The word "wine" can be countable, but it's often uncountable, i.e.
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It's dizzying, the uncountable options and variables Philadelphia is already mulling over.
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Or one of the uncountable, imaginative depictions of rape and psychological abuse?
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In the meantime, the tabletop-to-video game titles are practically uncountable.
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It is for all practical purposes uncountable, a tease of the infinite.
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Behind the horns, an uncountable number of people are glimpsed beyond a city gate.
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Open Book Musicians have written a nearly uncountable number of books in recent years.
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Of all the uncountable lies of this repulsive regime, this might be the biggest.
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Unlike the unique painting by Warhol, Robbins produced kits that were reproduced in uncountable numbers.
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So, what happens after 500 landings, seven retreadings, and uncountable "This is your captain speaking"s?
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Since then, Mama Ning estimates that she's made around 1,000 dolls and sold an "uncountable" number.
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Then there's the nearly uncountable smaller cons he orchestrated against the people he was close to.
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After all that math, NASA can only confidently say that say there all zillions of uncountable stars.
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The controversy stems from whether or not data is to be considered a countable or uncountable noun.
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Creepy footage from a farmer in Australia shows an uncountable number of rodents swarming after being disturbed.
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"In my presence a lot of women were slaughtered [and an] uncountable number of men," she said.
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Uncountable articles rushed to report about Prince and his legacy — his music, his religion, and his sexuality.
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The forest is also home to uncountable species of plants and animals, all threatened by the blaze.
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And not without good reasons: Synthetic materials have advanced human civilization, wealth and comfort in uncountable ways.
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The point is, downtime doesn't necessarily mean going into mind-jelly mode on Netflix for uncountable wasted hours.
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All of these outcomes are theoretically preventable, but then, the scale of the interactions in these systems is uncountable.
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The bold project of memorializing the uncountable number of Black lynching victims in the US is a hefty task.
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He has since sported an almost uncountable variety of looks that have us hoping he learned irony at Harvard.
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It is considered vital in slowing global warming, and it is home to uncountable species of fauna and flora.
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With them all offering uncountable hours of addictive programming, how is a listener or viewer supposed to keep up?
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It's important to understand that the core concepts in this subculture have been folded into uncountable layers of irony.
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But in this galaxy of endless twinkling wrongnesses and uncountable sucking black holes, there is a lodestar that shines brightest.
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Damage was done to Assad, a tyrant responsible for the deaths of an increasingly uncountable number of his own civilians.
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A project called Haze, which depicted a kitchen scene cluttered with an uncountable array of objects, went viral in 2013.
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Approach roads into cities were often a hideous jam-up of unregulated and uncountable billboards almost layered upon each other.
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Similar matches, in addition to an uncountable number of other sporting events, are taking place in New Zealand and South Africa.
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It will be humanity's greatest achievement, creating new levels of wealth, new social problems, and uncountable opportunities for billions of people.
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It forks with a complex network of paradoxes of free will and determinism that make up an uncountable number of possibilities.
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But past airstrikes, while they have killed an uncountable quantity of militants, have not prevented their organizations from regrouping and refilling their ranks.
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But if "First Sculpture" is correct, the impulse for art predates that of religions by uncountable millenniums, and might even predate humanity itself.
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The ever-populist YouTube contains streams of uncountable unofficial recordings, including multiple canons of classic bootlegs, and as a primary source for new recordings.
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It is considered vital in the ongoing efforts to slow down global warming and it is also home to uncountable species of fauna and flora.
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Things are humming along and then you have a baby or adopt a child, and all of a sudden there's an uncountable amount of work.
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An uncountable tally of civilians — many times the number of those who perished in the terrorist attacks in the United States in 2001 — were killed.
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Among these uncountable little blessings, I've always found, is the fact that we have not yet invented any non-human thing that convincingly replicates actual humans.
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Researchers aren't even sure how many land animals are out there, much less the numbers for plants, fungi or the most uncountable group of all: microbes.
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It's inspired by a process called "black MIDI," wherein a nigh-uncountable number of notes on a MIDI staff are programmed to trigger a sustained sound.
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No excuses for the physician who over-prescribed opioids but in these days of uncountable metrics and patient-centered care you be damned to say no.
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Anything-goes hellzone 4chan, Slenderman, Let's Plays and an uncountable number of memes and other web ephemera can all trace their ancestry to the SA forums.
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This Lego car is deferred enjoyment embodied, and that I haven't already watched it explode into an uncountable number of tiny pieces should be a crime.
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Not only did Kris' bash have Melanie Griffith, David Foster, flappers and an uncountable number of crystals, it allowed Khloé, finally, to be her old self again.
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Candidates named John in recent political life are practically uncountable: Kerry, Edwards, Kasich and McCain, along with Jeb Bush, whose full name is actually John Ellis Bush.
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The free animal feed made his pigs fat and happy, it saved uncountable pounds of perfectly edible food from being wasted, and best yet, it was free.
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The wave means: Come hither, and I will dig a burrow for us and our eggs, and we will populate the mud flats with fiddler crabs uncountable.
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Oliver Bruce, the micromobility expert, compares it to the prehistoric Cambrian Period, in which life on Earth evolved from simple-celled organisms into uncountable, wildly diverse lifeforms.
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Yet many — uncountable hundreds of thousands — of reservists who have honorably served their nation in peace and war cannot satisfy the requirement for issuance of the form.
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The gatherings vary from just a few individuals to uncountable thousands, and the hats and the songs and the booze are important aspects of a kräftskiva, or crayfish party.
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Since then the garbage plate, also known simply as "the plate" has flourished throughout the area, with an uncountable number of restaurants offering their own takes on the plate.
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Over the course of uncountable hours of recorded interviews, Kevin led the FBI patiently and thoroughly through every detail of Ernest's murder, including the name of the uncle who did it.
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Volvox, a Boston-bred and New York-based producer whose real name is Ariana Paoletti, played a near-uncountable number of her tough-as-nails, acid-drenched closing sets around NYC.
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He wrote some 22005 books, most recently "Leading from Behind: The Obama Doctrine and the U.S. Retreat from International Affairs" (19893, with Bryan Griffin), three plays and uncountable essays and articles.
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This concentration, Stoller argues, endangers democracy because it crushes small businesses, wrests concessions from workers, and allows corporations to amass uncountable fortunes that they can then throw into the electoral process.
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Between the two of them they've made colorful deconstructions of mainstage dance music, blistered pop punk, delirious club flips, grayscale sound art, and a nigh-uncountable amount of straight up uncharacterizeable tracks.
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Thousands of flyers, uncountable hours and giving hearts Joy VanLandschoot, who owns the store with her husband Gabe, says the demand for items related to the search for Mollie Tibbetts is overwhelming.
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They spent uncountable hours on YouTube channels that espoused white nationalism and denounced, as one alt-right ranter declared, the "feminization" and "mass, uncontrolled third-world immigration" that was destroying Western civilization.
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The Refuge provides vital habitat to bighorn sheep, elk, antelope, deer, porcupines, marmots, weasels, and uncountable numbers of smaller critters whose presence adds to the charm of the place… not to mention the food chain.
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For more than 19903 years, Nobuyoshi Araki has pushed the limits of production — he has taken an uncountable number of photographs, gathered into something like 500 books — and pushed the limits, too, of free expression.
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It also fails to address the far more present threat: Conventional weapons, including those distributed by the U.S. military, kill an uncountable number of people every year, and terrorists have had little trouble acquiring them.
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Is there anything new under the Spidey-sun that could possibly justify another iteration of a hero who has already been portrayed in nearly uncountable ways on page and screen and stage for over half a century?
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Not only are there some 75,000 phrases supported in Iconary, with more being added regularly, but there's no way to train the AI on them — the way that any one of them can be represented is uncountable.
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But as the National Highway Traffic Safety Administration's investigation into a fatal self-driving car accident should remind us, the automobile's centrality to the American way of life was an expensive and political battle with nearly uncountable human casualties.
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Even when they're taken seriously as a threat, which can be difficult to do with a group that spends as much time spewing hate as discussing sex toys like the vajankle, they're notoriously anonymous, potentially ironic, and largely uncountable.
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The latest attempt to put a hard-ish number on the uncountable comes from Carsten Menke, analyst at Julius Baer, who suggests that the legacy of the market's structural surpluses between 23 and 2013 is still something like seven million tonnes.
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By eating and being eaten, insects turn plants into protein and power the growth of all the uncountable species — including freshwater fish and a majority of birds — that rely on them for food, not to mention all the creatures that eat those creatures.
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A vision of the farther shore loomed up at the horizon, and she could see the land, solid walls of interlocking buildings to the shore, and the moving lights of machines building higher, higher, and the endless marching ants that were uncountable mobs of people.
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They've released sugar-rush EDM mutations alongside big U.S. labels, broken mutations of club music, delirious happy hardcore, voluminous big beat, proper house and techno bangers, borderline rap beats, and a nigh-uncountable number of other glittery buoyant forms—all under one roof raving.
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Advertise on Hyperallergic with Nectar Ads Maybe you have walked in Central Park, or taken the Staten Island Ferry, an uncountable number of times, but have you really listened to the birds, the voices of your fellow commuters, and the waves of the harbor?
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Angel Marcloid—a Chicago-based producer and songwriter who records as Fire-Toolz, among a nigh uncountable list of other monikers—has built a chaotic discography over the last couple of years, with this open-minded approach as part of the driving energy behind her work.
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Filthy families and souls in despair pressed flat against one another in the grip to survive, uncountable arms and legs, torn-open eyes, locked in the train all night waiting for dawn, a scene so much the antithesis of her own morning she cannot enter it.
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But as it nears, the swarm of locusts comes into focus: billions upon billions of them, thick as a blizzard, uncountable as raindrops, a jaw-dropping procession of the ravenous creatures of biblical infamy, flailing and flapping in the air, blocking out the sun like a bad omen.
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The judges found that some counties were voting with punch cards, similar to those that had produced uncountable "hanging chads" in the 2000 vote in Florida, a factor that had figured in the Supreme Court's decision to halt a vote recount there, insuring George W. Bush the presidency.
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" At an Army base in Germany, preparing to address the troops outdoors, she unbuttons her coat so it will blow open and "reveal a pair of knee-high boots tightly encasing two legs whose several inches of exposed thigh had been made shapely by uncountable hours on the elliptical.
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If any one person drove the United Kingdom out of the European Union, it was Angela Merkel, and her impulsive solo decision in the summer of 2015 to throw open Germany—and then all Europe—to 1.1 million Middle Eastern and North African migrants, with uncountable millions more to come.
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In the past five years there has been a positive deluge of victims speaking out — an uncountable number that represents not just the acute trauma of an unwanted touch or a dehumanizing comment, but the invisible ripples of confidence lost, jobs quit, careers stalled, women's influence diminished, men's power entrenched.
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It showed the snappin' and crackin' on the street, all right, but also his monastic digs in Carnegie Hall, where he slept on a pallet balanced on two of an almost uncountable number of file cabinets that held his life's considerable work, and used what seemed to be a public bathroom down the hall.
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There are more than 8 million registered motorcycles in this country, and there are groups from black biker clubs in Brooklyn to outlaw One Percenter motorcycle gangs in California and an uncountable number of other subcultures centered around the activity of riding two-wheeled motorized vehicles that cut across basically every line imaginable of age, ethnicity, class, and gender.
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You'd see her body, loping like an old coyote across a dirty sunlit freeway, and you'd see her stagger and vomit and grimace while she grabs a soap dish and bashes in the head of a bodyguard and then, like some sort of furious arachnid with an uncountable number of legs, pistol whips a crooked lawyer till his eye bleeds.
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After all, if it hadn't been for that bottle-throwing tantrum between McGregor and Nate Diaz during the pre-UFC 202 press conference (complete with an uncountable number of middle fingers coming from both sides), that event may have been a dud, instead of the biggest pay-per-view event the sport has ever known, which is what it became once video footage of the melee went viral.
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He'd only ever seen a big civilian city from the inside of planes or airports, and now he was outside on the tarmac—in his undress blues and carrying a sea bag, and the uncountable Falstaffs and Singapore slings were exacting their revenge on his head and guts—in a city, Da Nang, that was home to hundreds of thousands and was taking artillery fire, smoke rising like giant ghost trees from the rooftops.
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These splinter narratives tell us that massed antifa supersoldiers are about to start a new Civil War; that the World Trade Center was an inside job; that ISIS has an uncountable number of sleeper agents inside America, while vast swathes of European cities are no-go zones ruled by sharia law; that the Democratic Party is ruled by unspeakably corrupt pedophiles who keep a dungeon in a pizza joint in D.C.; that the European Union is vampirically draining the United Kingdom of 350 million pounds of wealth every single week; etcetera.
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Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable. Corollary. Every perfect, locally compact Hausdorff space is uncountable. Proof. Let X be a perfect, compact, Hausdorff space, then the theorem immediately implies that X is uncountable.
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If X is a perfect, locally compact Hausdorff space that is not compact, then the one-point compactification of X is a perfect, compact Hausdorff space. Therefore, the one point compactification of X is uncountable. Since removing a point from an uncountable set still leaves an uncountable set, X is uncountable as well.
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He then proved that the reals are uncountable: Assume that there is a sequence containing all the reals. Applying the theorem to this sequence produces a real not in the sequence, contradicting the assumption that the sequence contains all the reals. Hence, the reals are uncountable. The reverse-order proof starts by first proving the reals are uncountable.
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A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and a theory is categorical in some uncountable cardinal greater than or equal to then it is categorical in all cardinalities greater than .
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Some quantifiers are restricted to either countable or uncountable nouns, while others can be used in conjunction with both. The four quantifiers used in the Matis language are as follows: dadenpa: “many, in large quantity” (used with both countable and uncountable nouns) kimo: “much” (used only with uncountable nouns) dabɨtsɨk: “few” (used with countable nouns) papitsɨk: “little, small amount” (used with uncountable nouns) _Example 1:_ waka -n i dadenpa river -loc. ray intens.qtd “many rays in the river” _Example 2:_ ɨnbi waka -∅ papitsɨk -∅ ak- -nu 1sg.erg.
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No uncountable cover of a Lindelöf space can be locally finite, by essentially the same argument as in the case of compact spaces. In particular, no uncountable cover of a second-countable space is locally finite.
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Nouns in Nalik are categorized as being an uncountable noun, or a countable noun. Nouns can be part of a noun phrase or can be an independent subject referenced in a verbal complex. When used as subjects, some uncountable nouns are co-referential with plural subject markers however those are the exceptions and are usually marked with singular subject markers. With uncountable nouns, numerical markers cannot be used.
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On the other hand, it is a theorem of ZFC that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as Aronszajn trees. A κ-Suslin tree is a tree of height κ which has no chains or antichains of size κ. In particular, if κ is singular (i.e. not regular) then there exists a κ-Aronszajn tree and a κ-Suslin tree.
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However, there are some uncountable sets, such as the Cantor set, that are null.
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An uncountable limit ordinal may have either cofinality ω as does ωω or an uncountable cofinality. The cofinality of 0 is 0. The cofinality of any successor ordinal is 1. The cofinality of any nonzero limit ordinal is an infinite regular cardinal.
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Any countable product of a first-countable space is first-countable, although uncountable products need not be.
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"Nominal Reference, Temporal Constitution and Quantification in Event Semantics". In R. Bartsch, J. van Benthem, P. von Emde Boas (eds.), Semantics and Contextual Expression, Dordrecht: Foris Publication. Many nouns have both countable and uncountable uses; for example, soda is countable in "give me three sodas", but uncountable in "he likes soda".
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In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
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In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure.
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Every Vitali set V is uncountable, and v-u is irrational for any u,v \in V, u eq v.
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In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
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Her leading Single "Ku Mahu" has reached to its peak position, No.1, adding to her uncountable no.1 singles in Malaysia.
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4 the upward Löwenheim-Skolem theorem. But in fact Skolem didn't even believe it, because he didn't believe in the existence of uncountable sets.
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In particular, every uncountable Polish space has the cardinality of the continuum. Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces.
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In particular, this implies that the number of stochastic languages is uncountable. A p-adic language is regular if and only if \eta is rational.
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There is a tendency to use plural forms for uncountable nouns such as ‘staffs’, ‘equipments’, ‘informations’, ‘criterias’ and ‘phenomenons’. Additionally, some articles are often dropped.
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For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does \omega_\omega or an uncountable cofinality. The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1.
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Burushaski is a double-marking language and word order is generally subject–object–verb. Nouns in Burushaski are divided into four genders: human masculine, human feminine, countable objects, and uncountable ones (similar to mass nouns). The assignment of a noun to a particular gender is largely predictable. Some words can belong both to the countable and to the uncountable class, producing differences in meaning.
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In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets.
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Assuming the existence of some uncountable cardinal numbers analogous to \alef_0, they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).
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An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case \aleph_0 is strongly inaccessible). Weakly inaccessible cardinals were introduced by , and strongly inaccessible ones by and .
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A consequence is that any uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).
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In that case, Jensen's covering lemma holds: :For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the same cardinality as x. This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves \omega_1 and collapses \omega_2 to an ordinal of cofinality \omega.
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That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.
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In particular, no theory extending ZF can prove either the completeness or compactness theorems over arbitrary (possibly uncountable) languages without also proving the ultrafilter lemma on a set of same cardinality.
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Yan Liang's men became demoralized and fell into chaos, providing an opportunity for Cao Cao to attack. The battle of Boma was thus won with uncountable enemy dead and much plundering of supplies.
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Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no.
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The \Delta-lemma states that every uncountable collection of finite sets contains an uncountable \Delta- system. The \Delta-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proof showing that it is consistent with Zermelo–Fraenkel set theory that the continuum hypothesis does not hold. It was introduced by .
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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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The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal). The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic). Many larger epsilon numbers can be defined using the Veblen function.
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The word data has generated considerable controversy on whether it is an uncountable noun used with verbs conjugated in the singular, or should be treated as the plural of the now-rarely-used datum.
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In both cases, we find that the limitation on the weakened \psi function comes not so much from the operations allowed on the countable ordinals as on the uncountable ordinals we allow ourselves to denote.
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However S2 × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.
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In mathematics, Waraszkiewicz spirals are subsets of the plane introduced by . Waraszkiewicz spirals give an example of an uncountable family of pairwise incomparable continua, meaning that there is no continuous map from one onto another.
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It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC. The concept of a measurable cardinal was introduced by Stanislaw Ulam in 1930.
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In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable and quasi-finite, and for every subfield E, every absolutely entire E-algebra (regular field extension of E) of smaller cardinality than F can be embedded in F. They were introduced by . Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.
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Sejarah Melayu, 5.4: 47: So the king of Majapahit ordered his war commander to equip vessels for attacking Singapore, a hundred jong; other than that a few melangbing and kelulus; jongkong, cecuruh, tongkang all in uncountable numbers.
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The first record of jongkong comes from the 15th century Malay Annals, being used by Majapahit empire during the first Majapahit attack on Singapura (1350)Sejarah Melayu, 5.4: 47: So the king of Majapahit ordered his war commander to equip vessels for attacking Singapore, a hundred jong; other than that a few melangbing and kelulus; jongkong, cecuruh, tongkang all in uncountable numbers. and during the fall of Singapura (1398).Sejarah Melayu, 10.4:77: then His Majesty immediately ordered to equip three hundred jong, other than that kelulus, pelang, jongkong in uncountable numbers.
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Mass nouns are uncountable, i.e. no number can be assigned to them. In English, the difference between mass nouns and count nouns is distinct, contrary to other languages where the mass vs count distinctions may be neutralized.Nemoto, Naoko.
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Due to the spontaneous development of uncontrolled sprouting of vendors, uncountable fire outbreaks the government set out a well thought through redevelopment plan of the entire Kejetia market enclave. The redevelopment was to undertaken in three major phases.
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He was hired as a sales clerk working long hours. Despite his Catholic devotion and uncountable penances, along with all the restrictions imposed by the priest he confessed with, he never stopped having visions or communicating with spirits.
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While the set of complex numbers is uncountable, the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the complex numbers; in this sense, almost all complex numbers are transcendental.
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In life sciences, "viruses" generally refers to several distinct strains or species of virus. "Virus" is used in the original way as an uncountable mass noun, e.g. "a vial of virus". Individual, physical particles are called "virions" or "virus particles".
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Dartmouth Senior Theses Presentations. In addition to the widespread term wicked, the word pisser, often phonetically spelled pissa(h), is another Northeastern New England intensifier (plus sometimes an uncountable noun) for something that is very highly regarded by the speaker.
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For complete metrizable TVSs there is a converse: If a vector space has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.
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At Princeton he was always present in his combination laboratory/office until late in the evening, available to help his students untangle problems with experiments, as he tirelessly worked on his own research amid uncountable stacks of manuscripts and books.
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The Prüfer manifold can be constructed as follows . Take an uncountable number of copies Xa of the plane, one for each real number a, and take a copy H of the upper half plane (of pairs (x, y) with y > 0). Then glue the open upper half of each plane Xa to the upper half plane H by identifying (x,y)∈Xa for y > 0 with the point in H. The resulting quotient space Q is the Prüfer manifold. The images in Q of the points (0,0) of the spaces Xa under identification form an uncountable discrete subset.
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In English, such words are almost always mass nouns. Some uncountable nouns can be alternatively used as count nouns when meaning "a type of", and the plural means "more than one type of". For example, strength is uncountable in Strength is power, but it can be used as a countable noun to mean an instance of [a kind of] strength, as in My strengths are in physics and chemistry. Some words, especially proper nouns such as the name of an individual, are nearly always in the singular form because there is only one example of what that noun means.
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Annali d'Italia, dal principio dell'era volgare fino all'anno MDCCL. Vol. XXXI. Venezia, MDCCCXXXII, p. 170 the Chronicon of Regino of Prüm writes about the uncountable masses of the people killed with arrows,Chronicon of Regino of Prüm. In Györffy György, 2002 p.
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Moreover, Thai people present khao lam to monks to make merit. Furthermore, it is gradually becoming a Thai tradition. In the past, Thailand had an uncountable number of bamboo trees. Thai people thought about the utility of using bamboo for cooking purposes.
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The number of undocumented immigrants in Norway was estimated to roughly 20 thousand in 2009,CLANDESTINO Project (2009). Undocumented Migration: Counting the Uncountable. Data and Trends Across Europe. Clandistino Project, Final Report, 23 November 2009 and to between 18 and 56 thousand in 2017.
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Let G be an \omega-sequence cofinal on \omega_2^L and generic over L. Then no set in L of L-size smaller than \omega_2^L (which is uncountable in V, since \omega_1 is preserved) can cover G, since \omega_2 is a regular cardinal.
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We say G is a (\tau, \kappa) -disjoint collection if G is the union of at most \tau subcollections G_\alpha, where for each \alpha, G_\alpha is a disjoint collection of cardinality at most \kappa It was proven by Petr Simon that X is a Boolean space with the generating set G of CO(X) being (\tau, \kappa) -disjoint if and only if X is homeomorphic to a closed subspace of \alpha \kappa ^ \tau. The Ramsey-like property for polyadic spaces as stated by Murray Bell for Boolean spaces is then as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint.
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Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable while S is uncountable. Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.
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"Gustave Kahn. "Peinture: Exposition des Indépendants". Revue indépendante de littérature et d'art 7 (1888), p. 161 More recently, Meyer Schapiro wrote of Parade's "marvelous delicacy of tone, the uncountable variations within a narrow range, the vibrancy and soft luster, which make his canvases ... a joy to contemplate.
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However any uncountable discrete space is first-countable but not second-countable. Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implications do not hold.
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The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.
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The Spanish language has nouns that express concrete objects, groups and classes of objects, qualities, feelings and other abstractions. All nouns have a conventional grammatical gender. Countable nouns inflect for number (singular and plural). However, the division between uncountable and countable nouns is more ambiguous than in English.
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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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Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem showed in 1922 that every consistent theory of first-order predicate calculus, such as set theory, has an at most countable model. However, Cantor's theorem proves that there are uncountable sets. The root of this seeming paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in which the cardinality is measured. It is possible for a set to be uncountable in one model of set theory but countable in a larger model (because the bijections that establish countability are in the larger model but not the smaller one).
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In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property which Ramsey cardinals generalize to the uncountable case. Let [κ]<ω denote the set of all finite subsets of κ. An uncountable cardinal number κ is called Ramsey if, for every function :f: [κ]<ω → {0, 1} there is a set A of cardinality κ that is homogeneous for f. That is, for every n, f is constant on the subsets of cardinality n from A. A cardinal κ is called ineffably Ramsey if A can be chosen to be stationary subset of κ.
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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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Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.
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Alternatively there is a sharper form of the conjecture that states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel, in "On Vaught's conjecture". Cabal Seminar 76—77 (Proc. Caltech-UCLA Logic Sem., 1976—77), pp.
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In the Slovak language, the nouns "koruna" and "halier" both assume two plural forms. "Koruny"CIA - The World Factbook -- Slovakia. 15 May 2007; accessed 19 May 2007. and "haliere" appears after the numbers 2, 3 and 4 and in generic (uncountable) context, with "korún" and "halierov" being used after other numbers.
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A special scientific contribution was his work Altai shamanism (1991) with a rich collection of material brought from uncountable field research materials. Potapov joined the names of N.N. Poppe (1970), V.I. Tsintsius (1972), A.N. Kononov (1976), N.A. Baskakov (1980), A.M. Scherbak (1992) who were awarded "PIAK Gold medal" for Altaic studies.
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Oxymel (, ) is a mixture of honey and vinegar, used as a medicine. Its name is often found in Renaissance (and later) pharmacopoeiae in Late Latin form as either a countable or uncountable noun. As a countable noun, it is spelled variously as (singular) oxymellus and oxymellis, and plural oxymeli and oxymelli.
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The foundation has also given money to organizations that promote free-market economics, such as the Grand Rapids-based Acton Institute; the Heritage Foundation; and the Hudson Institute. Also the DeVos family has contributed much to Northwood University in Midland, Michigan. This includes various donations of scholarships, buildings and uncountable financial donations.
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The indefinite article of English takes the two forms a and an. Semantically, they can be regarded as meaning "one", usually without emphasis. They can be used only with singular countable nouns; for the possible use of some (or any) as an equivalent with plural and uncountable nouns, see Use of some below.
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On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure 0. An example of this is given by adding the first uncountable ordinal Ω to the previous example: the support of the measure is the single point Ω, which has measure 0.
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In 1922, Thoralf Skolem proved that if conventional axioms of set theory are consistent, then they have a countable model. Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox, and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the model. Thus the model considers its set of real numbers to be uncountable, or more precisely, the first-order sentence that says the set of real numbers is uncountable is true within the model.. In 1963, Paul Cohen used countable models to prove his independence theorems..
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Because it contains continuous paths, it is an uncountable set. Experiments also showed that for the same material, cubes with a Menger sponge structure could dissipate shocks five times better than a cubes without any pores. Cubes with Menger fractal structures after shockwave loading. The color indicates the temperature rise associated with plastic deformation.
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In probability theory, the Brownian web is an uncountable collection of one- dimensional coalescing Brownian motions, starting from every point in space and time. It arises as the diffusive space-time scaling limit of a collection of coalescing random walks, with one walk starting from each point of the integer lattice Z at each time.
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Every finite group is finitely generated since . The integers under addition are an example of an infinite group which is finitely generated by both 1 and −1, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated. For example, the group of real numbers under addition, (R, +).
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The symmetric Tsirelson space S(T) is polynomially reflexive and it has the approximation property. As with T, it is reflexive and no ℓ p space can be embedded into it. Since it is symmetric, it can be defined even on an uncountable supporting set, giving an example of non-separable polynomially reflexive Banach space.
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See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant. Similarly, most subsets of the natural numbers are not computable.
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49–84, IDEM 1994: "Aspeutos morfolóxicos del neutru n'asturianu", Editorial Complutense, Madrid, págs. 9–30, IDEM 1998: "Concordancias y referencias neutras en asturiano", Atti del XX/Congresso Internaziomale di Linguistica e Filologia Romanza (Palermo 18–24 settembre 1995), Max Niemeyer, Tübingen, v.II, págs. 39–47. in adjectives modifying uncountable nouns (lleche frío, carne tienro).
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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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In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field k is algebraic over k. In contrast to a version of Schur's lemma due to Dixmier, it does not require k to be uncountable. Quillen's original short proof uses generic flatness.
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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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Frequently this means that the ground field is uncountable and that the property is true except on a countable union of proper Zariski-closed subsets (i.e., the property holds on a dense Gδ set). For instance, this notion of very generic occurs when considering rational connectedness. However, other definitions of very generic can and do occur in other contexts.
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Since compact sets in X are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom. The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.
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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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Indeed, it is said that the different expansions of the Svayam bhagavan are uncountable and they cannot be fully described in the finite scriptures of any one religious community. Many of the Hindu scriptures sometimes differ in details reflecting the concerns of a particular tradition, while some core features of the view on Krishna are shared by all.
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If \kappa is a cardinal of uncountable cofinality, S \subseteq \kappa, and S intersects every club set in \kappa, then S is called a stationary set.Jech (2003) p.91 If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory.
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Only armed prophets, like Moses, succeed in bringing lasting change. Machiavelli claims that Moses killed uncountable numbers of his own people in order to enforce his will. Machiavelli was not the first thinker to notice this pattern. Allan Gilbert wrote: "In wishing new laws and yet seeing danger in them Machiavelli was not himself an innovator,"Gilbert.
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Uncountable (weak) limit cardinals that are also regular are known as (weakly) inaccessible cardinals. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily fixed points of the aleph function, though not all fixed points are regular.
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Drake, F. R. (1974). "Set Theory: An Introduction to Large Cardinals". Studies in Logic and the Foundations of Mathematics 76, Elsevier. Silver's original work involving large cardinals was perhaps motivated by the goal of showing the inconsistency of an uncountable measurable cardinal; instead he was led to discover indiscernibles in L assuming a measurable cardinal exists.
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One can consider the complement of each interval, written as (-\infty,a_n) \cup (b_n, \infty). By De Morgan's laws, the complement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there must be something between them. This shows that the intersection of (even an uncountable number of) nested, closed, and bounded intervals is nonempty.
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With this notation, the Rado graph is just G1. investigates the automorphism groups of this more general family of graphs. It follows from the classical model theory considerations of constructing a saturated model that under the continuum hypothesis CH, there is a universal graph with continuum many vertices. Of course, under CH, the continuum is equal to \aleph_1, the first uncountable cardinal.
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It is Hausdorff if and only if . ;Finest vector topology There exists a TVS topology on that is finer than every other TVS- topology on (that is, any TVS-topology on is necessarily a subset of ). Every linear map from into another TVS is necessarily continuous. If has an uncountable Hamel basis then is not locally convex and not metrizable.
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She is interrupted by Fenris's attack. She is then transported outside of creation to Yahweh, who then sets Lilith and her against each other to argue the merits of preserving or destroying all creation. Lilith advocates destruction, seeing creation as nothing but a prison. Elaine refuses to answer, claiming it is impossible for her to answer for uncountable other souls.
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Beyond the end of the universe exists The City of the Saved, an urban sprawl the size of a galaxy. Within it every human being that ever lived, from the first australopithecine to the last posthuman, has been inexplicably resurrected. For three hundred years, the uncountable inhabitants have enjoyed their unaging and invulnerable second lives. But now, the unthinkable has happened.
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In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. showed that they exist if the continuum hypothesis is true. On the other hand, they do not exist if Martin's axiom for ℵ1 is true.
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In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A more general notion of Følner nets can be defined analogously, and is suited for the study of uncountable groups. Følner sequences are named for Erling Følner.
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In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.
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If X is a compact Hausdorff space, then it coincides with its Stone–Čech compactification. Most other Stone–Čech compactifications lack concrete descriptions and are extremely unwieldy. Exceptions include: The Stone–Čech compactification of the first uncountable ordinal \omega_1, with the order topology, is the ordinal \omega_1 + 1. The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.
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As of December 2009 the current Polaqia members (11) are Hugo Covelo, Roque Romero, Brais Rodríguez, Diego Blanco, David Rubín, Sergio Covelo, Bernal Prieto, Álvaro López, Emma Ríos, Luís Sendón, Jose Domingo. Meanwhile, the anthology "Barsowia" has had uncountable contributors including Michael Bonfiglio (USA), Bouss (France), Louis Bertrand Devaud (France), Ken Niimura (Spain), Esteban Hernández (Spain), Susa Monteiro (Portugal), Paulo Monteiro (Portugal).
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Any non-trivial measure taking only the two values 0 and \infty is clearly non σ-finite. One example in \R is: for all A \subset \R, \mu(A) = \infty if and only if A is not empty; another one is: for all A \subset \R, \mu(A) = \infty if and only if A is uncountable, 0 otherwise. Incidentally, both are translation-invariant.
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This is the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. The real line can also be given the lower limit topology.
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This makes the transcendental numbers uncountable. No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals. Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument.
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For uncountable regular cardinals \kappa (and some other cardinals) this can be strengthened to \kappa\rightarrow(\kappa,\omega+1)^2; however, it is consistent that this strengthening does not hold for the cardinality of the continuum. The Erdős–Dushnik–Miller theorem has been called the first "unbalanced" generalization of Ramsey's theorem, and Paul Erdős's first significant result in set theory.
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The word Azakhel Payan is compound word; consisting of three words, i.e. "Aza", most probably the name of its founder. However, it is suggested that khel comes from the Avestan word khuail, meaning "uncountable" or "over-populated", such as the counting of stars or counting grains of rice. The word is a cognate of the Persian word kheleh, meaning "lots" or "too much".
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The passage ends with a plea for those guided here to make this world live again. Overwhelmed, the Alphans cannot reconcile the presence of human skeletons and an ancient human language uncountable light-years from Earth. Humans could not have travelled here over 25,000 years ago. With a strange intensity, Ferro astounds everyone by asserting it was the Arkadians who travelled to Earth.
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It is a smooth horrid tree, 4 to 10 meters high. Its trunk measures 40 to 60 cm in diameter and is highly branched and rigid, presenting uncountable spines. Leaves are small (3–5 mm long), bipinnate, tending to fall very early in spring after young sprouts become spines themselves. Inflorescence consists of lonely appearing racemes 3–7 cm long.
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Kelulus were used as transport vessel or war boat. Majapahit overseas invasion used kelulus, usually in uncountable numbers. The pati of Java had many war kelulus for raiding coastal villages. During the Demak Sultanate attack on Portuguese Malacca of 1513, kelulus were used as armed troop transports for landing alongside penjajap and lancaran, as the Javanese junks were too large to approach shore.
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In the countable sense, a verse is formally a single metrical line in a poetic composition. However, verse has come to represent any division or grouping of words in a poetic composition, with groupings traditionally having been referred to as stanzas. In the uncountable (mass noun) sense verse refers to "poetry" as contrasted to prose.Wiktionary, "Verse" (accessed 8 August 2014).
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The achromatic number of a graph G is the size of the largest clique that can be formed by contracting a family of independent sets in G. Uncountable clique minors in infinite graphs may be characterized in terms of havens, which formalize the evasion strategies for certain pursuit-evasion games: if the Hadwiger number is uncountable, then it equals the largest order of a haven in the graph. Every graph with Hadwiger number k has at most n2O(k log log k) cliques (complete subgraphs). defines a class of graph parameters that he calls S-functions, which include the Hadwiger number. These functions from graphs to integers are required to be zero on graphs with no edges, to be minor-monotoneIf a function f is minor- monotone then if H is a minor of G then f(H) ≤ f(G).
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In his paper on the problem, Erdős credited an anonymous mathematician with the observation that, when each x is mapped to a finite set of values, F is necessarily finite. However, as Erdős showed, the situation for countable sets is more complicated: the answer to Wetzel's question is yes if and only if the continuum hypothesis is false.. That is, the existence of an uncountable set of functions that maps each argument x to a countable set of values is equivalent to the nonexistence of an uncountable set of real numbers whose cardinality is less than the cardinality of the set of all real numbers. One direction of this equivalence was also proven independently, but not published, by another UIUC mathematician, Robert Dan Dixon. It follows from the independence of the continuum hypothesis, proved in 1963 by Paul Cohen,.
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Then the payoff to A is 0 if , 1 if and if . Thus each player seeks to choose the larger number, but there is a penalty of for choosing too large a number. A large number of variants have been studied, where the set may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.
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Jivdaya started this campaign because it encountered uncountable fatalities of birds during the Uttarayan festival. According to Ms Gira, many people start flying kites early in the morning and this is also the time when the birds fly out in the search of food. This leads to a number of fatalities. Ms Gira believes that if people fly kites during later hours, fatalities would be reduced.
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1; Dauben 1977, p. 89 15n. Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".
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Sejarah Melayu, 10.4:77: then His Majesty immediately ordered to equip three hundred jong, other than that kelulus, pelang, jongkong in uncountable numbers. The Javanese soldiers engaged with the defenders in a battle outside the fortress, before forcing them to retreat behind the walls. The invasion force laid a siege of the city and repeatedly tried to attack the fortress. However the fortress proved to be impregnable.
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To which is subjoined a short account of the natives of Madagascar, with suggestions as to their civilizations by J. Hatchard, L.B. Seeley and T. Hamilton, London, 1820. but has become much more common in the later 20th century, sometimes just meaning culture (itself in origin an uncountable noun, made countable in the context of ethnography)."Civilization" (1974), Encyclopædia Britannica 15th ed. Vol. II, Encyclopædia Britannica, Inc.
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The History of Ming work (the 明史, or Míng Shǐ) states only that there was a rain of uncountable stones of various sizes. The large objects were as big "as a goose egg, and the small ones were the size of the fruit of an aquatic plant". The date given was the third lunar month of 1490, which translates as March 21 to April 19, 1490.
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P Moe Nin (; 5 November 1883 – 6 January 1940) was one of Burma’s most prolific and treasured writers. His writing style differed from that prevalent in Burma at the time, writing concisely and clearly. Because of this, he is often regarded as the father of Burmese short story writing and the modern Burmese novel. He translated uncountable and valuable works of general knowledge from Western languages.
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A Turing machine can be empowered to store arbitrary rational numbers in a single tape symbol by making that finite alphabet arbitrarily large (in terms of a physical machine using transistor-based memory, building its memory locations of enough transistors to store the desired number), but this does not extend to the uncountable real numbers (for example, no number of transistors can accurately represent Pi).
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Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups. Even torsion-free abelian groups are vastly more varied in their characteristics than vector spaces. Torsion-free abelian groups of rank 1 are far more amenable than those of higher rank, and a satisfactory classification exists, even though there are an uncountable number of isomorphism classes.
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The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
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Then S \subseteq \kappa is stationary in \kappa if and only if \kappa\setminus S is bounded in \kappa. In particular, if the cofinality of \kappa is \omega=\aleph_0, then any two stationary subsets of \kappa have stationary intersection. This is no longer the case if the cofinality of \kappa is uncountable. In fact, suppose \kappa is regular and S \subseteq \kappa is stationary.
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Singapura was finally sacked by Majapahit in 1398, after approximately 1 month long siege by 300 jong and 200.000 soldiers.Sejarah Melayu, 10.4:77: then His Majesty immediately ordered to equip three hundred jong, other than that kelulus, pelang, jongkong in uncountable numbers. The last king, Sri Iskandar Shah, fled to the west coast of the Malay Peninsula to establish the Melaka Sultanate in 1400.
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His army was so largeat least 10,000 mounted soldiers and uncountable infantrymenthat most of it stayed behind when Andrew and his men embarked in Split two months later. The ships transported them to Acre, where they landed in October. The leaders of the crusade included John of Brienne, King of Jerusalem, Leopold of Austria, the Grand Masters of the Hospitallers, the Templars and the Teutonic Knights.
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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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A map satisfying Property 2 is sometimes called "chaotic in the sense of Li and Yorke". Property 2 is often stated succinctly as their article's title phrase "Period three implies chaos". The uncountable set of chaotic points may, however, be of measure zero (see for example the article Logistic map), in which case the map is said to have unobservable nonperiodicity or unobservable chaos.
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In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces [0,\omega_1] and [0,\omega], where \omega is the first infinite ordinal and \omega_1 the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point \infty = (\omega_1,\omega).
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Two hundred fedayeen between Таlvorik and Gelieguzan resisted until May 14 before retreating. The Turkish victory was accompanied by brutality: > Women have been stolen, their breasts cut off, their stomachs ripped, > children impaled, old men dismembered. Young girls withdrew in uncountable > set ... since May 5th, Turkish armies have wiped out one village after > another in Berdakh, Mkragom, Alikrpo, Avazakhiubr and Arnist.Correspondence > on events in Sasun.
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Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval (0, 1) is Polish. Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure.
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Shah’s compilation series has so far seen eight outings (2008, 2009, 2010, 2012, 2014, 2015, 2016, 2017). Encapsulating everything Shah stands for musically, their intrinsically White Island trance sound has sound-tracked an uncountable number of clubbers holidays. Roger typically uses the compilations as first-listen holders for future singles & collaborations with Vol.2 including ones with Signum, Judge Jules & Amanda O’Riordan and Tenishia.
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Skolem's paradox is the seeming contradiction that on the one hand, the set of real numbers is uncountable (and this is provable from ZFC, or even from a small finite subsystem ZFC' of ZFC), while on the other hand there are countable transitive models of ZFC' (this is provable in ZFC), and the set of real numbers in such a model will be a countable set. The paradox can be resolved by noting that countability is not absolute to submodels of a particular model of ZFC. It is possible that a set X is countable in a model of set theory but uncountable in a submodel containing X, because the submodel may contain no bijection between X and ω, while the definition of countability is the existence of such a bijection. The Löwenheim–Skolem theorem, when applied to ZFC, shows that this situation does occur.
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Basic invariants of a field include the characteristic and the transcendence degree of over its prime field. The latter is defined as the maximal number of elements in that are algebraically independent over the prime field. Two algebraically closed fields and are isomorphic precisely if these two data agree. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic.
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In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities.Cantor 1878, p. 242. In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.
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Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1 or \Omega.
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A summons from Main Mission calls Koenig away from Medical Centre. Waiting for him on the big screen is an image of a jumbled assortment of drifting alien spacecraft. Despite five years' time and a distance of uncountable light-years, Koenig uncannily knows it is the spaceship graveyard Cellini had described encountering behind Ultra. Further investigation reveals the main module of the Ultra Probe still docked with one of the derelicts.
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In set theory, Jensen's covering theorem states that if 0# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universe is close to the universe of all sets. The first proof appeared in . Silver later gave a fine structure free proof using his machines and finally gave an even simpler proof.
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In mathematics, a Bernstein set is a subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. A Bernstein set partitions the real line into two pieces in a peculiar way: every measurable set of positive measure meets both the Bernstein set and its complement, as does every set with the property of Baire that is not a meagre set..
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Such a poset is called algebraic. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an uncountable set. In some cases, however, the base for a poset is countable.
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An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.
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For example, when countable, báalt means 'apple' but when uncountable, it means 'apple tree' (Grune 1998). Noun morphology consists of the noun stem, a possessive prefix (mandatory for some nouns, and thus an example of inherent possession), and number and case suffixes. Distinctions in number are singular, plural, indefinite, and grouped. Cases include absolutive, ergative/oblique, genitive, and several locatives; the latter indicate both location and direction and may be compounded.
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Suppose that X is a subset of the reals, and each pair of elements of X is colored either black or white, with set of white pairs being open. The open coloring axiom states that either X has an uncountable subset such that any pair from this subset is white, or X can be partitioned into a countable number of subsets such that any pair from the same subset is black.
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The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, they cannot both be countable.
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The Durham Huskies were an ice hockey franchise based in the town of Durham, Ontario, Canada. The team is actually a series of teams that have spanned nine decades and through an uncountable series of leagues. The Huskies have existed under of couple short lived monikers before finding their name by accident in the 1950s. This team has spanned the Junior, Intermediate, and Senior levels of Ontario hockey.
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Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from within the model, but are countable to someone outside the model. The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory. In the other direction, model theory itself can be formalized within ZFC set theory.
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Valentina Harizanov is a Serbian-American mathematician and professor of mathematics at The George Washington University. Her main research contributions are in computable structure theory (roughly at the intersection of computability theory and model theory), where she introduced the notion of degree spectra of relations on computable structures and obtained the first significant results concerning uncountable, countable, and finite Turing degree spectra. Her recent interests include algorithmic learning theory and spaces of orders on groups.
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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 mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor. There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique.
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This posthumous novel presents the character Clara dos Anjos, a girl from a poor family that lives in the suburb of Rio de Janeiro. The story is about Clara's passion for Cassi Jones, an unscrupulous boy and son of a richer family. Cassi, who has made an uncountable number of women pregnant and abandoned them all, seduces Clara for his libidinous purposes. Clara, who is innocent due her parents' severe protectionism, ends up pregnant.
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Idarkopf Tower (), also known as Idarkopf Observation Tower () for long, is a wooden German lattice observation tower on the summit of Mt. Idarkopf in the state of Rhineland-Palatinate. The observation tower on the mountain of Idarkopf has a total height of . Built in 1980, the tower was constructed according to a certain kind of specialized wooden frame and was designed through triangular sketch. Staircases are uncountable because of the vast number of steps.
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"Trish Bradley crashed the 800 pound motorcycle 10 to 15 times a day, bruised herself uncountable times, and collided with her instructor.[...] 'I got so frustrated falling down again and again that sometimes I wondered if I would ever make it,' Mrs. Bradley said." Some current or former motor officers have come full circle by offering rider courses to the public based on the special skills and training methods used by police motorcyclists.
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With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hypothesis), there are still highly nontrivial ZFC theorems about cardinal exponentiation. Shelah constructed a Jónsson group, an uncountable group for which every proper subgroup is countable. He showed that Whitehead's problem is independent of ZFC. He gave the first primitive recursive upper bound to van der Waerden's numbers V(C,N).
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" Houston Press. December 6, 2006. In Houston Architectural Guide, Stephen Fox said the following about the stretch of Westheimer between Chimney Rock Road and South Gessner Road: Lomax said that the segment of Westheimer in Westchase is "virtually all chains -- a Geography of Nowhere wasteland of Boston Market, Borders, Kroger, Randalls, Taco Bell, Citgo and Sonic. Several of the six CVS outlets we passed are there, as are a few of the uncountable Starbucks.
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Examples of this are ~ducho-ro (grandmother) and ~duchʉ-ro (grandfather) (Stenzel, 2004, 129). A mass noun is a noun that has no plural form, not meaning singular but that it is an uncountable referent. For example, you cannot count water however you can weigh it to measure its mass. By adding the morpheme –ro to the root of a mass noun or verb in Wanano, it changes into a count noun (Stenzel, 2004, 139).
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While the Fundamental Category construction drastically reduces the size of the homsets of DX, it leaves its collection of objects unchanged. And yet, if X is the geometric model of some concurrent program P, this collection is uncountable. The Category of Components was introduced to find a full subcategory of the Fundamental category with as few objects as possible though it contains all the relevant information from the original.Components of the Fundamental Category.
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When there is an uncountably infinite collection of formulas, the Axiom of Choice (or at least some weak form of it) is needed. Using the full AC, one can well-order the formulas, and prove the uncountable case with the same argument as the countable one, except with transfinite induction. Other approaches can be used to prove that the completeness theorem in this case is equivalent to the Boolean prime ideal theorem, a weak form of AC.
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Sejarah Melayu, 5.4: 47: So the king of Majapahit ordered his war commander to equip vessels for attacking Singapore, a hundred jong; other than that a few melangbing and kelulus; jongkong, cecuruh, tongkang all in uncountable numbers. The fleet passed through the island of Bintan, from where the news spread to Singapura. The defenders immediately assembled 400 warships to face the invasion. Both sides clashed on the coast of Singapura in a battle that took place in three days.
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As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable.
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A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum. For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.
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The existential determinative (or determiner) some is sometimes used as a functional equivalent of a(n) with plural and uncountable nouns (also called a partitive). For example, Give me some apples, Give me some water (equivalent to the singular countable forms an apple and a glass of water). Grammatically this some is not required; it is also possible to use zero article: Give me apples, Give me water. The use of some in such cases implies some limited quantity.
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The study of butterfly migration was a passion for C. B. Williams. He made uncountable observations himself during his years in the Tropics and he had colleagues all over the world making new observations. He analysed and published the results in a long series of publications and became a world-leading authority on the subject. Through his research, he was able to shed light on many of the problems, which he had first formulated in his 1930 thesis.
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The "universal sorter" describes one way in which mass nouns are understood when they are used in the plural. Harry Bunt suggested the universal sorter in his 1981 doctoral dissertation. When an ordinarily uncountable noun such as wine appears with plural form (several wines), it can be understood as referring to various abstract kinds (for example, varieties of wine). The "universal packager" likewise describes how mass nouns are understood when they are used as countable nouns.
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It has no plural form.] :stone—as a unit of weight equal to 14 pounds (occasionally stones) Many names for Native American peoples are not inflected in the plural: :Cherokee :Cree :Comanche :Delaware :Hopi :Iroquois :Kiowa :Navajo :Ojibwa :Sioux :Zuni Exceptions include Algonquins, Apaches, Aztecs, Chippewas, Hurons, Incas, Mohawks, Oneidas, and Seminoles. English sometimes distinguishes between regular plural forms of demonyms/ethnonyms (e.g. "five Dutchmen", "several Irishmen"), and uncountable plurals used to refer to entire nationalities collectively (e.g.
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Many more, however, reluctantly attended services on Sunday with scowls or for as short a time as possible. The more identifiable of these were called 'Church Papists'; the less important, ordinary grumblers who merely talked of preferring the older ceremonies were uncountable. In the north and west, at least half the population outside the towns were Catholic to some degree. By this broad definition, Catholics would have numbered 10–15 percent of the total English population.
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Let N denote the natural numbers and R the reals. It follows from the theorem that the theory of (N, +, ×, 0, 1) (the theory of true first-order arithmetic) has uncountable models, and that the theory of (R, +, ×, 0, 1) (the theory of real closed fields) has a countable model. There are, of course, axiomatizations characterizing (N, +, ×, 0, 1) and (R, +, ×, 0, 1) up to isomorphism. The Löwenheim–Skolem theorem shows that these axiomatizations cannot be first- order.
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For example, in the theory of the real numbers, the completeness of a linear order used to characterize R as a complete ordered field, is a non- first-order property. Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable. This counterintuitive situation came to be known as Skolem's paradox; it shows that the notion of countability is not absolute.
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These forms include the different avataras of Krishna described in traditional Vaishnava texts, but they are not limited to these. Indeed, it is said that the different expansions of the Svayam bhagavan are uncountable and they cannot be fully described in the finite scriptures of any one religious community. Many of the Hindu scriptures sometimes differ in details reflecting the concerns of a particular tradition, while some core features of the view on Krishna are shared by all.
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If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be stationary. See also: Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose \kappa has countable cofinality.
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In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix. Thus, the probabilistic automaton also generalizes the concepts of a Markov chain and of a subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset. The number of stochastic languages is uncountable.
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Kechris has made contributions to the theory of Borel equivalence relations and the theory of automorphism groups of uncountable structures. His research interests cover foundations of mathematics, mathematical logic and set theory and their interactions with analysis and dynamical systems. Kechris earned his Ph.D. in 1972 under the direction of Yiannis N. Moschovakis, with a dissertation titled Projective Ordinals and Countable Analytic Sets. During his academic career he advised 23 PhD students and sponsored 20 postdoctoral researchers.
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383 – 390, 1967 Existence of a partition of the ordinal number \omega_2 into two colors with no monochromatic uncountable sequentially closed subset is independent of ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a Mahlo cardinal.Shelah, S., Proper and Improper Forcing, Springer 1992Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 – 606Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp.
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The Continuously Additive Model (CAM) assumes additivity in the time domain. The functional predictors are assumed to be smooth across the time domain since the times contained in an interval domain are an uncountable set, an unrestricted time- additive model is not feasible. This motivates to approximate sums of additive functions by integrals so that the traditional vector additive model be replaced by a smooth additive surface. CAM can handle generalized responses paired with multiple functional predictors.
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Therefore, cancer gains the evolutionary advantage over animals because of newly evolved animal traits that it can select against or for its own survival. This then places the selective pressure back upon animal species to evolve or forever succumb to cancer selective pressures. Most recently it has been theorized that all of the morphological and life history diversity seen today in animals, is the result of the uncountable deaths caused by cancer in ancestral animal lineages.
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In his 1975 paper "On the Singular Cardinals Problem", Silver proved that if a cardinal κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Prior to Silver's proof, many mathematicians believed that a forcing argument would yield that the negation of the theorem is consistent with ZFC. He introduced the notion of a master condition, which became an important tool in forcing proofs involving large cardinals.Cummings, James (2009).
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Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ < κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.
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The second largest military expedition, invasion of Singapura in 1398, Majapahit deployed 300 jong with no less than 200,000 men (more than 600 men in each jong).Sejarah Melayu, 10.4:77: then His Majesty immediately ordered to equip three hundred jong, other than that kelulus, pelang, jongkong in uncountable numbers. Among the smallest jong recorded, used by Chen Yanxiang to visit Korea, was 33-meter- long with an estimated capacity of 220 deadweight tons, with a crew of 121 people.
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An illustration of Cantor's diagonal argument for the existence of uncountable sets.This follows closely the first part of Cantor's 1891 paper. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above. The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").. English translation: Ewald 1996, pp. 840–843.
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In American English, "lichen" is pronounced the same as the verb "liken" (). In British English, both this pronunciation and one rhyming with "kitchen" ) are used.The Oxford English Dictionary cites only the "liken" pronunciation: English lichen derives from Greek leichēn ("tree moss, lichen, lichen-like eruption on skin") via Latin .. The Greek noun, which literally means "licker", derives from the verb leichein, "to lick".. Like the word moss, the word lichen is also used as an uncountable noun, as in, "Lichen grows on rocks".
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As this yields a notional cardinality of the continuum, Hartman advises that when setting out to describe a person, a continuum of properties would be most fitting and appropriate to bear in mind. This is the cardinality of intrinsic value in Hartman's system. Although they play no role in ordinary mathematics, Hartman deploys the notion of aleph number reciprocals, as a sort of infinitesimal proportion. This, he contends goes to zero in the limit as the uncountable cardinals become larger.
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They were trying to preserve the steam engine at Stretham that once drained the Waterbeach Level. Recovery from a climbing accident gave me the opportunity to research the history of this engine and the Fen area. After completing the Dip. Ed. at Cambridge followed by a brief teaching spell, a years research into fen drainage at the Imperial College, London, led to the award of its Diploma and the publication of his first book, Machines, Mills and Uncountable Costly Necessities.
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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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Where the set of component distributions is uncountable, the result is often called a compound probability distribution. The construction of such distributions has a formal similarity to that of mixture distributions, with either infinite summations or integrals replacing the finite summations used for finite mixtures. Consider a probability density function p(x;a) for a variable x, parameterized by a. That is, for each value of a in some set A, p(x;a) is a probability density function with respect to x.
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Another problem posed in the same year by Knaster, on the existence of an uncountable collection of dendroids with the property that no dendroid in the collection has a continuous surjection onto any other dendroid in the collection, was solved by and , who gave an example of such a family.. Previously announced in 2006.. A locally connected dendroid is called a dendrite. A cone over the Cantor set (called a Cantor fan) is an example of a dendroid that is not a dendrite.
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In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.. The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete..
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There have been uncountable volumes published on partnership taxation, and the subject is usually offered in advanced taxation courses in graduate school. A well recognized authority on this subject was the late Arthur Willis, whose work is still being carried on by his legal associates at Northwestern University, including Willis on Taxation (1971),Arthur B. Willis, "Willis on partnership taxation," McGraw- Hill, 1971. updated annually. A popular implementation guide is the book Understanding Partnership Accounting by Advent Software and American Express (2002).
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Kanoê is a polysynthetic language, where the more complex words are the verbs (Payne 1997). It is also primarily an agglutinative language, and many words are formed by simple roots, juxtaposition and suffixation. The gender can be expressed by suffixation or by a hyperonym, and while Kanoê does not make a distinction of number, it does make a distinction between uncountable and countable nouns, where the suffix {-te} is added . The syntax order of Kanoê follows SOV = subject + object + verb.
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Arratia set out to construct this limit, which is what we now call the Brownian web. Formally speaking, it is a collection of one-dimensional coalescing Brownian motions starting from every space-time point in \R^2. The fact that the Brownian web consists of an uncountable number of Brownian motions is what makes the construction highly non-trivial. Arratia gave a construction but was unable to prove convergence of coalescing random walks to a limiting object and characterize such a limiting object.
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When two compact Hausdorff spaces and are homeomorphic, the Banach spaces and are isometric. Conversely, when is not homeomorphic to , the (multiplicative) Banach–Mazur distance between and must be greater than or equal to , see above the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor.
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The expansion of Taranto was limited to the coast because of the resistance of the populations of inner Apulia. In 472 BC, Taranto signed an alliance with Rhegion, to counter the Iapygian tribes of the Messapians and Peucetians, and the Oscan-speaking Lucanians (see Iapygian- Tarentine Wars), but the joint armies of the Tarentines and Rhegines were defeated near Kailia, in what HerodotusHerodotus, vii 170. claims to be the greatest slaughter of Greeks in his knowledge, with 3,000 Reggians and uncountable Tarentines killed.
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That preference could also be seen in the image of the singer, her visual looks were very modern compared to the image sponsored by the communism. Her hair and her colorful clothes were qualified as vulgar by the most conservatives, but considered as fresh by the youth. She followed the trends of the '80s both musically and visually, which helped to make her popular. In the next two- year, Zhang sold more than 20 million copies with her uncountable cover albums.
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Mahākāśyapa then cleaned the monastery, and proceeded to Kukkuṭapāda, the place of burial he had selected. He gave a final teaching to the lay people, and performed supernatural accomplishments. Having settled in a cave there in the middle of three peaks, he covered himself in the robe he had received from the Buddha. The texts then state he took a vow that his body would stay there until the arriving of Maitreya Buddha, which is an uncountable number of years.
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Count nouns or countable nouns are common nouns that can take a plural, can combine with numerals or counting quantifiers (e.g., one, two, several, every, most), and can take an indefinite article such as a or an (in languages which have such articles). Examples of count nouns are chair, nose, and occasion. Mass nouns or uncountable (or non-count) nouns differ from count nouns in precisely that respect: they cannot take plurals or combine with number words or the above type of quantifiers.
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The term comics refers to the comics medium when used as an uncountable noun and thus takes the singular: "comics is a medium" rather than "comics are a medium". When comic appears as a countable noun it refers to instances of the medium, such as individual comic strips or comic books: "Tom's comics are in the basement." Panels are individual images containing a segment of action, often surrounded by a border. Prime moments in a narrative are broken down into panels via a process called encapsulation.
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The following are equivalent for any uncountable cardinal κ: # κ is weakly compact. # for every λ<κ, natural number n ≥ 2, and function f: [κ]n → λ, there is a set of cardinality κ that is homogeneous for f. # κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ. # Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
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Bendixson started out very much as a pure mathematician but later in his career he turned to also consider problems from applied mathematics. His first research work was on set theory and the foundations of mathematics, following the ideas which Georg Cantor had introduced. He contributed important results in point set topology. As a young student Bendixson made his name by proving that every uncountable closed set can be partitioned into a perfect set (the Bendixson derivative of the original set) and a countable set.
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For σ-compact spaces this is equivalent to Halmos's definition. For spaces that are not σ-compact the Baire sets under this definition are those under Halmos's definition together with their complements. However, in this case it is no longer true that a finite Baire measure is necessarily regular: for example the Baire probability measure that assigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co- countable subset is a Baire probability measure that is not regular.
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Twelve kinds of natural product such as otter, eoreumchi, mandarin duck, kestrel, and brown hawk-owl are living in Donggang River area. Also, uncountable kinds are the ones that are being preserved, and animals and plants that are indigenous to Korea are taking place. Also, so-called Donggang River windflow, that faces sky as it grows, found in this area has not been reported in the academics. In February 2000, white-tailed eagle, a rare bird of the world, at Hapsumeori, the lower reaches of Donggang River.
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In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".
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There are more extreme examples showing that second-order logic with standard semantics is more expressive than first-order logic. There is a finite second-order theory whose only model is the real numbers if the continuum hypothesis holds and which has no model if the continuum hypothesis does not hold (cf. Shapiro 2000, p. 105). This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality.
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An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0\. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable. Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.
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A poor, ignorant young boy in the outskirts of a small town, he is hopelessly limited in his possibilities, but (says Borges) his absurd projects reveal "a certain stammering greatness". Funes, we are told, is incapable of Platonic ideas, of generalities, of abstraction; his world is one of intolerably uncountable details. He finds it very difficult to sleep, since he recalls "every crevice and every moulding of the various houses which [surround] him". Borges spends the whole night talking to Funes in the dark.
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However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories. Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.
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The Greek Assembly has been accused for conducting a fraud against the state. According to the memorandum of an attorney: It acted as a criminal organisation, convincing citizens not to pay their insurance levies and taxes to the state, banks and severance funds, resulting in an uncountable damage for them. Certainly, the defendants of the organisation were ten. They were accused for various crimes, including the direction and affiliation in a criminal organisation, fraud against the state, banks and severance funds, money laundering etc.
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In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a countable open set. The axiom of choice implies the existence of sets of reals that do not have the perfect set property, such as Bernstein sets. However, in Solovay's model, which satisfies all axioms of ZF but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary. Every analytic set has the perfect set property.
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Every field of characteristic zero contains a unique subfield isomorphic to Q. Q is the field of fractions of the integers Z. The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the field of algebraic numbers. The set of all rational numbers is countable, while the set of all real numbers (as well as the set of irrational numbers) is uncountable. Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.
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Dem Deutschen Volke ("To the German People"), the dedication on the Reichstag building in Berlin. The German noun Volk () translates to people, both uncountable in the sense of people as in a crowd, and countable (plural Völker) in the sense of a people as in an ethnic group or nation (compare the English term folk). Within an English-language context, the German word is of interest primarily for its use in German philosophy, as in Volksseele ("national soul"), and in German nationalism – notably the derived adjective völkisch ("national, ethnic").
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If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X→M. Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family ZN of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f in ZN :f has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.
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Note that by Rice's theorem on index sets, most domains I are not recursive. Indeed, no effective map between all counting numbers \omega and the infinite (non-finite) indexing set I is asserted here, merely the subset relation I\subseteq\omega. Being dominated by a constructively non- countable set of numbers I, the name subcountable thus conveys that the uncountable set X is no bigger than \omega. The demonstration that X is subcountable also implies that it is classically (non-constructively) formally countable, but this does not reflect any effective countability.
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If Ω is uncountable, still, it may happen that p(ω) ≠ 0 for some ω; such ω are called atoms. They are an at most countable (maybe empty) set, whose probability is the sum of probabilities of all atoms. If this sum is equal to 1 then all other points can safely be excluded from the sample space, returning us to the discrete case. Otherwise, if the sum of probabilities of all atoms is between 0 and 1, then the probability space decomposes into a discrete (atomic) part (maybe empty) and a non-atomic part.
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Classic Almond Oriental Roller bred by Zeljko Talanga The key hallmark of the Oriental Roller is its flying style. They show a variety of different figures in the air, which are single somersaults, double somersaults, rolling (a number of uncountable somersaults), rotation with open wings, nose dives, sudden change of direction during flight and very rarely axial turns. Some breeds fly up to 1000 m high, others stay in the air for several hours. The aerobatics that these Oriental Rollers perform are comparable to those of the Galatz Roller and Birmingham Roller pigeons.
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In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".
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The village of Aintoura is known for having the first school of the Middle East. Saint-Joseph College of Aintoura was the first school of the Middle East, founded in 1653. For uncountable years, it has hosted sons of legions of notables from all over the region: Iran, Egypt, Cyprus, Syria and Turkey. Christians, Muslims, and Jews, sons of political foes, all grew side by side, became friends, shared the same values, thanks to an education that gave proper respect to ethical and moral values, while respecting each other's differences.
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The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process.
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The Maasai people become aware that a global crisis is approaching. Malevolent, unearthly creatures called shetani, which inhabit another dimension the Maasai know as the “Out Of” (because all things, such as humans, animals and plants, originally came "out of" it), are finding their way into the world. They are fomenting trouble between the superpowers, intent on causing mischief up to and including war. If not prevented, the barriers between the two dimensions will be breached and uncountable hordes of shetani will overrun the world, destroying all life.
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Divination tray There are sixteen major books in the Odu IfáSixteen major 'books in Odù Ifá literary corpus. When combined, there are a total of 256 Odu (a collection of sixteen, each of which has sixteen alternatives ⇔ 162, or 44) that are believed to reference all situations, circumstances, actions and consequences in life based on the uncountable ese (or "poetic tutorials") relative to the 256 Odu coding. These form the basis of traditional Yoruba spiritual knowledge and are the foundation of all Yoruba divination systems. Ifá proverbs, stories, and poetry are not written down.
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Certain words which were originally plural in form have come to be used almost exclusively as singulars (usually uncountable); for example billiards, measles, news, mathematics, physics, etc. Some of these words, such as news, are strongly and consistently felt as singular by fluent speakers. These words are usually marked in dictionaries with the phrase "plural in form but singular in construction" (or similar wording). Others, such as aesthetics, are less strongly or consistently felt as singular; for the latter type, the dictionary phrase "plural in form but singular or plural in construction" recognizes variable usage.
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The Löwenheim–Skolem theorem dealt a first blow to this hope, as it implies that a first-order theory which has an infinite model cannot be categorical. Later, in 1931, the hope was shattered completely by Gödel's incompleteness theorem. Many consequences of the Löwenheim–Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets.
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In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number. An uncountable cardinal number κ is said to be Jónsson if for every function f: [κ]<ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ. Every Rowbottom cardinal is Jónsson. By a theorem of Eugene M. Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.
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As noted in the previous paragraph, second- order comprehension axioms easily generalize to the higher-order framework. However, theorems expressing the compactness of basic spaces behave quite differently in second- and higher-order arithmetic: on one hand, when restricted to countable covers/the language of second-order arithmetic, the compactness of the unit interval is provable in WKL0 from the next section. On the other hand, given uncountable covers/the language of higher-order arithmetic, the compactness of the unit interval is only provable from (full) second-order arithmetic (). Other covering lemmas (e.g.
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In 1962 Everett accepted an invitation to present the relative-state formulation (as it was still called) at a conference on the foundations of quantum mechanics held at Xavier University in Cincinnati. In his exposition Everett presented his derivation of probability and also stated explicitly that observers in all branches of the wavefunction were equally "real." He also agreed with an observation from the floor that the number of branches of the universal wavefunction was an uncountable infinity. In August 1964, Everett and several WSEG colleagues started Lambda Corp.
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Continue this process whereby choosing a neighbourhood Un+1 ⊂ Un whose closure does not contain xn+1. Then the collection {Ui : i ∈ N} satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of X. Therefore, there is a point x in this intersection. No xi can belong to this intersection because xi does not belong to the closure of Ui. This means that x is not equal to xi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.
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All the conditions in the statement of the theorem are necessary: 1\. We cannot eliminate the Hausdorff condition; a countable set (with at least two points) with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable. 2\. We cannot eliminate the compactness condition, as the set of rational numbers shows. 3\. We cannot eliminate the condition that one point sets cannot be open, as any finite space with the discrete topology shows. Corollary.
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Articles and determiners agree in gender and number with the noun they determine; unlike with nouns, this inflection is made in speech as well as in writing. French has three articles: definite, indefinite, and partitive. The difference between the definite and indefinite articles is similar to that in English (definite: the; indefinite: a, an), except that the indefinite article has a plural form (similar to some, though English normally doesn't use an article before indefinite plural nouns). The partitive article is similar to the indefinite article but used for uncountable singular nouns.
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Higgins has received significant praise and support from human rights groups, journalists, and non-profit organisations. "Brown Moses is among the best out there when it comes to weapons monitoring in Syria," said Peter Bouckaert, emergencies director at Human Rights Watch. The New York Times war reporter C.J. Chivers said that fellow journalists owe a debt to Higgins' investigative reporting in Syria. "Many people, whether they admit or not, have been relying on that blog's daily labour to cull the uncountable videos that circulate from the conflict," he said.
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The term faasiqnimo is the uncountable noun form of the agent noun faasiq, which describes a corrupt or impious person, or a venial sinner. Each era has seen incremental changes in the Somali view of sexuality, from the pre-modern until well into the post- independence era wherein cross-clan relationships were encouraged to nurture tribal bond. In the contemporary era, the charge of faasiqnimo is sometimes levelled at those engaging in qabyaalad, i.e. mahrams refusing to officiate or arrange a marriage due to being from different clans or regional states.
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His speaking engagements took him to Russia, Belgium, Germany, Italy, France, and Washington D.C., where he spoke on numerous occasions, not to mention an uncountable number of engagements within his home state of Oklahoma. In 1984, he met in Frankfort, West Germany with 61 nations on providing food for the starving people in Africa. Upon returning to America, he contacted U.S. Senator David Boren and U.S. Senator Don Nickles in an attempt to gain their assistance in contacting President Ronald Reagan for his aid in sending helicopters to Africa to deliver food.
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Let V be an infinite-dimensional vector space over an uncountable field F. Then Con A isomorphic to Sub V implies that A has at least card F operations, for any algebra A. As V is infinite- dimensional, the largest element (unit) of Sub V is not compact. However innocuous it sounds, the compact unit assumption is essential in the statement of the result above, as demonstrated by the following result. Theorem (Lampe 1982). Every algebraic lattice with compact unit is isomorphic to the congruence lattice of some groupoid.
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It is 1915 days after leaving Earth orbit, and Moonbase Alpha is in the midst of a celebration. A rescue party from Earth travelling in a Superswift, an interstellar vessel equipped with a faster-than-light drive system, has arrived on the Moon. The Alpha castaways can now return home after more than five years of mad travel through a hostile universe. At this time, a team of Alphans—Alan Carter with nuclear physicists Jack Bartlett and Joe Ehrlich—have travelled uncountable light- years and are approaching Earth in the Superswift's compact pilot ship.
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He is invited to talks, seminars and conferences on sound, is a jury member or president at film festivals, but he is mostly solicited to institutes, centers and cinema schools in numerous countries; the Cinemathèque of Lisbonne devoted a week to him in 1985. In September 2000, being sick, he “goes down” to Montpellier to live with his wife Maryvon, near his beloved Mediterranean sea. That is there he dies in March 2006. Follow an uncountable number of messages of love and gratitude (both professional and private), and tributes.
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The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski (1948) established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. (He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic.) A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem, proved by Michael D. Morley (1965), states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e.
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In this case the equipment is operated like a scanning electron microscope, with a resolution of about 0,1 mm (limited by the beam diameter). In a similar mode the fine computer-controlled beam can "write" or "draw" a picture on the metal surface by melting a thin surface layer. ;Working chamber Since the appearance of the first electron-beam welding machines at the end of the 1950s, the application of electron-beam welding spread rapidly into industry and research in all highly developed countries. Up to now, uncountable numbers of various types of electron-beam equipment have been designed and realized.
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It is believed that the fort was built between 6000 and 8000 BC. It had been ruled over and possessed by uncountable kings from ancient times including Mughal, Macedonians, Arabs, Mongols or Ghaznivids. According to other accounts it is called Miri-Kalat and is related to prince Punnu a character of love tale Sassui Punhun narrated by many poets counting Shah Abdul Latif Bhitai of Sindh . Punnu was son of Jam Aali or Aari and his forefathers were rulers of this area during 12th century. The ancient archaeological site is located near the fort in which ancient graves are sited.
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The idea that all matter is composed of elementary particles dates to at least the 6th century BC. The Jains in ancient India were the earliest to advocate the particular nature of material objects between 9th and 5th century BCE. According to Jain leaders like Parshvanatha and Mahavira, the ajiva (non living part of universe) consists of matter or pudgala, of definite or indefinite shape which is made up tiny uncountable and invisible particles called permanu. Permanu occupies space-point and each permanu has definite colour, smell, taste and texture. Infinite varieties of permanu unite and form pudgala.
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According to specialists, the main effect is being caused by poisoning of the freshwater supplies and of the soil by saltwater infiltration and a deposit of a salt layer over arable land. It has been reported that in the Maldives, 16 to 17 coral reef atolls that were overcome by sea waves are without fresh water and could be rendered uninhabitable for decades. Uncountable wells that served communities were invaded by sea, sand, and earth; and aquifers were invaded through porous rock. Salted-over soil becomes sterile, and it is difficult and costly to restore for agriculture.
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Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined.
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Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.Wagon, Corollary 13.3 It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.
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Skolem (1934) pioneered the construction of non-standard models of arithmetic and set theory. Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.
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More than 2,000 rivers (with a length >10 km) and more than 100 main rivers belong to Vietnam. 13 of these rivers have a basin area of more than 10,000 km² with 10 being international ones. Nine rivers count as major ones accounting for ~93% of the total basin area; these are the Red River, Thái Bình River, Bằng Giang-Kỳ Cùng rivers, Ma River, Cả River, Vu Gia- Thu Bồn rivers, Đà Rằng River, Đồng Nai River and Cửu Long River. Beneath an uncountable number of lakes, ponds, lagoons and pools, there are water reservoirs with a total capacity of .
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A forcing or partially ordered set P is proper if for all regular uncountable cardinals \lambda , forcing with P preserves stationary subsets of [\lambda]^\omega . The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is a filter G \subseteq P such that Dα ∩ G is nonempty for all α<ω1. The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments show that if P is ccc or ω-closed, then P is proper.
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". In the 1920s and 30s, there were uncountable bars, cafés, and dance halls in Berlin. The most elegant could be found in West Berlin, near the area formed by the Bülowstraße, the Potsdamer Straße, and the Nollendorfplatz, reaching up to the Kurfürstendamm. No doubt, the most famous was Eldorado, that really was two, one on the Lutherstraße, and a second one in the Motzstraße. Curt Moreck (Konrad Haemmerling) described it in 1931, in his Führer durch das „lasterhafte“ Berlin ("Guide through the 'dissolute' Berlin"), as "an establishment of transvestites staged for the morbid fascination of the world metropolis.
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In a slightly higher register, Gerät represents a miscellaneous artifact or utensil, or, in casual German, may also refer to an item of remarkable size. The use of the word Teil (part) is a relatively recent placeholder in German that has gained great popularity since the late 1980s. Initially a very generic term, it has acquired a specific meaning in certain contexts. Zeug or Zeugs (compare Dings, can be loosely translated as 'stuff') usually refers to either a heap of random items that is a nuisance to the speaker, or an uncountable substance or material, often a drug.
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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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Of these, it lies near to both Kit's Coty House and the Coffin Stone on the eastern side of the river. Three further surviving long barrows, Addington Long Barrow, Chestnuts Long Barrow, and Coldrum Long Barrow, are located west of the Medway. Now a jumble of half-buried sarsen stones it is thought to have been a tomb similar to that of the Coldrum Stones. The name is derived from the belief that the chaotic pile of stones from the collapsed tomb were uncountable and various stories are told about the fate of those who tried.
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The reals are uncountable; that is: there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of N. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
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1\. If a noun ends with a vowel then the article is either án or wá if singular, or ún if plural or uncountable. Usually wá is used for round-fatty objects, and án for flat-thin objects. ( singular ) ( plural ) Kéti án (the farm) Kéti ún (the farms) Fothú án (the picture) Fothú ún (the pictures) Fata wá (the leaf) Fata ún (the leaves) Boro wá (the large) Boro ún (the large) Lou ún (the blood) 2\. If a noun ends with a consonant then the article is the end-consonant plus án or wá for singular or ún for plural.
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There are a dozen couples of northern lapwing and eastern yellow wagtail, together with couples of meadow pipit, whinchat, red-backed shrike, bearded reedling, goshawk, spotted nutcracker, Eurasian wryneck, European honey buzzard, thrush nightingale, long-tailed tit, lesser spotted woodpecker, wood warbler, hawfinch, and Eurasian hobby. In the night time sedge warbler and reed warbler are regularly heard, while grasshopper warbler, river warbler, marsh warbler, and great reed warbler are reported now and then. Osprey are regularly seen fishing in the lake. Uncountable numbers of resting species are reported by the lake, including various swans, hawks, eagles, cormorants, and sparrows.
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Other such demands have been issued by ETAN/US, TAPOL, and—with qualifications—Human Rights Watch and Amnesty International. A 2001 editorial by the East Timor NGO La'o Hamutuk said: > An uncountable number of Crimes Against Humanity were committed during the > 1975–1999 period in East Timor. Although an international court could not > pursue all of them, it ... [would] confirm that the invasion, occupation and > destruction of East Timor by Indonesia was a long-standing, systematic, > criminal conspiracy, planned and ordered at the highest levels of > government. Many of the perpetrators continue to wield authority and > influence in East Timor’s nearest neighbour.
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According to the historian Ammianus Marcellinus, the Goths immediately marched to the city of Adrianople and attempted to take it; Ammianus gives a detailed account of their failure. Ammianus refers to a great number of Roman soldiers who had not been let into the city and who fought the besieging Goths below the walls. A third of the Roman army succeeded in retreating, but the losses were uncountable. Many officers, among them the general Sebastian, were killed in the worst Roman defeat since the Battle of Edessa, the high point of the Crisis of the Third Century.
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In mathematics, particularly in set theory, Fodor's lemma states the following: If \kappa is a regular, uncountable cardinal, S is a stationary subset of \kappa, and f:S\rightarrow\kappa is regressive (that is, f(\alpha)<\alpha for any \alpha\in S, \alpha eq 0) then there is some \gamma and some stationary S_0\subseteq S such that f(\alpha)=\gamma for any \alpha\in S_0. In modern parlance, the nonstationary ideal is normal. The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called "The Pressing Down Lemma".
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Sejarah Melayu, 10.4:77: then His Majesty immediately ordered to equip three hundred jong, other than that kelulus, pelang, jongkong in uncountable numbers. During the Majapahit conquest, small breech-loading swivel gun called cetbang is used in naval warfare, against more traditional boarding tactics employed by other kingdom of the archipelago. During the reign of Hayam Wuruk, Majapahit was involved in a battle against the royal family of Sunda Kingdom in the Battle of Bubat. However, the Paregreg war of 1404 to 1406 drained the coffers of the Majapahit kingdom, and led to its decline in the following years.
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As the Jin account describes, when they attacked the city's Xuanhua Gate, their "fire bombs fell like rain, and their arrows were so numerous as to be uncountable." The Jin captured Kaifeng despite the appearance of the molten metal bomb and secured another 20,000 fire arrows for their arsenal. The Ming dynasty scholar Mao Yuanyi states in his Wubei Zhi, written in 1628: > The Song people used the turntable trebuchet, the single-pole trebuchet and > the squatting-tiger trebuchet. They were all called 'fire trebuchets' > because they were used to project fire-weapons like the (fire-)ball, > (fire-)falcon, and (fire-)lance.
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In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers... Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers.. Cantor's construction builds a one-to-one correspondence between the set of transcendental numbers and the set of real numbers. In this article, Cantor only applies his construction to the set of irrational numbers. Cantor's work established the ubiquity of transcendental numbers.
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In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tychonoff cube is named after Andrey Tychonoff, who first considered the arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact. Tychonoff later generalized this to the product of collections of arbitrary compact spaces. This result is now known as Tychonoff's theorem and is considered one of the most important results in general topology.
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Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ci interpreted as the uncountable cardinal ℵi. (Here ℵi means ℵi in the full universe, not the constructible universe.) There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory.
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Built by Aníbal González between 1904 and 1906 commissioned by Manuel Suárez, full of rooms of large mirrors and chairs with red upholstery. There were arguing supporters of bullfighting, the football derby in the city. There were also uncountable meetings for discuss the Ibero- American Exposition of 1929, were held the masquerade dances of the Carnivals and was made the last stop before it raised the curtain of the Teatro San Fernando. During the Spanish Civil War the Gran Café de París was renamed as Café de Roma to avoid confusion with the café of Avenue de l'Opéra in Paris.
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1); "Unlike the standard upright symbols, which always correspond to the orthodox pieces, there is no strict one-to-one correspondence between rotated symbols and particular piece types: the number of fairy pieces in use is uncountable, and the number of possible pieces is infinite. Instead, rotated symbols are assigned to pieces as needed, and the composer has wide latitude in choosing which ones they feel are appropriate, with only a few very common ones fixed by convention..." (p. 2); "The use of distinct symbols for these pieces is more common among players of the aforementioned variants than among problem enthusiasts" (p. 6).
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In linguistics, a mass noun, uncountable noun, or non-count noun is a noun with the syntactic property that any quantity of it is treated as an undifferentiated unit, rather than as something with discrete elements. Non- count nouns are distinguished from count nouns. Given that different languages have different grammatical features, the actual test for which nouns are mass nouns may vary between languages. In English, mass nouns are characterized by the fact that they cannot be directly modified by a numeral without specifying a unit of measurement, and that they cannot combine with an indefinite article (a or an).
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An example is the set of natural numbers, In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers is larger than , because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable".
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The laic and the reformer factions were constituted mainly by Italian Liberal Party (PLI), other Republicans, Social Democrats and some autonomous socialists affiliated to the same political faction within PSI led by Giuseppe Romita. After the birth of LCIGL they remained in the CIGL, but not for long. The increasing political strikes of CGIL against Italian membership in NATO and the violent events of 17 May 1949 in MolinellaThat day, in Molinella the communists opposed to the results of the election for the local (trade congress) won regurarly by socialdemocrat faction, assaulting congress during its first meeting. At the end of the uncountable a woman died and many were wounded.
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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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According to historian Wang Zhaochun, the account of this battle provided the "earliest truly detailed descriptions of the use of gunpowder weapons in warfare." Records show that the Jin utilized gunpowder arrows and trebuchets to hurl gunpowder bombs while the Song responded with gunpowder arrows, fire bombs, thunderclap bombs, and a new addition called the "molten metal bomb" (金汁炮). As the Jin account describes, when they attacked the city's Xuanhua Gate, their "fire bombs fell like rain, and their arrows were so numerous as to be uncountable." The Jin captured Kaifeng despite the appearance of the molten metal bomb and secured another 20,000 fire arrows for their arsenal.
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Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality. Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal \alpha: A subset of a limit ordinal \alpha is said to be unbounded (or cofinal) under \alpha provided any ordinal less than \alpha is less than some ordinal in the set.
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Kargaly have been again opened in the third time in 1740th years, thanks to uncountable traces of workings out of the Bronze Age. All copper ore mining was conducted up to the end of 19th centuries in the area of ancient ore developments. The extracted minerals went on the north, to mountain areas of Southern Urals Mountains rich with the different forests. The nearest metallurgical works where Kargaly copper ore melted, have defended from this mining complex approximately on 180–200 km; the most remote – to 500 km. During third period tens of million tons of extracted Kargaly’s ore delivered to these distant metallurgical plants by the horse cartage.
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The number systems for English nouns is a simple singular-plural distinction, of which the singular is the base form – meaning that the singular is changed somehow to form a plural, in English this is usually the addition of '-s' ('cat' > 'cats'). Any English noun can be placed into one of three sub-classes within this two-way system: # Nouns that can be used in either the singular (sing.) or plural (pl.), these make up the vast majority of non-abstract things – 'cat', 'star', 'tree'. # Nouns that can normally only be used in the sing., these are mainly abstract ideas and uncountable things – 'honesty', 'milk', 'health', 'flour'.
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In 1957, Peixoto went to research the subject with Lefschetz at the Princeton University, where he spent uncountable hours talking to the Russian professor about Mathematics and other subjects. Despite of the great age difference (Peixoto was 36 years old and Lefschetz 73), they became good friends. With Lefschetz incentive, Peixoto wrote his first paper on structural stability, that would be later published on the Annals of Mathematics, of which Lefschetz was editor. In 1958, they went to the International Mathematical Congress, in Edinburgh, Scotland, where Lefschetz introduced Peixoto to the Russian mathematician Lev Pontryagin, whose work on dynamical systems was used by Peixoto as a basis for his studies.
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For locally compact Hausdorff topological spaces that are not σ-compact the three definitions above need not be equivalent, A discrete topological space is locally compact and Hausdorff. Any function defined on a discrete space is continuous, and therefore, according to the first definition, all subsets of a discrete space are Baire. However, since the compact subspaces of a discrete space are precisely the finite subspaces, the Baire sets, according to the second definition, are precisely the at most countable sets, while according to the third definition the Baire sets are the at most countable sets and their complements. Thus, the three definitions are non-equivalent on an uncountable discrete space.
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One approach for avoiding mathematical construction issues of stochastic processes, proposed by Joseph Doob, is to assume that the stochastic process is separable. Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set. Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied. Another approach is possible, originally developed by Anatoliy Skorokhod and Andrei Kolmogorov, for a continuous-time stochastic process with any metric space as its state space.
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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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Classifiers are not generally a feature of English or other European languages, although classifier-like constructions are found with certain nouns. A commonly cited English example is the word head in phrases such as "five head of cattle": the word cattle (for some speakers) is an uncountable (mass) noun, and requires the word head to enable its units to be counted. The parallel construction exists in French: une tête de bétail ("one head of cattle") and in Spanish: una cabeza de ganado ("one head of cattle"). Note the difference between "five head of cattle" (meaning five animals), and "five heads of cattle" (identical to "five cattle's heads", meaning specifically their heads).
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Calculation of the number of casualties is difficult since the very primary sources were contradictory. Serbian military reports talk about "several hundreds", same number as what Kosta Novaković gives. The Serbian historians later placed the number to 198, and 31 wounded. The narrative stories of the Luma highlanders place the number as far as 16,000 or even up to 18,000, much more than the 12,000 which mostly circulates, also placed in the Albania memorial dedicated to the battle. An Ottoman telegram sent from Ohrid to Elbasan, dated 20 November 1912, reported 3 officers and an "uncountable" number of soldiers, with around 1,000 riffles captured by the Albanians.
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Moose wintering in Rama, Saskatchewan 2013 Nearby the farmland gives way to nearly endless lakes and rivers fed by uncountable streams, swamps and other wetlands. Ducks, geese and many other water-loving migratory birds congregate here and in the nearby Quill Lakes area during the spring and fall migrations. In the fall, hunters from the local area as well as visitors from other provinces, the United States and around the world come to this area for the birds and many other game animals. Good Spirit Lake Provincial Park, Whitesand Regional Park, Camp Whitesand, Leslie Beach, Canora Beach and many other parks and recreational opportunities nearby.
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Whilst it is possible to define so-called gap-n morasses for n > 1, they are so complex that focus is usually restricted to the gap-1 case, except for specific applications. The "gap" is essentially the cardinal difference between the size of the "small approximations" used and the size of the ultimate structure. A (gap-1) morass on an uncountable regular cardinal κ (also called a (κ,1)-morass) consists of a tree of height κ + 1, with the top level having κ+-many nodes. The nodes are taken to be ordinals, and functions between these ordinals are associated to the edges in the tree order.
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Pi () is a well known transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial with rational coefficients. The best known transcendental numbers are and e. Though only a few classes of transcendental numbers are known, in part because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers compose a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set.
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Already in his youth he developed a passion for football and attended countless major and minor events. Later he supported other sports codes as well, notably boxing and athletics, and he was a noticeable member of the audience in uncountable high- profile Namibian social events. This was to an extent that on several occasions his unlikely absence would be noticed, for instance at a concert of The Rockets in Windhoek that was opened with the question "Where is Robbie?" Although a poor man throughout his life, Savage's role as mascot of the Brave Warriors enabled him to tour the African continent for several sports events.
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The area was described by Francisco Álvares during his stay in Ethiopia as being five days' journey (ca. 100 km) in length, and extending far into Muslim Afar territory. One of their largest towns, Manadeley, situated on the edge of the Ethiopian Highlands and overlooking the Afar lowlands, was a market town of great size. Álvares describes it as a town of "very great trade, like a city or seaport", where any good could be found, and with merchants from a number of areas, such as Jeddah, Fez, elsewhere in Morocco, Tunis, Greece, Ormus, Cairo, and India, as well as an uncountable number of people from surrounding regions in Ethiopia.
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Writing for NPR, Ann Powers found it altogether contemplative, joyful, and mythological. Jonathan Bogart of The Atlantic wrote that, with her Tolkien-inspired lyrics, Richard "remains true to the oldest and most important standards of R&B;, which, more than any other musical genre, charts the uncountable intricacies of the human heart." Grantland critic Steven Hyden felt that the album blurs R&B; conventions like Frank Ocean's Channel Orange (2012) and Janelle Monáe's The ArchAndroid (2010), and as "an ambitious, singular work", it demands repeated listens. AllMusic's Andy Kellman called Goldenheart "sumptuous and grand" with enough exceptional songs to compensate for its intensity and indulgence.
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To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe. More generally, the definition of 0# works provided that there is an uncountable set of indiscernibles for some Lα, and the phrase "0# exists" is used as a shorthand way of saying this. There are several minor variations of the definition of 0#, which make no significant difference to its properties. There are many different choices of Gödel numbering, and 0# depends on this choice.
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Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe L into itself. Donald A. Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#. It follows from Jensen's covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in the constructible universe L. Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of 0#.
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Let X0 be the countable subset given by the finitely many Fn-orbits of the fixed points hi ±∞, the fixed points of the hi and all their conjugates. Since X is uncountable, there is an element of g with fixed points outside X0 and a point w outside X0 different from these fixed points. Then for some subsequence (gm) of (gn) :gm = h1n(m,1) ··· hkn(m,k), with each n(m,i) constant or strictly monotone. On the one hand, by successive use of the rules for computing limits of the form hn·wn, the limit of the right hand side applied to x is necessarily a fixed point of one of the conjugates of the hi's.
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The Löwenheim–Skolem theorem shows that if a first-order theory of cardinality λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in model theory, it implies that it is not possible to characterize countability or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ(x) such that an arbitrary structure M satisfies φ if and only if the domain of discourse of M is countable (or, in the second case, uncountable). The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic.
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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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In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who published this theorem for countable linear orders in 1940. More strongly, they showed that in the countable case there exists an order embedding into a proper subset of the given order; however, they provided examples showing that this strengthening does not always hold for uncountable orders. In reverse mathematics, the Dushnik–Miller theorem for countable linear orders has the same strength as the arithmetical comprehension axiom (ACA0), one of the "big five" subsystems of second-order arithmetic.
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Choose a collection of 2ℵ0 measure 0 subsets of R such that every measure 0 subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α. For each countable ordinal β choose a real number xβ that is not in any of the sets Sα for α < β, which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set X of all these real numbers xβ has only a countable number of elements in each set Sα, so is a Sierpiński set. It is possible for a Sierpiński set to be a subgroup under addition.
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Freedman's main theorem about Casson handles states that they are all homeomorphic to D^2\times \R^2; or in other words they are topological 2-handles. In general they are not diffeomorphic to D^2\times \R^2 as follows from Donaldson's theorem, and there are an uncountable infinite number of different diffeomorphism types of Casson handles. However the interior of a Casson handle is diffeomorphic to \R^4; Casson handles differ from standard 2 handles only in the way the boundary is attached to the interior. Freedman's structure theorem can be used to prove the h-cobordism theorem for 5-dimensional topological cobordisms, which in turn implies the 4-dimensional topological Poincaré conjecture.
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In 1654, a gunpowder explosion in Delft destroyed the studio of Dutch artist Carel Fabritius along with most of his paintings. The artist himself died in the explosion.Karel Fabricius biography in De groote schouburgh der Nederlantsche konstschilders en schilderessen (1718) by Arnold Houbraken, courtesy of the Digital library for Dutch literature On 26 September 1687, the Parthenon and its sculptures were severely damaged by the explosion of an Ottoman Empire ammunition dump stored inside the building, which was ignited by Venetian bombardment. On 24 December 1734, a fire in the Royal Alcázar of Madrid destroyed over 400 paintings, uncountable sculptures and thousands of documents, including the music collection of the royal chapel.
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The Guardian distinguishes two meanings of the term welcoming culture: Originally it was meant to attract people from abroad to Germany to compensate for a huge shortage of skilled workers, particularly in sparsely populated areas. Since the beginning of the European refugee crisis in 2015, the term was being used to promote assistance for millions of refugees coming to Germany, who were received by highly visible posters "refugees welcome", and by actual help of any kind, mainly on private initiative of uncountable German citizens. Translation of the word "welcome" are found at many places, which are visited by tourists. The French daily Libération adds that the word "welcome culture" was originally created decades ago in the tourism industry.
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Kramer took part successfully in the second night of the Viña de Mar Festival in 2008. In a comedy routine that lasted almost 90 minutes, Kramer impersonated around 33 characters. The audience acclaimed and awarded him with two Antorchas (Oro and Plata – English: Gold and Silver Torchs) and a Gaviota de Plata (English: Silver Seagull). Many Chilean TV shows have taken advantage of the good rating that produces Kramer's routine and they have repeated it in uncountable occasions, without the comedian receives some money remuneration. On 12 July 2009, Kramer appeared on TVN's Animal Nocturno, where he personified several presidential candidates, such as Marco Enríquez-Ominami, Eduardo Frei Ruiz-Tagle, and Sebastián Piñera.
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After that, under the able and aggressive leader Gajah Mada, Majapahit spread its influence beyond Java and Sumatra, to the rest of the Nusantara archipelago. In 1350 Majapahit launched its largest military expedition, the invasion of Pasai, with 400 large jong and uncountable malangbang and kelulus.Chronicle of the Kings of Pasai, 3: 98: After that, he is tasked by His Majesty to ready all the equipment and all weapons of war to come to that country of Pasai, about four hundred large jongs and other than that much more of malangbang and kelulus. The second largest military expedition, invasion of Singapura in 1398, Majapahit deployed 300 jong with no less than 200,000 men.
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Next they are taken as prisoner by the "pink elephant": they are inhabitants of Utopia who are trying to undo the various changes to history, the people of Elysium was able to reach Utopia eventually without destroying Deponia, but Rufus' interference of the timeline caused the destruction of the Deponia which caused Elysium to crash into Utopia wiping out their species. They were the ones using McChronicle car and are the owners of the time-pod which was hidden in the mall. Not much later, the Rufus of this timeline also turns up. According to the Utopians there are multiple time loops which interfere with each other resulting the past already repeated uncountable number of times.
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In set theory, there is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an initial segment of the Natural numbers where the domain of the enumerating function can assume any ordinal. Under this definition, an enumeration of a set S is any surjection from an ordinal α onto S. The more restrictive version of enumeration mentioned before is the special case where α is a finite ordinal or the first limit ordinal ω. This more generalized version extends the aforementioned definition to encompass transfinite listings. Under this definition, the first uncountable ordinal \omega_1 can be enumerated by the identity function on \omega_1 so that these two notions do not coincide.
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In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them. While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable. Here, for any given computable number x, the well ordering principle provides that there is a minimal element in S which corresponds to x, and therefore there exists a subset consisting of the minimal elements, on which the map is a bijection.
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So ω can be identified with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since, for example, \aleph_0^2 = \aleph_0 whereas \omega^2 > \omega). Also, \omega_1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and \omega_1 is the order type of that set), \omega_2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and \omega_\omega is the limit of the \omega_n for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_n).
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The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a free abelian group, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator. Consider first the set of all possible singular n-simplices \sigma_n(X) on a topological space X. This set may be used as the basis of a free abelian group, so that each singular n-simplex is a generator of the group. This set of generators is of course usually infinite, frequently uncountable, as there are many ways of mapping a simplex into a typical topological space. The free abelian group generated by this basis is commonly denoted as C_n(X).
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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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To describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed. If g is an element of the cartesian product ∏{Hi} of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups {Hi} (written as ∑E{Hi}) is the subset of ∏{Hi}, where, for each element g of ∑E{Hi}, gi is the identity e_{H_i} for all but a finite number of gi (equivalently, only a finite number of gi are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
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She also turned to fantasy, such as Graeme and the Dragon (1954, Graeme Mitchison being a grandson through Denis), science fiction such as Memoirs of a Spacewoman (1962) and Solution Three (1975), fantasy such as the humorous Arthurian novel To the Chapel Perilous (1955), non-fiction such as African Heroes (1968), and also children's novels, poetry, travel and a three-volume autobiography. She was never sure exactly how many books she had written, often claiming there were about 70. The articles were uncountable, from book reviews for the old Time and Tide magazine and the New Statesman to practical essays on farming, campaigning articles, recollections and reflections. After her husband's death, Mitchison wrote several memoirs, published as separate titles between 1973 and 1985.
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In Alan Dean Foster's 1986 horror/fantasy novel, Into the Out Of, elders of the Maasai people become aware that from the south of them in the Ruaha wilderness of Tanzania a global crisis is approaching. Malevolent shetani, which originate from a dimensional portal known to the Maasai as the “Out Of” (because all things, such as humans, animals and plants, originally came "out of" it), are finding their way into this world. In addition to general sabotage, the shetani are fomenting trouble between the superpowers, intent on inciting war. If not prevented, the barriers between the two dimensions will be permanently breached and uncountable hordes of shetani will overrun the world, enslaving the few humans they do not exterminate.
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Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open. Then X is uncountable. Proof. We will show that if U ⊆ X is non-empty and open, and if x is a point of X, then there is a neighbourhood V ⊂ U whose closure does not contain x (x may or may not be in U). Choose y in U different from x (if x is in U, then there must exist such a y for otherwise U would be an open one point set; if x is not in U, this is possible since U is non-empty). Then by the Hausdorff condition, choose disjoint neighbourhoods W and K of x and y respectively.
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According to the Nagarakretagama, canto XIII and XIV mentioned several states in Sumatra, Malay Peninsula, Borneo, Sulawesi, Nusa Tenggara islands, Maluku, New Guinea, Mindanao, Sulu Archipelago, Luzon and some parts of the Visayas islands as under the Majapahit realm of power. The Hikayat Raja Pasai, a 14th- century Aceh chronicle describe a Majapahit naval invasion on Samudra Pasai in 1350. The attacking force consisted of 400 large jong and an uncountable number of malangbang and kelulus.Chronicle of the Kings of Pasai, 3: 98: After that, he is tasked by His Majesty to ready all the equipment and all weapons of war to come to that country of Pasai, about four hundred large jongs and other than that much more of malangbang and kelulus.
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His primary research area is Ramsey theory of infinite sets. He is known for solutions to the basis problem for uncountable linear orders and to the L space problem from general topologyJustin Tatch Moore: A SOLUTION TO THE L SPACE PROBLEM, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 19, Number 3, Pages 717–736 and for his work in determining the consequences of relating the continuum to certain values of the aleph function. Moore, together with his PhD student Yash Lodha, produced the first torsion-free counterexample to the von Neumann-Day problem, originally described by mathematician John von Neumann in 1929. Lodha presented this solution at the London Mathematical Society's Geometric and Cohomological Group Theory symposium in August 2013.
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The track is a conceptual song centered around a gangster family formed by the two performers, Rosalía and Travis Scott. The song sees Scott being murdered in a gang war between drug networks, while Rosalía becomes a widow who still dresses in black in order to express mourning. According to XXLs Trent Fitzgerald "The gang could either be Italian or Brazilian due to the uncountable Sicilian references, from 'capo' to 'omertá' and for the use of the word 'brazuca', which references someone originary from Brazil". Rosalía sings about how she cannot trust anyone anymore, while Scott's lyrics contains "a conversation with and about the woman of his life, his wife, and how she had nothing to do with this business, that he assumes all the responsibility".
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Kane became pregnant with his child, and she traveled with the infant through time to a cliff side near Raine. She wrote a book about her life in the language of thorns and because of her enchantments, no one but her daughter could read it—by the book's climax, she has. The final words of the book open the Gates of Time that admit Axis and Kane, Nepenthe's parents, and their uncountable legions of followers near Raine, three thousand years in their future. When Queen Tessera learns of the planned invasion, with the help of Vevay, her own mage, and the magicians of the Floating School, she uses magic to make Raine seem a dilapidated ruin and thereby protect it from Axis.
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Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existence of 0# contradicts the axiom of constructibility: V = L. If 0# exists, then it is an example of a non-constructible Δ set of integers. This is in some sense the simplest possibility for a non- constructible set, since all Σ and Π sets of integers are constructible. On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered.
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On the other hand, a vector space of uncountable dimension, endowed with the finest vector topology, then this is a topological vector spaces with the Hahn-Banach extension property that is neither locally convex nor metrizable. A vector subspace of a TVS has the separation property if for every element of such that , there exists a continuous linear functional on such that and for all . Clearly, the continuous dual space of a TVS separates points on if and only if } has the separation property. In 1992, Kakol proved that any infinite dimensional vector space , there exist TVS-topologies on that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on .
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One of Baumgartner's results is the consistency of the statement that any two \aleph_1-dense sets of reals are order isomorphic (a set of reals is \aleph_1-dense if it has exactly \aleph_1 points in every open interval). With András Hajnal he proved the Baumgartner–Hajnal theorem, which states that the partition relation \omega_1\to(\alpha)^2_n holds for \alpha<\omega_1 and n<\omega. He died in 2011 of a heart attack at his home in Hanover, New Hampshire. The mathematical context in which Baumgartner worked spans Suslin's problem, Ramsey theory, uncountable order types, disjoint refinements, almost disjoint families, cardinal arithmetics, filters, ideals, and partition relations, iterated forcing and Axiom A, proper forcing and the proper forcing axiom, chromatic number of graphs, a thin very-tall superatomic Boolean algebra, closed unbounded sets, and partition relations.
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Characteristic late work in which all detail dissolves in uncountable specks of pigment in a seemingly random way: "The Beheading of Saint Catherine", 1791, study for an altar painting for Brno Cathedral that came not to be executed. Feuchtmüller (1989), catalogue raisonné No. 945 The Virgin (1755), main altar of the parish church of Waizenkirchen (Upper Austria) Martin Johann Schmidt, called Kremser Schmidt or Kremserschmidt, (25 September 1718 – 28 June 1801), was one of the outstanding Austrian painters of the late Baroque/Rococo along with Franz Anton Maulbertsch. He was born at Grafenwörth, lower Austria, a son of the sculptor Johannes Schmidt. A pupil of Gottlieb Starmayr, he spent most of his life at Stein, where he mostly worked in the numerous churches and monasteries of his Lower Austrian homeland.
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Not every infinite graph has a normal spanning tree. For instance, a complete graph on an uncountable set of vertices does not have one: a normal spanning tree in a complete graph can only be a path, but a path has only a countable number of vertices. However, every graph on a countable set of vertices does have a normal spanning tree.. See in particular Theorem 3, p. 193. Even in countable graphs, a depth-first search might not succeed in eventually exploring the entire graph, and not every normal spanning tree can be generated by a depth-first search: to be a depth-first search tree, a countable normal spanning tree must have only one infinite path or one node with infinitely many children (and not both).
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If an infinite graph G has a normal spanning tree, so does every connected graph minor of G. It follows from this that the graphs that have normal spanning trees have a characterization by forbidden minors. One of the two classes of forbidden minors consists of bipartite graphs in which one side of the bipartition is countable, the other side is uncountable, and every vertex has infinite degree. The other class of forbidden minors consists of certain graphs derived from Aronszajn trees.. The details of this characterization depend on the choice of set-theoretic axiomatization used to formalize mathematics. In particular, in models of set theory for which Martin's axiom is true and the continuum hypothesis is false, the class of bipartite graphs in this characterization can be replaced by a single forbidden minor.
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Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X that defines the topology of X and that makes X a complete separable metric space. Then X as a Borel space is Borel isomorphic to one of (1) R, (2) Z or (3) a finite space. (This result is reminiscent of Maharam's theorem.) It follows that a standard Borel space is characterized up to isomorphism by its cardinality, and that any uncountable standard Borel space has the cardinality of the continuum. Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.
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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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In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. English translation: Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, Extract of page 73 including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem.
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Norman later explained that this restriction of the term's meaning had been unintended, and in his 2013 update of The Design of Everyday Things, he added the concept "signifiers". In the digital age, designers were learning how to indicate what actions were possible on a smartphone's touchscreen, which didn't have the physical properties that Norman intended to describe when he used the word "affordances". However, the definition from his original book has been widely adopted in HCI and interaction design, and both meanings are now commonly used in these fields. Following Norman's adaptation of the concept, affordance has seen a further shift in meaning where it is used as an uncountable noun, referring to the easy discoverability of an object or system's action possibilities, as in "this button has good affordance".
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Professor Kuupole is from Nandom-Kogle in the Upper West Region of Ghana where he started his primary middle school education from 1962 to 1971. He proceeded to Nandom Secondary School and later to the University of Cape Coast to pursue higher education and had his MPhil in French in France among numerous other courses both home and abroad and serves several boards with uncountable publications and research works to his credit. Prior to his appointment as Vice Chancellor in October 2012, the Professor supervised undergraduate and postgraduate theses and has served in various capacities, including Head of the Department of French, Dean of the Faculty of Arts, and National President of the Alumni Association. Professor Kuupole holds a BA (Hons) in French and a Diploma in Education from the University of Cape Coast.
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In 1915, Hartogs could use neither von Neumann-ordinals nor the replacement axiom, and so his result is one of Zermelo set theory and looks rather different from the modern exposition above. Instead, he considered the set of isomorphism classes of well-ordered subsets of X and the relation in which the class of A precedes that of B if A is isomorphic with a proper initial segment of B. Hartogs showed this to be a well-ordering greater than any well-ordered subset of X. (This must have been historically the first genuine construction of an uncountable well-ordering.) However, the main purpose of his contribution was to show that trichotomy for cardinal numbers implies the (then 11 year old) well-ordering theorem (and, hence, the axiom of choice).
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Combined with Solovay's result, this shows that the statements "There is an inaccessible cardinal" and "Every set of reals has the perfect set property" are equiconsistent over ZF. Finally, showed that consistency of an inaccessible cardinal is also necessary for constructing a model in which all sets of reals are Lebesgue measurable. More precisely he showed that if every Σ set of reals is measurable then the first uncountable cardinal ℵ1 is inaccessible in the constructible universe, so that the condition about an inaccessible cardinal cannot be dropped from Solovay's theorem. Shelah also showed that the Σ condition is close to the best possible by constructing a model (without using an inaccessible cardinal) in which all Δ sets of reals are measurable. See and and for expositions of Shelah's result.
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English translation: . In 1972, Irving Kaplansky wrote: "It is often said that Cantor's proof is not 'constructive,' and so does not yield a tangible transcendental number. This remark is not justified. If we set up a definite listing of all algebraic numbers … and then apply the diagonal procedure …, we get a perfectly definite transcendental number (it could be computed to any number of decimal places).". This proof is not only constructive, but it is also simpler than the non-constructive proof that Perron provides because that proof takes the unnecessary detour of first proving that the set of all reals is uncountable.. Cantor's diagonal argument has often replaced his 1874 construction in expositions of his proof. The diagonal argument is constructive and produces a more efficient computer program than his 1874 construction.
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In 1322 friar Odoric of Pordenone reported that the during his voyage from India to China he boarded a vessel of the zunc[um] type carried at least 700 people, either sailors or merchants. The Majapahit Empire used jongs as its main source of naval power. It is unknown how many exactly the total number of jong used by Majapahit, but they are grouped into 5 fleets. The largest number of jong deployed in an expedition is about 400 jongs accompanied with uncountable malangbang and kelulus, when Majapahit attacked Pasai.Chronicle of the Kings of Pasai, 3: 98: After that, he is tasked by His Majesty to ready all the equipment and all weapons of war to come to that country of Pasai, about four hundred large jongs and other than that much more of malangbang and kelulus.
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Originally, he had planned to feature a "virtually uncountable number of planets," which would be procedurally generated as the player explored them. However, this concept was later modified to a smaller number of planets, each with multiple locations. He revealed a major selling point of the game would be no loading times between exploration and combat, or when moving from one location to another, something which had never before been accomplished on a disc-based RPG, and something which had been a long-time personal goal of his. He also went into detail about the backgrounds of Jaster, Kisala, Zegram and Lilika, and he explained the basic combat system (a hack and slash system using three party members, with the player able to issue commands to the two NPCs), the Revelation Flow system, and some basic information about the Factory system.
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This rule can be seen in the examples "there is less flour in this canister" and "there are fewer cups (grains, pounds, bags, etc.) of flour in this canister", which are based on the reasoning that flour is uncountable whereas the unit used to measure the flour (cup, etc.) is countable. Nevertheless, even most prescriptivists accept the most common usage "there are less cups of flour in this canister" and prescribe the rule addition that "less" should be used with units of measurement (other examples: "less than 10 pounds/dollars"). Prescriptivists would, however, consider "fewer cups of coffee" to be correct in a sentence such as "there are fewer cups of coffee on the table now", where the cups are countable separate objects. In addition, "less" is recommended in front of counting nouns that denote distance, amount, or time.
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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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The image of f is the countable set whose members are themselves infinite (and possibly uncountable) sets. By using the axiom of countable choice we may choose one member from each of these sets, and this member is itself a finite subset of X. More precisely, according to the axiom of countable choice, a (countable) set exists, so that for every natural number n, g(n) is a member of f(n) and is therefore a finite subset of X of size n. Now, we define U as the union of the members of G. U is an infinite countable subset of X, and a bijection from the natural numbers to U, , can be easily defined. We may now define a bijection that takes every member not in U to itself, and takes h(n) for every natural number to .
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Having marched through the lands of the Cenél Conaill and Cenél nEógain, Boru led his army across the River Bann at Fersat Camsa (Macosquin) and into Ulaid, where he accepted submissions from the Ulaid at Craeb Telcha, before marching south and through the traditional assembly place of the Conaille Muirtheimne at i n-oenach Conaille. Flaithbertach Ua Néill continued his attacks on Ulaid in 1007, attacking the Conaille Muirtheimne. In 1011, the same year Boru finally achieved hegemony over the entire of Ireland, Flaithbertach launched an invasion of Ulaid, and after destroying Dún Echdach (Duneight, south of Lisburn) and the surrounding settlement, took the submission of the Dál Fiatach, who had the Ulaid kingship, thus removing them from Boru's over-lordship. The next year, Flaithbertach raided the Ards peninsula and took an uncountable number of spoils.
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Referring to the contribution of the Spanish Maquis to the French resistance movement, Martha Gellhorn wrote in The Undefeated (1945): Also during World War II, Spaniards assassinated the German generals von Schaumberg (commandant of the region around Paris) and von Ritter (a recruiter of forced labor). In October 1944 a group of 6,000 maquis including Antonio Téllez Solà invaded Spain via the Aran Valley but were driven back after ten days. Few details of the maquis' actions in Spain have been made public because of the secrecy of the Franco government, but guerrillas, including Francesc Sabaté Llopart, Jose Castro Veiga, and Ramon Vila Capdevila were responsible for the deaths of hundreds of Guardia Civil (Civil Guard) officers, and uncountable acts of industrial sabotage. Between 1943 and 1952, 2,166 maquis were reported arrested by the Civil Guard, nearly wiping out the movement.
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A random variable has a probability distribution, which specifies the probability of Borel subsets of its range. Random variables can be discrete, that is, taking any of a specified finite or countable list of values (having a countable range), endowed with a probability mass function that is characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals (having an uncountable range), via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both. Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
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She in turn nearly shoots Sin before the Brigadier can stop her and impose order on his troops once again. The Doctor enters the cattle truck, only to find himself lost on a hellish moor where the tarns are filled with blood and severed heads and limbs float along streams of gore. In the middle of a moat of blood, raised on an altar of dead flesh, a standing stone pulses with the same energies the Doctor detected at Cirbury; and here, the Ragman is waiting for him. The Ragman shows him things; an asteroid travelling through uncountable light years of space, bathed in alien radiation and given life without sentience, drawn to Earth and nourished by the planet's ley- lines, until the being within was given birth in an eruption of class-based violence and death.
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Rim driven Scoop wheel of the Stretham Old Engine, Cambridgeshire Scoop wheel of a Dutch mill Shaft driven Scoop wheel of the Dogdyke Engine, Lincolnshire A scoop wheel or scoopwheel is a pump, usually used for land drainage. A scoop wheel pump is similar in construction to a water wheel, but works in the opposite manner: a waterwheel is water-powered and used to drive machinery, a scoop wheel is engine-driven and is used to lift water from one level to another. Principally used for land drainage, early scoop wheels were wind- driven National monument record for typical but surviving wind driven scoop wheel at Turf Fen but later steam-powered beam engines were used.Most of this section taken from 'Machines, Mills & uncountable costly necessities', R L Hills, Goose & Co (Norwich), 1967 It can be regarded as a form of pump.
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In this algorithm the termination probability of the process may depend on the current state rather than being a fixed factor. A faster algorithm was proposed in 2007 by Niño-Mora by exploiting the structure of a parametric simplex to reduce the computational effort of the pivot steps and thereby achieving the same complexity as the Gaussian elimination algorithm. Cowan, W. and Katehakis (2014), provide a solution to the problem, with potentially non-Markovian, uncountable state space reward processes, under frameworks in which, either the discount factors may be non-uniform and vary over time, or the periods of activation of each bandit may be not be fixed or uniform, subject instead to a possibly stochastic duration of activation before a change to a different bandit is allowed. The solution is based on generalized restart-in-state indices.
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In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X may be written uniquely as the disjoint union of a perfect set and a countable set.
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In July the division entered Berlin—the first American unit to enter the German capital city. During World War II, the 2nd Armored Division took 94,151 POWs, liberated 22,538 Allied POWs, shot down or damaged on the ground 266 enemy aircraft, and destroyed or captured uncountable thousands of enemy tanks and other equipment and supplies. Members of the Division received 9,369 individual awards, including two Medals of Honor, twenty-three Distinguished Service Crosses, and 2,302 Silver Stars as well as nearly 6,000 Purple Hearts; among those receiving the Silver Star were Edward H. Brooks, Hugh Armagio, Stan Aniol, Staff Sergeant John J. Henry, William L. Giblin, Neil J. Garrison, Morton Eustis, son of William Corcoran Eustis, and Sgt Kenneth J. White. The division was twice cited by the Belgian government and division soldiers for the next 50 years wore the fourragere of the Belgian Croix de Guerre.
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In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
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Like most other Western Oti–Volta languages, it has lost the complicated noun class agreement system still found in e.g. the more distantly related Gurmanche, and has only a natural gender system, human/non-human. The noun classes are still distinguishable in the way nouns distinguish singular from plural by paired suffixes: nid(a) "person" plural nidib(a) buug(a) "goat" plural buus(e) nobir(e) "leg, foot" plural noba(a) fuug(o) "item of clothing" plural fuud(e) molif(o) "gazelle" plural moli(i) A unpaired suffix -m(m) is found with many uncountable and abstract nouns, e.g. ku'om(m) "water" The bracketed final vowels in the examples occur because of the feature which most strikingly separates Kusaal from its close relatives: the underlying forms of words, such as buuga "goat" are found only when the word in question is the last word in a question or a negated statement.
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Solovay suggested in his paper that the use of an inaccessible cardinal might not be necessary. Several authors proved weaker versions of Solovay's result without assuming the existence of an inaccessible cardinal. In particular showed there was a model of ZFC in which every ordinal-definable set of reals is measurable, Solovay showed there is a model of ZF + DC in which there is some translation-invariant extension of Lebesgue measure to all subsets of the reals, and showed that there is a model in which all sets of reals have the Baire property (so that the inaccessible cardinal is indeed unnecessary in this case). The case of the perfect set property was solved by , who showed (in ZF) that if every set of reals has the perfect set property and the first uncountable cardinal ℵ1 is regular then ℵ1 is inaccessible in the constructible universe.
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Suppose that X is the first uncountable ordinal, with the finite measure where the measurable sets are either countable (with measure 0) or the sets of countable complement (with measure 1). The (non-measurable) subset E of X×X given by pairs (x,y) with x The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory. The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while showed that it is consistent with ZFC that a strong Fubini-type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.
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By joining the single arrows together and the double arrows together, one obtains a torus with seven mutually touching regions; therefore seven colors are necessary This construction shows the torus divided into the maximum of seven regions, each one of which touches every other. The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs (possibly with an uncountable number of vertices) for which every finite subgraph is planar. To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn–Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable . This can also be seen as an immediate consequence of Kurt Gödel's compactness theorem for first-order logic, simply by expressing the colorability of an infinite graph with a set of logical formulae.
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A concrete Boolean algebra or field of sets is any nonempty set of subsets of a given set X closed under the set operations of union, intersection, and complement relative to X. (As an aside, historically X itself was required to be nonempty as well to exclude the degenerate or one-element Boolean algebra, which is the one exception to the rule that all Boolean algebras satisfy the same equations since the degenerate algebra satisfies every equation. However this exclusion conflicts with the preferred purely equational definition of "Boolean algebra," there being no way to rule out the one-element algebra using only equations— 0 ≠ 1 does not count, being a negated equation. Hence modern authors allow the degenerate Boolean algebra and let X be empty.) Example 1. The power set 2X of X, consisting of all subsets of X. Here X may be any set: empty, finite, infinite, or even uncountable.
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Dejter showedDejter I. J. "Perfect domination in regular grid graphs", Austral. Jour. Combin., 42 (2008), 99-114 that there is an uncountable number of parallel total perfect codes in the planar integer lattice graph L; in contrast, there is just one 1-perfect code, and just one total perfect code in L, the latter code restricting to total perfect codes of rectangular grid graphs (which yields an asymmetric, Penrose, tiling of the plane); in particular, Dejter characterized all cycle products Cm x Cn containing parallel total perfect codes, and the d-perfect and total perfect code partitions of L and Cm x Cn, the former having as quotient graph the undirected Cayley graphs of the cyclic group of order 2d2+2d+1 with generator set {1,2d2}. In 2012, Araujo and DejterDejter I. J.; Araujo C. "Lattice-like total perfect codes", Discussiones Mathematicae Graph Theory, 34 (2014) 57–74, doi:10.7151/dmgt.1715.
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Formally, if \kappa is a limit ordinal, then a set C\subseteq\kappa is closed in \kappa if and only if for every \alpha<\kappa, if \sup(C\cap \alpha)=\alpha e0, then \alpha\in C. Thus, if the limit of some sequence from C is less than \kappa, then the limit is also in C. If \kappa is a limit ordinal and C\subseteq\kappa then C is unbounded in \kappa if for any \alpha<\kappa, there is some \beta\in C such that \alpha<\beta. If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
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In a more concise (although more obscure) way: :\psi(\alpha) is the smallest ordinal which cannot be expressed from 0, 1, \omega and \Omega using sums, products, exponentials, and the \psi function itself (to previously constructed ordinals less than \alpha). Here is an attempt to explain the motivation for the definition of \psi in intuitive terms: since the usual operations of addition, multiplication and exponentiation are not sufficient to designate ordinals very far, we attempt to systematically create new names for ordinals by taking the first one which does not have a name yet, and whenever we run out of names, rather than invent them in an ad hoc fashion or using diagonal schemes, we seek them in the ordinals far beyond the ones we are constructing (beyond \Omega, that is); so we give names to uncountable ordinals and, since in the end the list of names is necessarily countable, \psi will “collapse” them to countable ordinals.
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A cardinal κ is called virtually Ramsey if for every function :f: [κ]<ω → {0, 1} there is C, a closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ which is homogenous for f; slightly weaker is the notion of almost Ramsey where homogenous sets for f are required of order type λ, for every λ < κ. The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of 0#, or indeed that every set with rank less than κ has a sharp. Every measurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal. A property intermediate in strength between Ramseyness and measurability is existence of a κ-complete normal non-principal ideal I on κ such that for every and for every function :f: [κ]<ω → {0, 1} there is a set B ⊂ A not in I that is homogeneous for f.
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Nusantaran thalassocracies made extensive use of naval power and technologies. This enabled the seafaring Malay people to attack as far as the coast of Tanganyika and Mozambique with 1000 boats and attempted to take the citadel of Qanbaloh, about 7,000 km to their West, in 945-946 AD. In 1350 AD Majapahit launched its largest military expedition, the invasion of Pasai, with 400 large jong and innumerable smaller vessels.Chronicle of the Kings of Pasai, 3: 98: After that, he is tasked by His Majesty to ready all the equipment and all weapons of war to come to that country of Pasai, about four hundred large jongs and other than that much more of malangbang and kelulus. The second largest military expedition, invasion of Singapura in 1398, Majapahit deployed 300 jong with no less than 200,000 men.Sejarah Melayu, 10.4:77: then His Majesty immediately ordered to equip three hundred jong, other than that kelulus, pelang, jongkong in uncountable numbers.
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After the first several installments of the unofficial Outcesticide collection came through, the official live album From the Muddy Banks of the Wishkah was released in 1996, containing various performances recorded between 1989 and 1994. It should also be noted that the official With the Lights Out box set was released in 2004, which included many rare, unreleased recordings, while the official Live at Reading debuted Nirvana's now classic 1992 Reading Festival performance in 2009, after years of endless bootlegging. Due to the fact that there is an uncountable number of unofficial releases and their accompanying versions, below is a partial list of the most sought-after and talked about bootleg CD recordings, including the A Season in Hell Part One box set, the Into the Black box set, and the complete Outcesticide series. Since bootleg CDs are notorious for including erroneously listed information pertaining to song titles, dates and venues, the information below has omitted any false listings and replaced them with the correct information.
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John O'Brien sums up the problem of trying to deliver person centredness through formal service systems that have a very different culture thus: > Many human service settings are zones of compliance in which relationships > are subordinated to and constrained by complex and detailed rules. In those > environments, unless staff commit themselves to be people's allies and treat > the rules and boundaries and structures as constraints to be creatively > engaged as opposed to simply conforming, person centred work will be limited > to improving the conditions of people's confinement in services. He calls for leadership to challenge these boundaries: > Most service organisations have the social function of putting people to > sleep, keeping them from seeing the social reality that faces people with > disabilities...People go to sleep when the slogan that "we are doing the > best that is possible for 'them'" distracts from noticing and taking > responsibility for the uncountable losses imposed by service activities that > keep people idle, disconnected and alienated from their own purposes in > life. One way to understand leadership is to see it as waking up to people's > capacities and the organisational and systemic practices that devalue and > demean those capacities.
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If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω. If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β<α such that xι is in U for all ι≥β. Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 (omega-one, the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.
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