" As a first step, her team is enumerating all possible knittable stitches: "There's going to be a countably infinite number.
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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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Any subspace of a countably generated space is again countably generated.
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Every barrelled space is countably barrelled. However, there exist semi-reflexive countably barrelled spaces that are not barrelled. The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.
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Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi- barrelled and thus also σ-quasi-barrelled space. The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled. Every σ-barrelled space is a σ-quasi-barrelled space. Every DF-space is countably quasi-barrelled.
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Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-qasui-barrelled space, and a sequentially barrelled space. An H-space is a TVS whose strong dual space is countably barrelled. Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled. Every σ-barrelled space is a σ-quasi-barrelled space.
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The set of all integers, {..., -1, 0, 1, 2, ...} is a countably infinite set. The set of all even integers is also a countably infinite set, even if it is a proper subset of the integers. The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers.
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Every sequential space (in particular, every metrizable space) is countably generated. An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.
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There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled. There exist sequentially barrelled spaces that are not σ-quasi- barrelled. There exist quasi-complete locally convex TVSs that are not sequentially barrelled.
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In the case of countable languages, all prime models are at most countably infinite.
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There exist σ-barrelled spaces that are not countably barrelled. There exist normed DF-spaces that are not countably barrelled. There exists a quasi-barrelled space that is not a 𝜎-barrelled space. There exist σ-barrelled spaces that are not Mackey spaces.
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So there is no utilitarian-optimal division. The problem with the above example is that the value measure of partner 2 is finitely-additive but not countably-additive. The compactness part of the DS theorem immediately implies that: :::If all value measures are countably- additive and nonatomic, :::then a utilitarian-optimal division exists. In this special case, non-atomicity is not required: if all value measures are countably-additive, then a utilitarian-optimal partition exists.
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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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We build an uncountably categorical but not countably categorical theory whose only computably presentable model is the saturated one.
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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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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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Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable.
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They can be constructed as induced subgraphs of the Rado graph. The Rado graph, the Henson graphs and their complements, disjoint unions of countably infinite cliques and their complements, and infinite disjoint unions of isomorphic finite cliques and their complements are the only possible countably infinite homogeneous graphs. The universality property of the Rado graph can be extended to edge-colored graphs; that is, graphs in which the edges have been assigned to different color classes, but without the usual edge coloring requirement that each color class form a matching. For any finite or countably infinite number of colors χ, there exists a unique countably-infinite χ-edge-colored graph Gχ such that every partial isomorphism of a χ-edge-colored finite graph can be extended to a full isomorphism.
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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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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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It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n (2 times n), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.
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Over the rational numbers it is isomorphic as a Hopf algebra to the universal enveloping algebra of the free Lie algebra on countably many variables.
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Starting from n vector spaces, or a countably infinite collection of them, each with the same field, we can define the product space like above.
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Hilbert spaces generalize finite-dimensional vector spaces to countably-infinite dimensions. The tensor product is still defined; it is the tensor product of Hilbert spaces.
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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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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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The Baer–Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.
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Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the same graph by Wilhelm Ackermann, Paul Erdős, and Alfréd Rényi. In combinatorial set theory, the Erdős–Rado theorem extends Ramsey's theorem to infinite sets.
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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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If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by adding more edges and vertices of G) to a finite Eulerian graph. In particular, every countably infinite graph with only one end and with no odd vertices can be written as a union of disjoint cycles .
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Kowalsky's theorem, named after Hans-Joachim Kowalsky, states that any metrizable space of weight K can be represented as a topological subspace of the product of countably many K-hedgehog spaces.
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The dimensionality of F∞ is countably infinite. A standard basis consists of the vectors ei which contain a 1 in the i-th slot and zeros elsewhere. This vector space is the coproduct (or direct sum) of countably many copies of the vector space F. Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by FN \- see below.
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Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatized. Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if they are less practical for deductive work.
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The Rado graph can also be constructed as the block intersection graph of an infinite block design in which the number of points and the size of each block are countably infinite.
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In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph. The theorem was first published by , in both the form stated above and an equivalent complementary form: every infinite graph contains either a countably infinite clique or an independent set with equal cardinality to the whole graph. In their paper, they credited Paul Erdős with assistance in its proof. They applied these results to the comparability graphs of partially ordered sets to show that each partial order contains either a countably infinite antichain or a chain of cardinality equal to the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order. The same theorem can also be stated as a result in set theory, using the arrow notation of , as \kappa\rightarrow(\kappa,\alef_0)^2.
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In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.
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When formatting is difficult, this can also be written "A1 ∩ A2 ∩ A3 ∩ ...". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.
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This method forms the basis for a proof by induction, with the 0-vertex subgraph as its base case, that every finite or countably infinite graph is an induced subgraph of the Rado graph.
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By extension, any convergent infinite [must be provably infinite] series would work. Assuming that the infinite series converges to a value n, the Zeno machine would complete a countably infinite execution in n time units.
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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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Doing this yields A2 (iteration 2). This process can be repeated a countably infinite number of times to create an An for all n. Antoine's necklace A is defined as the intersection of all the iterations.
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In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets.
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For the generalized case, the Ornstein isomorphism theorem still holds if the group G is a countably infinite amenable group. D. Ornstein and B. Weiss. "Entropy and isomorphism theorems for actions of amenable groups." J. Analyse Math.
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Groups whose center, derived subgroup, and Frattini subgroup are all equal are called special groups. Infinite special groups whose derived subgroup has order p are also called extraspecial groups. The classification of countably infinite extraspecial groups is very similar to the finite case, , but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in . The nilpotent groups whose center is cyclic and derived subgroup has order p and whose conjugacy classes are at most countably infinite are classified in .
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A 0-dimensional manifold (or differentiable or analytic manifold) is nothing but a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group. A product of countably infinite copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinite copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product.
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If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn. The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. Several additional examples are given in the article on the initial topology.
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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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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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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 the problem is bounded (i.e. it is defined on a finite section of space) there are countably many normal modes (usually numbered n = 1, 2, 3, ...). If the problem is not bounded, there is a continuous spectrum of normal modes.
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In discrete mathematics, one uses the term almost all to mean cofinite (all but finitely many), cocountable (all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graphs.
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In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler Whitney.
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Vladimir Kanovei and Shelah give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it. In its most general form, transfer is a bounded elementary embedding between structures.
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For example, Lebesgue measure on the real numbers is not finite, but it is σ-finite. Indeed, consider the intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line.
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Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle. Theorem. Let F be a countably (topologically) generated profinite group. Then # F is projective if and only if any finite embedding problem for F is solvable.
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The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in . Though theoretically important, the conditions are countably infinite in number and no finite subset will suffice, as shown in . (Accessed on 11 May 2009) A different (but also countably infinite) set of necessary and sufficient conditions were given in , where it was shown that a semigroup can be embedded in a group if and only if it is cancellative and satisfies a so-called "polyhedral condition". The two embedding theorems by Malcev and Lambek were later compared in .
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However in general the Exp ring can be much larger than the group ring: for example, the group ring of the integers is the ring of Laurent polynomials in 1 variable, while the exp ring is a polynomial ring in countably many generators.
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In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.Steen, p.19; Willard, p. 126. A space is said to be σ-locally compact if it is both σ-compact and locally compact.
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Finally, consider the signature σ consisting of a single unary relation symbol P. Every σ-structure is partitioned into two subsets: Those elements for which P holds, and the rest. Let K be the class of all σ-structures for which these two subsets have the same cardinality, i.e., there is a bijection between them. This class is not elementary, because a σ-structure in which both the set of realisations of P and its complement are countably infinite satisfies precisely the same first- order sentences as a σ-structure in which one of the sets is countably infinite and the other is uncountable.
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In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that allows a countably infinite number of algorithmic steps to be performed in finite time. These machines are ruled out in most models of computation. More formally, a Zeno machine is a Turing machine that takes 2−n units of time to perform its n-th step; thus, the first step takes 0.5 units of time, the second takes 0.25, the third 0.125 and so on, so that after one unit of time, a countably infinite (i.e. ℵ0) number of steps will have been performed.
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If z is an integer, then the value of wz is independent of the choice of , and it agrees with the earlier definition of exponentiation with an integer exponent. If z is a rational number m/n in lowest terms with , then the countably infinitely many choices of yield only n different values for wz; these values are the n complex solutions s to the equation . If z is an irrational number, then the countably infinitely many choices of lead to infinitely many distinct values for wz. The computation of complex powers is facilitated by converting the base w to polar form, as described in detail below.
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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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A measure \mu is called a s-finite measure if it is the sum of at most countably many finite measures. Every σ-finite measure is s-finite, the converse is not true. For a proof and counterexample see s-finite measure#Relation to σ-finite measures.
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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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However, m is usually not countably additive, and hence does not define a measure in the usual sense. For a filter F that is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere.
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For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types, see Section #Natural numbers for a simple example. For a countably infinite set, the set of possible order types is even uncountable.
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For a strictly convex function, the Legendre transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (For a function of one variable, the tangents are well-defined at all but at most countably many points, since a convex function is differentiable at all but at most countably many points.) The equation of a line with slope and -intercept is given by . For this line to be tangent to the graph of a function at the point requires :f\left(x_0\right) = p x_0 + b and :p = f'(x_0). The function f'is strictly monotone as the derivative of a strictly convex function.
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The only finite homogeneous graphs are the cluster graphs mKn formed from the disjoint unions of isomorphic complete graphs, the Turán graphs formed as the complement graphs of mKn, the 3 × 3 rook's graph, and the 5-cycle. The only countably infinite homogeneous graphs are the disjoint unions of isomorphic complete graphs (with the size of each complete graph, the number of complete graphs, or both numbers countably infinite), their complement graphs, the Henson graphs together with their complement graphs, and the Rado graph. If a graph is 5-ultrahomogeneous, then it is ultrahomogeneous for every k. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement.
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The function type in programming languages does not correspond to the space of all set-theoretic functions. Given the countably infinite type of natural numbers as the domain and the booleans as range, then there are an uncountably infinite number (2ℵ0 = c) of set-theoretic functions between them. Clearly this space of functions is larger than the number of functions that can be defined in any programming language, as there exist only countably many programs (a program being a finite sequence of a finite number of symbols) and one of the set-theoretic functions effectively solves the halting problem. Denotational semantics concerns itself with finding more appropriate models (called domains) to model programming language concepts such as function types.
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The existence of many noncomputable sets follows from the facts that there are only countably many Turing machines, and thus only countably many computable sets, but according to the Cantor's theorem, there are uncountably many sets of natural numbers. Although the halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. Thus the halting problem is an example of a recursively enumerable set, which is a set that can be enumerated by a Turing machine (other terms for recursively enumerable include computably enumerable and semidecidable). Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.
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Two distinct kinds of supernumbers commonly appear in the literature: those with a finite number of generators, typically = 1, 2, 3 or 4, and those with a countably-infinite number of generators. These two situations are not as unrelated as they may seem at first. First, in the definition of a supermanifold, one variant uses a countably-infinite number of generators, but then employs a topology that effectively reduces the dimension to a small finite number. In the other case, one may start with a finite number of generators, but in the course of second quantization, a need for an infinite number of generators arises: one each for every possible momentum that a fermion might carry.
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If the chromatic number of a graph is uncountable, then the graph necessarily contains as a subgraph a half graph on the natural numbers. This half graph, in turn, contains every complete bipartite graph in which one side of the bipartition is finite and the other side is countably infinite.
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When the self-adjoint operator in question is compact, this version of the spectral theorem reduces to something similar to the finite- dimensional spectral theorem above, except that the operator is expressed as a finite or countably infinite linear combination of projections, that is, the measure consists only of atoms.
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In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. For more, see Cantor's first uncountability proof.
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For example, there are only countably many \Delta^0_2 sets, so one might think that these should be non-random. However, the halting probability Ω is \Delta^0_2 and 1-random; it is only after 2-randomness is reached that it is impossible for a random set to be \Delta^0_2.
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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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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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Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of finitely generated modules need not be finitely generated. As an example, consider the ring R = Z[X1, X2, ...] of all polynomials in countably many variables. R itself is a finitely generated R-module (with {1} as generating set).
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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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Numerical univariate data consist observations that are numbers. They are obtained using either interval or ratio scale of measurement. This type of univariate data can be classified even further into two subcategories: discrete and continuous. A numerical univariate data is discrete if the set of all possible values is finite or countably infinite.
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Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.
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Finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists. Leopold Kronecker The most famous proponent of finitism was Leopold Kronecker,From an 1886 lecture at the 'Berliner Naturforscher-Versammlung', according to H. M. Weber's memorial article, as quoted and translated in Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker", Jahresberichte der Deutschen Mathematiker Vereinigung, vol ii (1893), pp. 5-31. Cf. page 19.
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For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word, specifically, 1. Technically, a measure is a function that assigns a non-negative real number or to (certain) subsets of a set (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
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A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space. There exist σ-barrelled spaces that are not Mackey spaces. There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces. There exist sequentially complete Mackey spaces that are not σ-quasi- barrelled.
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Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo measure zero sets is complete, it is a Maharam algebra. solved a long-standing problem by constructing a Maharam algebra that is not a measure algebra, i.e., that does not admit any countably additive strictly positive finite measure.
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In mathematics, a Countryman line is an uncountable linear ordering whose square is the union of countably many chains. The existence of Countryman lines was first proven by Shelah. Shelah also conjectured that, assuming PFA, every Aronszajn line contains a Countryman line. This conjecture, which remained open for three decades, was proven by Justin Moore.
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More formally, the conclusion of Adian–Rabin theorem means that set of all finite presentations :\langle x_1, x_2, x_3, \dots \mid R\rangle (where x_1, x_2, x_3, \dots is a fixed countably infinite alphabet, and R is a finite set of relations in these generators and their inverses) defining groups with property P, is not a recursive set.
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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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Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive. If is a dense proper vector subspace of a reflexive Banach space then is a normed space that not semi- reflexive but its strong dual space is a reflexive Banach space. There exists a semi-reflexive countably barrelled space that is not barrelled.
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A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable.
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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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Core models are constructed by transfinite recursion from small fragments of the core model called mice. An important ingredient of the construction is the comparison lemma that allows giving a wellordering of the relevant mice. At the level of strong cardinals and above, one constructs an intermediate countably certified core model Kc, and then, if possible, extracts K from Kc.
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That is, a complete type specifies the subgraph that a particular set of vertex variables induces. A saturated model is a model that realizes all of the types that have a number of variables at most equal to the cardinality of the model. The Rado graph has induced subgraphs of all finite or countably infinite types, so it is saturated.
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It can be shown van der Aalst et al. 2003 that in the case of a complete workflow log generated by a sound SWF net, the net generating it can be reconstructed. Complete means that its \succ_W relation is maximal. It is not required that all possible traces be present (which would be countably infinite for a net with a loop).
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A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled. A locally convex quasi-barreled space that is also a 𝜎-barrelled space is a barrelled space.
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Temporal logics such as computation tree logic (CTL) can be used to specify some LT properties. All linear temporal logic (LTL) formulae are LT properties. By a counting argument, we see that any logic in which each formula is a finite string cannot represent all LT properties, as there must be countably many formulae but there are uncountably many LT properties.
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The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.
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All continuous state spaces can be described by a corresponding continuous function and are therefore infinite. Discrete state spaces can also have (countably) infinite size, such as the state space of the time-dependent "counter" system, similar to the system in queueing theory defining the number of customers in a line, which would have state space {0, 1, 2, 3, ...}.
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In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M.E. Rudin constructed oneM.E. Rudin, A normal space X for which X × I is not normal, Fundam. Math. 73 (1971) 179-186. Zbl. 0224.54019 in 1971.
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Magidor, Matthew Foreman, and Saharon Shelah formulated and proved the consistency of Martin's maximum, a provably maximal form of Martin's axiom. Magidor also gave a simple proof of the Jensen and the Dodd-Jensen covering lemmas. He proved that if 0# does not exist then every primitive recursive closed set of ordinals is the union of countably many sets in L.
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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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The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense). The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.
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One can take , where is the Cantor–Bendixson rank of , and is the finite number of points in the β-th derived set of . See Mazurkiewicz, Stefan; Sierpiński, Wacław (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Mathematicae 1: 17–27. The Banach space is then isometric to . When are two countably infinite ordinals, and assuming , the spaces and are isomorphic if and only if .
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Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet }). The real numbers are uncountable so most real numbers are not computable.
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Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical. Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable. The definitions of arithmetical and analytical reals can be stratified into the arithmetical hierarchy and analytical hierarchy.
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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 the same issue of Annals of Mathematics and immediately after Haar's paper, the Haar theorem was used to solve Hilbert's fifth problem for compact groups by John von Neumann. Unless G is a discrete group, it is impossible to define a countably additive left- invariant regular measure on all subsets of G, assuming the axiom of choice, according to the theory of non-measurable sets.
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This reduction can be attempted in a one-sorted theory by adding unary predicates that tell whether an element is a number or a set, and taking the domain to be the union of the set of real numbers and the power set of the real numbers. But notice that the domain was asserted to include all sets of real numbers. That requirement cannot be reduced to a first-order sentence, as the Löwenheim–Skolem theorem shows. That theorem implies that there is some countably infinite subset of the real numbers, whose members we will call internal numbers, and some countably infinite collection of sets of internal numbers, whose members we will call "internal sets", such that the domain consisting of internal numbers and internal sets satisfies exactly the same first-order sentences as are satisfied by the domain of real numbers and sets of real numbers.
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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 interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. Given a topological space X, a subset A of X that can be expressed as the union of countably many nowhere dense subsets of X is called meagre. The rational numbers, while dense in the real numbers, are meagre as a subset of the reals.
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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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In 1949 Schubert, H. Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57-104. Horst Schubert proved that every oriented knot in S^3 decomposes as a connect-sum of prime knots in a unique way, up to reordering, making the monoid of oriented isotopy-classes of knots in S^3 a free commutative monoid on countably-infinite many generators.
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The set of polynomials with coefficients in F is a vector space over F, denoted F[x]. Vector addition and scalar multiplication are defined in the obvious manner. If the degree of the polynomials is unrestricted then the dimension of F[x] is countably infinite. If instead one restricts to polynomials with degree less than or equal to n, then we have a vector space with dimension n + 1\.
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If f : I → R is continuous and has an approximate derivative everywhere on I except for at most countably many points, then f is, in fact, generalized absolutely continuous, so it is the (indefinite) Khinchin-integral of its approximate derivative. This result does not hold if the set of points where f is not assumed to have an approximate derivative is merely of Lebesgue measure zero, as the Cantor function shows.
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The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1.
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1980 The Kleene Symposium, North-Holland Publishing Company. that blindly follows its instructions.A "robot": "A computer is a robot that performs any task that can be described as a sequence of instructions." cf Stone 1972:3 Melzak's and Lambek's primitive modelsLambek's "abacus" is a "countably infinite number of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads, etc). The locations are distinguishable, the counters are not".
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This sequence of sequences will form the second stratum. To each disk in the second stratum another sequence of disks with analogously defined properties can be assigned. This process continuous for countably many strata. A strand is a sequence of disks, with the first disk being selected from the first stratum, say D_i, and the second being selected from the sequence that was associated with D_i, and so on.
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A meromorphic function is a complex function that is holomorphic and therefore analytic everywhere in its domain except at a finite, or countably infinite, number of points. The points at which such a function cannot be defined are called the poles of the meromorphic function. Sometimes all of these poles lie in a straight line. In that case mathematicians may say that the function is "holomorphic on the cut plane".
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Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of ZFC form a field. Each set model M of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within M (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M. Thus, if M has uncountably many real numbers, we can prove from "outside" M that not every real number of M is definable over M. This argument becomes more problematic if it is applied to class models of ZFC, such as the von Neumann universe . The argument that applies to set models cannot be directly generalized to class models in ZFC because the property "the real number x is definable over the class model N" cannot be expressed as a formula of ZFC.
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In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. They are named after Clifford Hugh Dowker. The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as mathematical objects) helped mathematicians better understand the nature and variety of topological spaces. Topological spaces are sets together with some subsets (designated as "open sets") satisfying certain properties.
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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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This extends to a (finite or countably infinite) sequence of events. However, the probability of the union of an uncountable set of events is not the sum of their probabilities. For example, if Z is a normally distributed random variable, then P(Z=x) is 0 for any x, but P(Z∈R) = 1. The event A∩B is referred to as “A and B”, and the event A∪B as “A or B”.
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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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A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience. The logarithm has a jump discontinuity of 2i when crossing the branch cut. The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut. On each sheet, the value of the log differs from its principal value by a multiple of 2i.
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Standard measure theory takes the third option. One defines a family of measurable sets, which is very rich, and almost any set explicitly defined in most branches of mathematics will be among this family. It is usually very easy to prove that a given specific subset of the geometric plane is measurable. The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property called σ-additivity.
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From the definition of an Actor, it can be seen that numerous events take place: local decisions, creating Actors, sending messages, receiving messages, and designating how to respond to the next message received. However, this article focuses on just those events that are the arrival of a message sent to an Actor. This article reports on the results published in Hewitt [2006]. :Law of Countability: There are at most countably many events.
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There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers". Non-algebraic numbers are called transcendental numbers. Specific examples of transcendental numbers include π and Euler's number e.
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In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms. Gδ spaces are also called perfect spaces.
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When a cluster graph is formed from cliques that are all the same size, the overall graph is a homogeneous graph, meaning that every isomorphism between two of its induced subgraphs can be extended to an automorphism of the whole graph. With only two exceptions, the cluster graphs and their complements are the only finite homogeneous graphs,. and infinite cluster graphs also form one of only a small number of different types of countably infinite homogeneous graphs..
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Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .
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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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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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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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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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In a simplification of the thought experiment, the monkey could have a typewriter with just two keys: 1 and 0. The infinitely long string thusly produced would correspond to the binary digits of a particular real number between 0 and 1. A countably infinite set of possible strings end in infinite repetitions, which means the corresponding real number is rational. Examples include the strings corresponding to one-third (010101...), five-sixths (11010101...) and five- eighths (1010000...).
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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 second proof is more reminiscent of R.D. Laing:R.D. Laing (1970) Your concept of your concept is not my concept of your concept—a reproduced concept is not the same as the original concept. Pask defined concepts as persisting, countably infinite, recursively packed spin processes (like many cored cable, or skins of an onion) in any medium (stars, liquids, gases, solids, machines and, of course, brains) that produce relations. Here we prove A(T) ≠ B(T).
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In 1905, the Hungarian mathematician Julius König published a paradox based on the fact that there are only countably many finite definitions. If we imagine the real numbers as a well- ordered set, those real numbers which can be finitely defined form a subset. Hence in this well-order there should be a first real number that is not finitely definable. This is paradoxical, because this real number has just been finitely defined by the last sentence.
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An apeirogon can be defined as a partition of the Euclidean line into infinitely many equal-length segments. In geometry, an apeirogon (from the Greek word ἄπειρος apeiros, "infinite, boundless" and γωνία gonia, "angle") or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.
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In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups.
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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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There are stronger versions of Frucht's theorem that show that certain restricted families of graphs still contain enough graphs to realize any symmetry group. Frucht proved that in fact countably many 3-regular graphs with the desired property exist; for instance, the Frucht graph, a 3-regular graph with 12 vertices and 18 edges, has no nontrivial symmetries, providing a realization of this type for the trivial group. Gert Sabidussi showed that any group can be realized as the symmetry groups of countably many distinct k-regular graphs, k-vertex-connected graphs, or k-chromatic graphs, for all positive integer values k (with k\ge 3 for regular graphs and k\ge 2 for k-chromatic graphs). From the facts that every graph can be reconstructed from the containment partial order of its edges and vertices, that every finite partial order is equivalent by Birkhoff's representation theorem to a finite distributive lattice, it follows that every finite group can be realized as the symmetries of a distributive lattice, and of the graph of the lattice, a median graph.
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Continuous-time quantum walks arise when one replaces the continuum spatial domain in the Schrödinger equation with a discrete set. That is, instead of having a quantum particle propagate in a continuum, one restricts the set of possible position states to the vertex set V of some graph G = (V,E) which can be either finite or countably infinite. Under particular conditions, continuous-time quantum walks can provide a model for universal quantum computation.Andrew M. Childs, "Universal Computation by Quantum Walk".
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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 decomposable measure is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite measures. There are several theorems in measure theory such as the Radon–Nikodym theorem that are not true for arbitrary measures but are true for σ-finite measures. Several such theorems remain true for the more general class of decomposable measures.
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The Rado graph arises almost surely in the Erdős–Rényi model of a random graph on countably many vertices. Specifically, one may form an infinite graph by choosing, independently and with probability 1/2 for each pair of vertices, whether to connect the two vertices by an edge. With probability 1 the resulting graph is isomorphic to the Rado graph. This construction also works if any fixed probability p not equal to 0 or 1 is used in place of 1/2.
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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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In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order : 2463205976112133171923293141475971 : = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 : ≈ 8. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients.
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Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non- isomorphic two generator groups.
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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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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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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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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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When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.
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Two of his earlier results (1963 and 1965) investigate whether or not and under what circumstances a Bayesian learning approach is consistent, i.e. when does the prior converge to the true probability distribution given sufficiently many observed data. In particular the 1965 paper with the innocent title "On the asymptotic behaviour of Bayes estimates in the discrete case II" finds the rather disappointing answer that when sampling from a countably infinite population the Bayesian procedure fails almost everywhere, i.e. one does not obtain the true distribution asymptotically.
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In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi- measure, or CSM) is a kind of prototype for a measure on an infinite- dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
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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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A computer with access to an infinite tape of data may be more powerful than a Turing machine: for instance, the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data is called a Turing oracle. Even a Turing oracle with random data is not computable (with probability 1), since there are only countably many computations but uncountably many oracles. So a computer with a random Turing oracle can compute things that a Turing machine cannot.
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In contrast, a discrete variable over a particular range of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The number of permitted values is either finite or countably infinite. Common examples are variables that must be integers, non- negative integers, positive integers, or only the integers 0 and 1. Methods of calculus do not readily lend themselves to problems involving discrete variables.
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Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large n. Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for 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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Assigning a Gödel number to each Turing machine definition produces a subset S of the natural numbers corresponding to the computable numbers and identifies a surjection from S to the computable numbers. There are only countably many Turing machines, showing that the computable numbers are subcountable. The set S of these Gödel numbers, however, is not computably enumerable (and consequently, neither are subsets of S that are defined in terms of it). This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals.
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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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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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There are two distinct senses of the word "undecidable" in contemporary use. The first of these is the sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set.
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This generalized lexicographical order is a total order if each factor set is totally ordered. Unlike the finite case, an infinite product of well-orders is not necessarily well-ordered by the lexicographical order. For instance, the set of countably infinite binary sequences (by definition, the set of functions from non-negative integers to }, also known as the Cantor space ) is not well-ordered; the subset of sequences that have precisely one (i.e. }) does not have a least element under the lexicographical order induced by , because is an infinite descending chain.
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In particular, definite integrals of algebraic functions, known as periods, can be transcendental numbers. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general. Example: For a smooth complex projective K3 surface X, the group is isomorphic to Z22, and H1,1(X) is isomorphic to C20. Their intersection can have rank anywhere between 1 and 20; this rank is called the Picard number of X. The moduli space of all projective K3 surfaces has a countably infinite set of components, each of complex dimension 19.
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Completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval , which is not complete. In topology one considers completely metrizable spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space.
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One advantage of using Δ-sets in this way is that the resulting chain complex is generally much simpler than the singular chain complex. For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated. One drawback of this method is that one must prove that the geometric realization of the Δ-set is actually homeomorphic to the topological space in question. This can become a computational challenge as the Δ-set increases in complexity.
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If there is a surjection from to that is not injective, then no surjection from to is injective. In fact no function of any kind from to is injective. This is not true for infinite sets: Consider the function on the natural numbers that sends 1 and 2 to 1, 3 and 4 to 2, 5 and 6 to 3, and so on. There is a similar principle for infinite sets: If uncountably many pigeons are stuffed into countably many pigeonholes, there will exist at least one pigeonhole having uncountably many pigeons stuffed into it.
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The plot is similar to that of the multivalued complex logarithm function except that the spacing between sheets is not constant and the connection of the principal sheet is different There are countably many branches of the function, denoted by , for integer ; being the main (or principal) branch. is defined for all complex numbers z while with is defined for all non-zero z. We have and for all . The branch point for the principal branch is at , with a branch cut that extends to along the negative real axis.
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A graph is defined to be k-ultrahomogeneous if every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. If a graph is 5-ultrahomogeneous, it is ultrahomogeneous for every k; the only finite connected graphs of this type are complete graphs, Turán graphs, 3 × 3 rook's graphs, and the 5-cycle. The infinite Rado graph is countably ultrahomogeneous. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement.
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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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It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not to think of it as a product of copies of [0,1], but instead as :[0,1] × [0,1/2] × [0,1/3] × ···; as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence :(xn) that satisfies :0 ≤ xn ≤ 1/n.
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The dimension of the coordinate space is , by the basis exhibited above. The dimension of the polynomial ring F[x] introduced above is countably infinite, a basis is given by , , , A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.The indicator functions of intervals (of which there are infinitely many) are linearly independent, for example. Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation.
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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 Frenkel–Lepowsky–Meurman construction starts with two main tools: # The construction of a lattice vertex operator algebra VL for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string compactified on a torus Rn/L. It can be described roughly as the tensor product of the group ring of L with the oscillator representation in n dimensions (which is itself isomorphic to a polynomial ring in countably infinitely many generators). For the case in question, one sets L to be the Leech lattice, which has rank 24.
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Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (). Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given.
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See , Fact 1 and its proof. This result, shown by , justifies the definite article in the common alternative name "the random graph" for the Rado graph. Repeatedly drawing a finite graph from the Erdős–Rényi model will in general lead to different graphs; however, when applied to a countably infinite graph, the model almost always produces the same infinite graph. For any graph generated randomly in this way, the complement graph can be obtained at the same time by reversing all the choices: including an edge when the first graph did not include the same edge, and vice versa.
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For example, the box dimension of a single point is 0, but the box dimension of the collection of rational numbers in the interval [0, 1] has dimension 1. The Hausdorff measure by comparison, is countably additive. An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If A and B are two sets in a Euclidean space then A + B is formed by taking all the pairs of points a,b where a is from A and b is from B and adding a+b.
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Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.
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Clearly the number of distinct subsets that can be constructed this way is as . Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum).
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Since every infinite well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well- ordered, clearly the general AC implies that every infinite set is Dedekind- infinite. However, the equivalence of the two definitions is much weaker than the full strength of AC. In particular, there exists a model of ZF in which there exists an infinite set with no countably infinite subset. Hence, in this model, there exists an infinite, Dedekind-finite set. By the above, such a set cannot be well-ordered in this model.
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For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.
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They can also be finite, countably infinite, or uncountably infinite. For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}, commonly written {H, T}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written {HH, HT, TH, TT}. If the sample space is unordered, it becomes {{head,head}, {head,tail}, {tail,tail}}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).
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In Jainism, godliness is said to be the inherent quality of every soul (or every living organism) characterizing infinite bliss, infinite power, Kevala Jnana (pure infinite knowledge), infinite perception, and perfect manifestations of (countably) infinite other attributes. There are two possible views after this point. One is to look at the soul from the perspective of the soul itself. This entails explanations of the properties of the soul, its exact structure, composition and nature, the nature of various states that arise from it and their source attributes as is done in the deep and arcane texts of Samayasāra, Niyamasara and Pravachanasara.
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The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed. The difference between dimensions 1 and 2 on the one hand, and 3 and higher on the other hand, is due to the richer structure of the group of Euclidean motions in 3 dimensions. For the group is solvable, but for it contains a free group with two generators. John von Neumann studied the properties of the group of equivalences that make a paradoxical decomposition possible, and introduced the notion of amenable groups.
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While a graduate student at the University of Chicago, Dubins surprised his teacher Leonard Jimmie Savage with a mathematical demonstration that this is not true. Dubins and Savage wrote a book that appeared in 1965 titled How to Gamble if You Must (Inequalities for Stochastic Processes) which presented a mathematical theory of gambling processes and optimal behavior in gambling situations, pointing out their relevance to traditional approaches to probability. Under the influence of the work of Bruno de Finetti, Dubins and Savage worked in the context of finitely additive rather than countably additive probability theory, thereby bypassing some technical difficulties. Dubins was the author of nearly a hundred scholarly publications.
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Only a subset of such real number strings (albeit a countably infinite subset) contains the entirety of Hamlet (assuming that the text is subjected to a numerical encoding, such as ASCII). Meanwhile, there is an uncountably infinite set of strings which do not end in such repetition; these correspond to the irrational numbers. These can be sorted into two uncountably infinite subsets: those which contain Hamlet and those which do not. However, the "largest" subset of all the real numbers are those which not only contain Hamlet, but which contain every other possible string of any length, and with equal distribution of such strings.
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Dominos can tile the plane in a countably infinite number of ways. The number of tilings of a 2×n rectangle with dominoes is F_n, the nth Fibonacci number.Concrete Mathematics by Graham, Knuth and Patashnik, Addison- Wesley, 1994, p. 320, Domino tilings figure in several celebrated problems, including the Aztec diamond problem In which large diamond-shaped regions have a number of tilings equal to a power of two, with most tilings appearing random within a central circular region and having a more regular structure outside of this "arctic circle", and the mutilated chessboard problem, in which removing two opposite corners from a chessboard makes it impossible to tile with dominoes..
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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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FN is the product of countably many copies of F. By Zorn's lemma, FN has a basis (there is no obvious basis). There are uncountably infinite elements in the basis. Since the dimensions are different, FN is not isomorphic to F∞. It is worth noting that FN is (isomorphic to) the dual space of F∞, because a linear map T from F∞ to F is determined uniquely by its values T(ei) on the basis elements of F∞, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case.
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There are several ways to construct a measure μ from a content λ on a topological space. This section gives one such method for locally compact Hausdorff spaces such that the content is defined on all compact subsets. In general the measure is not an extension of the content, as the content may fail to be countably additive, and the measure may even be identically zero even if the content is not. First restrict the content to compact sets. This gives a function λ of compact sets C with the following properties: # \lambda(C)\in\ [0, \infty] for all compact sets C # \lambda(\varnothing) = 0.
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The monoid of smooth structures on n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere, taken up to orientation- preserving diffeomorphism. The monoid operation is the connected sum. Provided n e 4, this monoid is a group and is isomorphic to the group \Theta_n of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on Gluck twists.
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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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In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number. An uncountable cardinal number κ is said to be Rowbottom if for every function f: [κ]<ω -> λ (where λ < κ) there is a set H of order type κ that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsets of H has countably many elements. Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.
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There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).
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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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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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In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. This holds more generally for any algebra R over an uncountable algebraically closed field k and for any simple module M that is at most countably-dimensional: the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A simple module over k-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k.
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They can be used to define the tree-depth of a graph, and as part of the left-right planarity test for testing whether a graph is a planar graph. A characterization of Trémaux trees in the monadic second-order logic of graphs allows graph properties involving orientations to be recognized efficiently for graphs of bounded treewidth using Courcelle's theorem. Not every infinite connected graph has a Trémaux tree, and the graphs that do have them can be characterized by their forbidden minors. A Trémaux tree exists in every connected graph with countably many vertices, even when an infinite form of depth-first search would not succeed in exploring every vertex of the graph.
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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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In particular, it satisfies a sort of least-upper-bound axiom that says, in effect: :Every nonempty internal set that has an internal upper bound has a least internal upper bound. Countability of the set of all internal numbers (in conjunction with the fact that those form a densely ordered set) implies that that set does not satisfy the full least-upper-bound axiom. Countability of the set of all internal sets implies that it is not the set of all subsets of the set of all internal numbers (since Cantor's theorem implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related to Skolem's paradox.
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As there are no squares in the hyperbolic plane, their role needs to be taken by regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles). There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).
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Shelah, S., Whitehead groups may not be free even assuming CH I, Israel Journal of Mathematics (28) 1972Shelah, S., Whitehead groups may not be free even assuming CH II, Israel Journal of Mathematics (350 1980 Consider the ring A = R[x,y,z] of polynomials in three variables over the real numbers and its field of fractions M = R(x,y,z). The projective dimension of M as A-module is either 2 or 3, but it is independent of ZFC whether it is equal to 2; it is equal to 2 if and only if CH holds. A direct product of countably many fields has global dimension 2 if and only if the continuum hypothesis holds.
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From the essential perspective, the soul of every living organism is perfect in every way, is independent of any actions of the organism, and is considered God or to have godliness. But the epithet of God is given to the soul in whom its properties manifest in accordance with its inherent nature. There are countably infinite souls in the universe. According to Ratnakaranda śrāvakācāra (a major Jain text): :आप्तेनो च्छिनदोषेण सर्वज्ञेनागमेशिना। :भवितव्यं नियोगेन नान्यथा ह्याप्तता भवेत्।।५। :In the nature of things the true God should be free from the faults and weaknesses of the lower nature; [he should be] the knower of all things and the revealer of dharma; in no other way can divinity be constituted.
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However, the language S(x) may not even be a recursive language, since there are uncountably many such x, but only countably many recursive languages. A function f on ordered pairs (x,y) is a selector for a set S if f(x,y) is equal to either x or y and if f(x,y) is in S whenever at least one of x, y is in S. A set is semi-recursive if it has a recursive selector, and is P-selective or semi-feasible if it is semi-recursive with a polynomial time selector. Semi- feasible sets have small circuits; they are in the extended low hierarchy; and cannot be NP-complete unless P=NP.
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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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In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.
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Idempotent relations have been used as an example to illustrate the application of Mechanized Formalisation of mathematics using the interactive theorem prover Isabelle/HOL. Besides checking the mathematical properties of finite idempotent relations, an algorithm for counting the number of idempotent relations has been derived in Isabelle/HOL. Idempotent relations defined on weakly countably compact spaces have also been shown to satisfy "condition Γ": that is, every nontrivial idempotent relation on such a space contains points \langle x,x\rangle, \langle x,y\rangle,\langle y,y\rangle for some x,y. This is used to show that certain subspaces of an uncountable product of spaces, known as Mahavier products, cannot be metrizable when defined by a nontrivial idempotent relation.
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A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its complement satisfies some (context-dependent) notion of smallness. For example, a property which holds on a dense Gδ (intersection of countably many open sets) is said to hold generically. In algebraic geometry, one says that a property of points on an algebraic variety that holds on a dense Zariski open set is true generically; however, it is usually not said that a property which holds merely on a dense set (which is not Zariski open) is generic in this situation. ; in general: In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, with an eye towards identifying a unifying principle.
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When a group G acts on a set S, the action may be extended naturally to the Cartesian product Sn of S, consisting of n-tuples of elements of S: the action of an element g on the n-tuple (s1, ..., sn) is given by : g(s1, ..., sn) = (g(s1), ..., g(sn)). The group G is said to be oligomorphic if the action on Sn has only finitely many orbits for every positive integer n.Oligomorphic permutation groups - Isaac Newton Institute preprint, Peter J. Cameron (This is automatic if S is finite, so the term is typically of interest when S is infinite.) The interest in oligomorphic groups is partly based on their application to model theory, for example when considering automorphisms in countably categorical theories.
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For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively.. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used.
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For a given set A, whether a subset of Aω will be determined depends to some extent on its topological structure. For the purposes of Gale-Stewart games, the set A is endowed with the discrete topology, and Aω endowed with the resulting product topology, where Aω is viewed as a countably infinite topological product of A with itself. In particular, when A is the set {0,1}, the topology defined on Aω is exactly the ordinary topology on Cantor space, and when A is the set of natural numbers, it is the ordinary topology on Baire space. The set Aω can be viewed as the set of paths through a certain tree, which leads to a second characterization of its topology.
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In standard probability theory, one begins with a set, Ω, of outcomes (or, in alternate terminology, a set of possible worlds) and a set, F, of some (not necessarily all) subsets of Ω, such that F is closed under the countably infinite versions of the operations of basic set theory: union (∪), intersection (∩), and complementation ( ′). A member of F is called an event (or, alternatively, a proposition), and F, the set of events, is the domain of the algebra. Ω is, necessarily, a member of F, namely the trivial event "Some outcome occurs." A probability function P assigns to each member of F a real number, in such a way as to satisfy the following axioms: : For any event E, P(E) ≥ 0.
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In general terms, a calculus is a formal system that consists of a set of syntactic expressions (well-formed formulas), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions. When the formal system is intended to be a logical system, the expressions are meant to be interpreted as statements, and the rules, known to be inference rules, are typically intended to be truth-preserving. In this setting, the rules, which may include axioms, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements. The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see axiom schema).
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This argument allows us to discuss specific features of the model that we may otherwise miss—for example, a bound on a specific increasing sequence cn can be expressed as realizing the type which uses countably many parameters. If the sequence is not definable, this fact about the structure cannot be described using the base language, so a weakly saturated structure may not bound the sequence, while an ω-saturated structure will. The reason we only require parameter sets that are strictly smaller than the model is trivial: without this restriction, no infinite model is saturated. Consider a model M, and the type Each finite subset of this type is realized in the (infinite) model M, so by compactness it is consistent with M, but is trivially not realized.
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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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Instead of a general statement, the following is a sketch of a methodology to differentiate between structures that can and cannot be discriminated. 1\. The core idea is that whenever one wants to see if a property P can be expressed in FO, one chooses structures A and B, where A does have P and B doesn't. If for A and B the same FO sentences hold, then P cannot be expressed in FO. In short: :A \in P, B ot\in P and A \equiv B, where A \equiv B is shorthand for A \models \alpha \Leftrightarrow B \models \alpha for all FO-sentences α, and P represents the class of structures with property P. 2\. The methodology considers countably many subsets of the language, the union of which forms the language itself.
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The book begins with five chapters that discuss the field of reverse mathematics, which has the goal of classifying mathematical theorems by the axiom schemes needed to prove them, the big five subsystems of second-order arithmetic into which many theorems of mathematics have been classified. These chapters also review some of the tools needed in this study, including computability theory, forcing, and the low basis theorem. Chapter six, "the real heart of the book", applies this method to an infinitary form of Ramsey's theorem: every edge coloring of a countably infinite complete graph or complete uniform hypergraph, using finitely many colors, contains a monochromatic infinite induced subgraph. The standard proof of this theorem uses the arithmetical comprehension axiom, falling into one of the big five subsystems, ACA0.
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Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. Ulam showed that the smallest cardinal κ that admits a non-trivial countably- additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.
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The Rado graph, as numbered by and . In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non- randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
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Generalized to directed graphs, the conjecture has simple counterexamples, as observed by . Here, the chromatic number of a directed graph is just the chromatic number of the underlying graph, but the tensor product has exactly half the number of edges (for directed edges g→g' in G and h→h' in H, the tensor product G × H has only one edge, from (g,h) to (g',h'), while the product of the underlying undirected graphs would have an edge between (g,h') and (g',h) as well). However, the Weak Hedetniemi Conjecture turns out to be equivalent in the directed and undirected settings . The problem cannot be generalized to infinite graphs: gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors.
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Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converse needn't hold in this case. Quotient spaces of locally compact Hausdorff spaces are compactly generated.
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An example of this is given by the derivative g of the (differentiable but not absolutely continuous) function f(x)=x²·sin(1/x²) (the function g is not Lebesgue-integrable around 0). The Denjoy integral corrects this lack by ensuring that the derivative of any function f that is everywhere differentiable (or even differentiable everywhere except for at most countably many points) is integrable, and its integral reconstructs f up to a constant; the Khinchin integral is even more general in that it can integrate the approximate derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition that is weaker than absolute continuity but is satisfied by any approximately differentiable function. This is the concept of generalized absolute continuity; generalized absolutely continuous functions will be exactly those functions which are indefinite Khinchin integrals.
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This means that, for every set S of cardinality \kappa, and every partition of the ordered pairs of elements of S into two subsets P_1 and P_1, there exists either a subset S_1\subset S of cardinality \kappa or a subset S_2\subset S of cardinality \alef_0, such that all pairs of elements of S_i belong to P_i. Here, P_1 can be interpreted as the edges of a graph having S as its vertex set, in which S_1 (if it exists) is a clique of cardinality \kappa, and S_2 (if it exists) is a countably infinite independent set. If S is taken to be the cardinal number \kappa itself, the theorem can be formulated in terms of ordinal numbers with the notation \kappa\rightarrow(\kappa,\omega)^2, meaning that S_2 (when it exists) has order type \omega.
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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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By contrast, a breadth-first search will never reach the grandchildren, as it seeks to exhaust the children first. A more sophisticated analysis of running time can be given via infinite ordinal numbers; for example, the breadth-first search of the depth 2 tree above will take ω·2 steps: ω for the first level, and then another ω for the second level. Thus, simple depth-first or breadth-first searches do not traverse every infinite tree, and are not efficient on very large trees. However, hybrid methods can traverse any (countably) infinite tree, essentially via a diagonal argument ("diagonal"—a combination of vertical and horizontal—corresponds to a combination of depth and breadth). Concretely, given the infinitely branching tree of infinite depth, label the root (), the children of the root (1), (2), …, the grandchildren (1, 1), (1, 2), …, (2, 1), (2, 2), …, and so on.
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This result may fail for continuous functions F that admit a derivative f(x) at almost every point x, as the example of the Cantor function shows. However, if F is absolutely continuous, it admits a derivative F′(x) at almost every point x, and moreover F′ is integrable, with equal to the integral of F′ on Conversely, if f is any integrable function, then F as given in the first formula will be absolutely continuous with F′ = f a.e. The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function F(x) admits a derivative f(x) at all but countably many points, then f(x) is Henstock–Kurzweil integrable and is equal to the integral of f on The difference here is that the integrability of f does not need to be assumed.
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The resulting extended number system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a non-Archimedean system, and the Archimedean principle can be expressed by quantification over sets. One can conservatively extend any theory including reals, including set theory, to include infinitesimals, just by adding a countably infinite list of axioms that assert that a number is smaller than 1/2, 1/3, 1/4 and so on. Similarly, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism. We can distinguish three levels at which a non-Archimedean number system could have first-order properties compatible with those of the reals: # An ordered field obeys all the usual axioms of the real number system that can be stated in first-order logic.
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The notion of a closed vector measure stimulated much research, especially by W. Graves and his students at Chapel Hill, North Carolina. In intervening years it turned out that this notion is not only a basic tool in the study of algebras of operators generated by Boolean algebras of projections but lies at the very core of the major theorems in this area, even throwing a new perspective on the classical results in this field. As successful as the theory of integration with respect to countably additive vector measures has been in various branches of mathematics, such as mathematical physics, functional analysis and operator theory, for example, it is also known that there are fundamental problems which cannot be treated in this way. Nevertheless, these problems still seem to require for their solution "some sort of integration process" that Kluvánek pursued to the end of his career.
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A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form (\lambda K + I)u=f \, (where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves much like as in finite dimensions. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz (1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limit point. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities (so that K − λI has a finite-dimensional kernel for all complex λ ≠ 0).
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In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element matches exactly one in the other set) f\colon X \to Y such that both and its inverse are monotonic (preserving orders of elements). In the special case when is totally ordered, monotonicity of implies monotonicity of its inverse. For example, the set of integers and the set of even integers have the same order type, because the mapping n\mapsto 2n is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them.
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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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The extension property can be used to build up isomorphic copies of any finite or countably infinite graph G within the Rado graph, as induced subgraphs. To do so, order the vertices of G, and add vertices in the same order to a partial copy of G within the Rado graph. At each step, the next vertex in G will be adjacent to some set U of vertices in G that are earlier in the ordering of the vertices, and non-adjacent to the remaining set V of earlier vertices in G. By the extension property, the Rado graph will also have a vertex x that is adjacent to all the vertices in the partial copy that correspond to members of U, and non-adjacent to all the vertices in the partial copy that correspond to members of V. Adding x to the partial copy of G produces a larger partial copy, with one more vertex., Proposition 6.
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For any partition of the vertices of the Rado graph into two sets A and B, or more generally for any partition into finitely many subsets, at least one of the subgraphs induced by one of the partition sets is isomorphic to the whole Rado graph. gives the following short proof: if none of the parts induces a subgraph isomorphic to the Rado graph, they all fail to have the extension property, and one can find pairs of sets Ui and Vi that cannot be extended within each subgraph. But then, the union of the sets Ui and the union of the sets Vi would form a set that could not be extended in the whole graph, contradicting the Rado graph's extension property. This property of being isomorphic to one of the induced subgraphs of any partition is held by only three countably infinite undirected graphs: the Rado graph, the complete graph, and the empty graph.
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The name we put on things is very important: it sets the norm for how we judge them. He introduces three basic dimensions of value, systemic, extrinsic and intrinsic for sets of properties—perfection is to systemic value what goodness is to extrinsic value and what uniqueness is to intrinsic value—each with their own cardinality: finite, \aleph_0 and \aleph_1. In practice, the terms "good" and "bad" apply to finite sets of properties, since this is the only case where there is a ratio between the total number of desired properties and the number of such properties possessed by some object being valued. (In the case where the number of properties is countably infinite, the extrinsic dimension of value, the exposition as well as the mere definition of a specific concept is taken into consideration.) Hartman quantifies this notion by the principle that each property of the thing is worth as much as each other property, depending on the level of abstraction.
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The easiest (though somewhat heavy) construction starts with the ring of formal power series RX_1,X_2,... over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal infinite sums of terms, each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates. One defines ΛR as its subring consisting of those power series S that satisfy #S is invariant under any permutation of the indeterminates, and #the degrees of the monomials occurring in S are bounded. Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term X1 should also contain a term Xi for every i > 1 in order to be symmetric.
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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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A mathematical coincidence often involves an integer, and the surprising feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'. Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance.
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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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Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11–12.Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma,Mathoverflow discussion it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K. The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.
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Many different versions of infinitary logic were proposed in the late 20th century. One reason that has been given for believing in the axiom of determinacy is that it can be written as follows (in a version of infinite logic): \forall G \subseteq Seq(S): \forall a \in S :\exists a' \in S :\forall b \in S :\exists b' \in S :\forall c \in S :\exists c' \in S ... : (a,a',b,b',c,c'...) \in G OR \exists a \in S :\forall a' \in S :\exists b \in S :\forall b' \in S :\exists c \in S :\forall c' \in S ... :(a,a',b,b',c,c'...) otin G Note: Seq(S) is the set of all \omega-sequences of S. The sentences here are infinitely long with a countably infinite list of quantifiers where the ellipses appear. In an infinitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantifiers that are true for finite formulas, such as \forall a:\exists b : \forall c: \exists d : R(a,b,c,d) OR \exists a: \forall b: \exists c: \forall d: \lnot R(a,b,c,d).
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