ1 on X and µ2 on Y. In general there may be many different product measures on X×Y. Fubini's theorem and Tonelli's theorem both need technical conditions to avoid this complication; the most common way is to assume all measure spaces are σ-finite, in which case there is a unique product measure on X×Y. There is always a unique maximal product measure on X×Y, where the measure of a measurable set is the inf of the measures of sets containing it that are countable unions of products of measurable sets.
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Consider a circle within the ball, containing the point at the center of the ball. Using an argument like that used to prove the Claim, one can see that the full circle is equidecomposable with the circle minus the point at the ball's center. (Basically, a countable set of points on the circle can be rotated to give itself plus one more point.) Note that this involves the rotation about a point other than the origin, so the Banach–Tarski paradox involves isometries of Euclidean 3-space rather than just SO(3). Use is made of the fact that if A ~ B and B ~ C, then A ~ C. The decomposition of A into C can be done using number of pieces equal to the product of the numbers needed for taking A into B and for taking B into C. The proof sketched above requires 2 × 4 × 2 + 8 = 24 pieces - a factor of 2 to remove fixed points, a factor 4 from step 1, a factor 2 to recreate fixed points, and 8 for the center point of the second ball.
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The nodes are thus in a one-to-one correspondence with finite (possibly empty) sequences of positive numbers, which are countable and can be placed in order first by sum of entries, and then by lexicographic order within a given sum (only finitely many sequences sum to a given value, so all entries are reached—formally there are a finite number of compositions of a given natural number, specifically 2n−1 compositions of ), which gives a traversal. Explicitly: 0: () 1: (1) 2: (1, 1) (2) 3: (1, 1, 1) (1, 2) (2, 1) (3) 4: (1, 1, 1, 1) (1, 1, 2) (1, 2, 1) (1, 3) (2, 1, 1) (2, 2) (3, 1) (4) etc. This can be interpreted as mapping the infinite depth binary tree onto this tree and then applying breadth-first search: replace the "down" edges connecting a parent node to its second and later children with "right" edges from the first child to the second child, from the second child to the third child, etc.
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In 1970, Solovay demonstrated that the existence of a non-measurable set for the Lebesgue measure is not provable within the framework of Zermelo–Fraenkel set theory in the absence of an additional axiom (such as the axiom of choice), by showing that (assuming the consistency of an inaccessible cardinal) there is a model of ZF, called Solovay's model, in which countable choice holds, every set is Lebesgue measurable and in which the full axiom of choice fails. The axiom of choice is equivalent to a fundamental result of point-set topology, Tychonoff's theorem, and also to the conjunction of two fundamental results of functional analysis, the Banach–Alaoglu theorem and the Krein–Milman theorem. It also affects the study of infinite groups to a large extent, as well as ring and order theory (see Boolean prime ideal theorem). However, the axioms of determinacy and dependent choice together are sufficient for most geometric measure theory, potential theory, Fourier series and Fourier transforms, while making all subsets of the real line Lebesgue-measurable.
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In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then “collapse” them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system “runs out of fuel” and cannot name a certain ordinal, a much larger ordinal is brought “from above” to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e.
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Shakespeare also alluded to the petty school that children attended at age 5 to 7 to learn to read, a prerequisite for grammar school.. John Fletcher and Shakespeare Beginning in 1987, Ward Elliott, who was sympathetic to the Oxfordian theory, and Robert J. Valenza supervised a continuing stylometric study that used computer programs to compare Shakespeare's stylistic habits to the works of 37 authors who had been proposed as the true author. The study, known as the Claremont Shakespeare Clinic, was last held in the spring of 2010.. The tests determined that Shakespeare's work shows consistent, countable, profile-fitting patterns, suggesting that he was a single individual, not a committee, and that he used fewer relative clauses and more hyphens, feminine endings, and run-on lines than most of the writers with whom he was compared. The result determined that none of the other tested claimants' work could have been written by Shakespeare, nor could Shakespeare have been written by them, eliminating all of the claimants whose known works have survived—including Oxford, Bacon, and Marlowe—as the true authors of the Shakespeare canon.. Shakespeare's style evolved over time in keeping with changes in literary trends.
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The 1916 United States presidential election in Washington took place on November 2, 1920, as part of the 1916 United States presidential election in which all contemporary 48 states participated. Voters chose seven electors to represent them in the Electoral College via a popular vote pitting Democratic incumbents Woodrow Wilson Thomas R. Marshall, against Republican challengers Associate Justice Charles Evans Hughes and his running mate, former Vice- President Charles W. Fairbanks. Washington had been a one-party Republican bastion for twenty years before this election.Burnham, Walter Dean; ‘The System of 1896’, in Kleppner, Paul (editor), The Evolution of American Electoral Systems, pp. 176-179 Democratic representation in the Washington legislature would during this period at times be countable on one hand,Schattschneider, Elmer Eric; The Semisovereign People: A Realist's View of Democracy in America, pp. 76-84 and neither Alton B. Parker nor William Jennings Bryan in his third presidential run carried even one county in the state. Republican primaries had taken over as the chief mode of political competition when introduced in the late 1900s.Murray, Keith; ‘Issues and Personalities of Pacific Northwest Politics, 1889-1950’, The Pacific Northwest Quarterly, vol.
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In mathematical logic and set theory, an ordinal notation is a partial function from the set of all finite sequences of symbols from a finite alphabet to a countable set of ordinals, and a Gödel numbering is a function from the set of well-formed formulae (a well-formed formula is a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each wff with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties: # the subset of natural numbers is a recursive set # the induced well-ordering on the subset of natural numbers is a recursive relation There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen.
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The results above give a complete description of which complex manifolds are Kobayashi hyperbolic in complex dimension 1. The picture is less clear in higher dimensions. A central open problem is the Green–Griffiths–Lang conjecture: if X is a complex projective variety of general type, then there should be a closed algebraic subset Y not equal to X such that every nonconstant holomorphic map C → X maps into Y.Demailly (1997), Conjecture 3.7. Clemens and Voisin showed that for n at least 2, a very general hypersurface X in CPn+1 of degree d at least 2n+1 has the property that every closed subvariety of X is of general type.Voisin (1996). ("Very general" means that the property holds for all hypersurfaces of degree d outside a countable union of lower-dimensional algebraic subsets of the projective space of all such hypersurfaces.) As a result, the Green–Griffiths–Lang conjecture would imply that a very general hypersurface of degree at least 2n+1 is Kobayashi hyperbolic. Note that one cannot expect all smooth hypersurfaces of a given degree to be hyperbolic, for example because some hypersurfaces contain lines (isomorphic to CP1). Such examples show the need for the subset Y in the Green–Griffiths–Lang conjecture.
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