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Topic: Von Neumann cardinal assignment

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	Von Neumann cardinal assignment - Wikipedia, the free encyclopedia
		The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers.
		For a well-ordered set U, we define its cardinal number to be the smallest ordinal number equinumerous to U.
		With the full Axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers.
	en.wikipedia.org /wiki/Von_Neumann_cardinal_assignment (138 words)

	Cardinality - Wikipedia, the free encyclopedia
		In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
		Any set that has the same cardinality as the set of the natural numbers is said to be a countably infinite set.
		The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.
	en.wikipedia.org /wiki/Cardinality (682 words)

	Cardinal number - Open Encyclopedia (Site not responding. Last check: 2007-10-21)
		In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third).
		This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle.
		It can also be proved that the cardinal $\aleph_0 (aleph-0, where aleph is the first letter in the Hebrew alphabet, represented by the Unicode character א) of the set of natural numbers is the smallest infinite$ cardinal, i.e., that any infinite set admits a subset of cardinality $\aleph_0 .$
	open-encyclopedia.com /Cardinal_number (1899 words)

	Cardinal number
		He called these cardinal numbers transfinite cardinal numbers, and defined all sets that had a one-to-one correspondence with N to be denumerably infinite setss.
		Formally, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.
		It can also be proved that the cardinal (aleph-0, where aleph is the first letter in the Hebrew alphabet, represented by the Unicode character א) of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality.
	pedia.newsfilter.co.uk /wikipedia/c/ca/cardinal_number.html (1913 words)

	Station Information - Cardinal assignment
		In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets which are equinumerous).
		This is in accordance to Cantor's original vision of a cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about this set is its size.
		In modern set theory, we usually use the Von Neumann cardinal assignment which uses the theory of ordinal numbers and the full power of the Axioms of choice and replacement.
	www.stationinformation.com /encyclopedia/c/ca/cardinal_assignment.html (343 words)

	Cardinal number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		In (The scientific study of language) linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to (The number designating place in an ordered sequence) ordinal numbers, words that are used for order (first, second, third).
		He called these cardinal numbers (Click link for more info and facts about transfinite cardinal numbers) transfinite cardinal numbers, and defined all sets that had a one-to-one correspondence with N to be (Click link for more info and facts about denumerably infinite set) denumerably infinite sets.
		The latter cardinal number is also often denoted by c; it is the (Click link for more info and facts about cardinality of the continuum) cardinality of the continuum (the set of (Any rational or irrational number) real numbers).
	www.absoluteastronomy.com /encyclopedia/C/Ca/Cardinal_number.htm (2281 words)

	Cardinal number - Wikipedia, the free encyclopedia
		In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of numbers used to denote the size of a set.
		A cardinal which is not infinite is called finite; it can then be proved that the finite cardinals are just the natural numbers, i.e., that a set X is finite if and only if X
		The latter cardinal number is also often denoted by c; it is the cardinality of the continuum (the set of real numbers).
	www.wikipedia.com /wiki/cardinal+number (1907 words)

	Learn more about Cardinal number in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		The intuitive idea of a cardinal is to create some notion of the relative size or "bigness" of a set without reference to the kind of members which it has.
		This is easily visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.
		It can also be proved that the cardinal (aleph-0, where aleph is the first letter in the Hebrew alphabet, represented by the Unicode character and#1488;) of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality.
	www.onlineencyclopedia.org /c/ca/cardinal_number.html (1208 words)

	Station Information - Successor cardinal
		In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers.
		Using the von Neumann cardinal assignment and the axiom of choice (AC), this successor operation is easy to define: for a cardinal number κ we have
		Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals.
	www.stationinformation.com /encyclopedia/s/su/successor_cardinal.html (217 words)

	cardinal
		In Catholicism, a cardinal is a prince of the Church; cardinals are appointed by the Pope and collectively elect the next Pope if they are under the age of eighty at the time of the election.
		In biology, a cardinal is the songbird Cardinalis cardinalis.
		Cardinal is a bright red color like the color of the robe of a cardinal.
	www.fact-library.com /cardinal.html (225 words)

	Cardinal number (Site not responding. Last check: 2007-10-21)
		Since mathematics is concerned with infinite objects, a study of cardinality tries to discuss the size of infinite sets.
		It is also a area studied for its own sake as part of set theory, particularly in trying to describe the properties of large cardinals.
		The cardinal numbers were invented by George Cantor, when he was developing the Naïve set theory in 1874–1884.
	www.sciencedaily.com /encyclopedia/cardinal_number (1869 words)

	From Frege To Godel: von Heijenoort (Site not responding. Last check: 2007-10-21)
		von Neumann was the first to define the ordinals in the modern way.
		von Neumann introduced his own axiomatization of set theory based on functions as objects (some of which correspond to sets).
		von Neumann carries this out assuming the notions of well-orderedness and similarity, though this is unnecessary because the ordinals can be defined directly using only the membership relation.
	www.andrew.cmu.edu /~cebrown/notes/vonHeijenoort.html (8419 words)

	More on Cardinal Numbers
		Naming this cardinal number \aleph_0, aleph-null, Cantor proved that many subsets of N have the same cardinality as N, even if this might be against intuition at first.
		Cantor also developed a lot of the general theory of cardinal numbers; he proved that there is a transfinite cardinal number that is the smallest (\aleph_0, aleph-null) and that for every cardinal number, there is a next-larger cardinal (\aleph_1, \aleph_2, \aleph_3, \cdots).
		It can also be proved that the cardinal \aleph_0 (aleph-0, where aleph is the first letter in the Hebrew alphabet, represented א) of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality \aleph_0.
	www.artilifes.com /cardinal-numbers.htm (2039 words)

	Introduction (Site not responding. Last check: 2007-10-21)
		von Neumann then realized that sets having these two properties had exactly the properties of 'ordinal numbers' as originally defined by Cantor, so that (i) and (ii) can be taken as the definition of the notion of ordinal number.
		The von Neumann representation ties the ordinal concept very directly to the most basic concepts of set theory, allowing the properties of ordinals to be established by reasoning that uses only elementary properties of sets and set formers, with occasional use of transfinite induction.
		The cardinality of a set is defined as the smallest ordinal which can be put into 1-1 correspondence with the set, and it is proved that (a) there is only one such ordinal, and (b) this is also the smallest ordinal which can be mapped onto s by a single-valued map.
	www.settheory.com /intro.html (18848 words)

	Knowledge King - Limit ordinal (Site not responding. Last check: 2007-10-21)
		In general, we always get a limit ordinal when taking the union of a set of ordinals that has no maximum element.
		Limit ordinals are usually a kind of "turning point" in which we have to use limiting operations such as taking the union over all preceding ordinals (technically we could do anything at limit ordinals, but taking the union is continuous in the order topology and usually this is what we want).
		Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level!).
	www.knowledgeking.net /encyclopedia/l/li/limit_ordinal.html (416 words)

	CS107, Fall 2003, Cardinal Stritch University
		Students of a college age realize that education is important; students who attend Cardinal Stritch University have made a commitment to their education; thus I expect that students will attend class as a matter of course.
		This especially impacts group assignments: everyone is expected to do their part for a group assignment; those who do not will not get credit.
		Cardinal Stritch University and the instructor wish to positively affirm the intent of federal law, the Rehabilitation Act of 1973, Sec.
	faculty.stritch.edu /atamulis/Fa03/CSIntro (1133 words)

	Large cardinal -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Therefore the discussion of large cardinals takes place in a realm of (Click link for more info and facts about conditional proof) conditional proofs, which (according to the consensus view of logicians) will remain so.
		The following is a list of some types of large cardinals; it is arranged in order of the consistency strength.
		Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for all listed cardinal descriptions φ of lesser consistency strength, V(κ) satisfies "there are unboundedly many cardinals satisfying φ".
	www.absoluteastronomy.com /encyclopedia/l/la/large_cardinal.htm (568 words)

	Game Theory
		von Neumann and Morgenstern (1947), and was an essential aspect of their invention of game theory.
		Prior to the work of von Neumann and Morgenstern (1947), situations of this sort were inherently baffling to analysts.
		A crucial aspect of von Neumann and Morgenstern's (1947) work was the solution to this problem.
	plato.stanford.edu /entries/game-theory (20493 words)

	1.9 Abstractions for Instructing Machines--Program Structures
		The fundamental concepts of the modern computing machine were enunciated by John Von Neumann in the late 1940s.
		The execution of simultaneous von Neumann machines all cooperating toward the achievement of a single controlling master task is called parallelism or parallel processing.
		The cardinal sin in programming is sitting down at the terminal or microcomputer with nothing but the statement of the problem, and beginning to write code.
	www.arjaybooks.com /Modula-2/Text/Ch1/Ch1.9.html (2634 words)

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	Cardinal Numbers Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-21)
		Looking For cardinal numbers - Find cardinal numbers and more at Lycos Search.
		Find cardinal numbers - Your relevant result is a click away!
		Look for cardinal numbers - Find cardinal numbers at one of the best sites the Internet has to offer!
	www.karr.net /encyclopedia/Cardinal_numbers (2085 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The assignment problem is the allocation problem where n indivisible objects are to be allocated among n agents, and each agent is to receive exactly one object.
		We introduce the new concept of ordinal efficiency in the random assignment problem: it relies on the stochastic dominance relation induced by individual preferences over sure objects; it is more demanding than ex post efficiency and less demanding than ex ante (i.e., with respect to expected utility functions) efficiency.
		Moreover, the PS assignment is nonenvious ex ante, whereas the RP one is nonenvious in the ordinal sense only.
	wwwpub.utdallas.edu /%7Edarcy/SEM/seminarspring2000.html (1456 words)

	GÖDEL, CANTOR AND PLATO
		There is no intermediate cardinality for it to have (and in general there are no cardinal numbers other than Alephs), nor has the full set of real numbers a higher cardinality than Aleph one.
		Since this is an axiom, a set theory can be propounded that does not accept it, and so the space for cardinal numbers that are not Alephs coincides quite simply with ‘non-well-orderable’ sets (which being sets are consistent) in a set theory which rejects the axiom of choice.
		As such, it must have a genuine cardinality, and this must be greater than (since it is different from and cannot be less than) Aleph zero.
	www.wittgenstein.internet-today.co.uk /plato.html (8934 words)

	Cardinality Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-21)
		Looking For cardinality - Find cardinality and more at Lycos Search.
		Find cardinality - Your relevant result is a click away!
		Look for cardinality - Find cardinality at one of the best sites the Internet has to offer!
	www.karr.net /encyclopedia/Cardinality (854 words)

	Game Theory
		Game theory is the mathematical theory of bargaining, the essentials of which were developed by John Von Neumann and Oskar Morgenstern in their book The Theory of Games and Economic Behavior (1947).
		Von Neumann and Morgenstern restricted their attention to zero-sum games, that is, to games in which no player can gain except at another's expense.
		Then we may examine the ratios between the probabilities associated with her maximizing the acquisition of bundles high on her ordinal utility function and the amounts she is willing to pay for each gamble, and derive her cardinal utility function from these ratios.
	setis.library.usyd.edu.au /stanford/archives/fall1997/entries/game-theory (2528 words)

	Star Wars and Littleton, by Lyndon H. LaRouche, Jr. (Jun. 11, 1999)
		The big difference is, that the market players, also using John von Neumann's game theory, usually kill many more, and much more efficiently, not by the sword or gun, but demographically.
		It is the influence of the perverted, late John von Neumann's "theory of games," in fostering the use of intrinsically immoral practices in the form of so-called zero-sum games.
		That is the reason why the efforts of the followers of Norbert Wiener and John von Neumann to develop "artificial intelligence" will always remain anti-scientific quackery, whether at Massachusetts Institute of Technology (MIT), or elsewhere.
	www.larouchepub.com /lar/1999/lar_littleton_2627.html (15177 words)

	Management to Mathematics
		Three-term course on business practices employing a standard case study approach and including a major assignment incorporating oral and written reports and small group language tutorials gauged to individual student needs.
		Axiomatic set theory as framework for mathematical concepts; relations and functions, numbers, cardinality, axiom of choice, transfinite numbers.
		Set theory: Zermelo/Fraenkel and von Neumann/Gödel axioms; cardinal and ordinal numbers; continuum hypothesis; constructible sets; independence results and forcing.
	www.registrar.ucla.edu /Archive/catalog/1995_97/M2.htm (2671 words)

	Encyclopedia: Successor cardinal (Site not responding. Last check: 2007-10-21)
		Updated 266 days 22 hours 39 minutes ago.
		Cardinals which are not successor cardinals are called limit cardinals; and by the above definition, if λ is a limit ordinal, then
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/successor-cardinal (262 words)

	SUCCESSOR OPERATION
		When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one.
		Using von Neumann's ordinal numbers (the standard ordinals used in set theory), we have, for any ordinal number,
		It is immediate that there is no ordinal number between α and S(α) and with the ordering on the ordinal numbers α
	www.websters-online-dictionary.org /definition/SUCCESSOR+OPERATION (321 words)

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