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	PlanetMath: von Neumann-Bernays-Gödel set theory
		This theory is essentially stronger than ZFC or NBG, as it can prove their consistency (in addition to everything they already prove).
		by means of the Burali-Forti paradox, that the class of all ordinals is not a set, and hence there is a bijection between the class of ordinals and the class of all sets.
		This is version 12 of von Neumann-Bernays-Gödel set theory, born on 2003-06-25, modified 2004-09-22.
	www.planetmath.org /encyclopedia/VonNeumannBernausGodelSetTheory.html (802 words)

	Axiomatic set theory - Wikipedia, the free encyclopedia
		Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century.
		Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.
		The most frequent objection to set theory is the constructivist view that mathematics is loosely related to computation and that naive set theory is being formalised with the addition of noncomputational elements.
	en.wikipedia.org /wiki/Axiomatic_set_theory (2633 words)

	Set theory - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-15)
		Set theory is the mathematical theory of sets, which represent collections of abstract objects.
		Naive set theory is the original set theory developed by mathematicians at the end of the 19th century
		Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's Paradox) in naive set theory.
	encyclopedia.learnthis.info /s/se/set_theory_1.html (139 words)

	Proof Explorer - Home Page - Metamath
		Set theorists often study the consequences of additional axioms that assert, for example, the existence of larger and larger infinities beyond those postulated by ZFC or even Grothendieck's Axiom, to the point of flirting with inconsistency (although Gödel also showed that we can never know whether even the ZFC axioms are consistent).
		In the set theory database file, set.mm, we adopt the convention that at least one set variable always appears in a distinct variable pair, so these are the only cases you will see.
		In particular, the (red) "set variables" in Metamath's axioms are not the individual variables of actual axioms; instead, they are metavariables ranging over these individual variables (which in turn range over the individuals in the logic's universe of discourse—in our case of set theory, the universe of discourse is the collection of all sets).
	metamath.planetmirror.com /mpegif/mmset.html (8567 words)

	Zermelo-Fraenkel set theory - Wikipedia, the free encyclopedia
		The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory.
		Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
		On the other hand, the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal.
	en.wikipedia.org /wiki/Zermelo-Fraenkel_set_theory (538 words)

	Set at opensource encyclopedia (Site not responding. Last check: 2007-10-15)
		If a set has n elements, where n is a natural number (possibly 0), then the set is said to be a finite set with cardinality n; otherwise it is said to be an infinite set.
		The "number of elements" in a certain set is called the cardinal number of the set and denoted $A$ for a set $A$ (for a finite set this is an ordinary number, for an infinite set it differentiates between different "degrees of infiniteness", named $\aleph_0$ (aleph zero), $\aleph_1, \aleph_2...$ ).
		The set of functions from a set A to a set B is sometimes denoted by B
	wiki.tatet.com /Set.html (1265 words)

	PlanetMath: von Neumann-Bernays-Gödel set theory
		This theory is essentially stronger than ZFC or NBG, as it can prove their consistency (in addition to everything they already prove).
		by means of the Burali-Forti paradox, that the class of all ordinals is not a set, and hence there is a bijection between the class of ordinals and the class of all sets.
		This is version 12 of von Neumann-Bernays-Gödel set theory, born on 2003-06-25, modified 2004-09-22.
	planetmath.org /encyclopedia/VonNeumannBernausGodelSetTheory.html (795 words)

	2 Set Theory
		Axiomatic set theory has its origins in the paradoxes that plagued the naive set theory used at the turn of the century, and was first created by Zermelo in 1908 to avoid these paradoxes, but it has a greater use and appeal than its origins might suggest.
		Set theory has become the basis for almost all mathematics, so its axiomatization has a foundational importance.
		The group of axioms for both the prominent set theories is small, and mostly states the assumptions of naive set theory precisely, but there are fundamental differences, eliminating the paradoxes of the past.
	www.u.arizona.edu /~miller/thesis/node5.html (445 words)

	Encyclopedia: ZFC set theory (Site not responding. Last check: 2007-10-15)
		The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations.
		An alternative, finite system is given by the von Neumann-Bernays-Gödel axioms (NBG), which add the concept of a classes in addition to that of a set; it is "equivalent" in the sense that any theorem about sets which can be proved in one system can be proven in the other.
		Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y ∪ {y}.
	www.nationmaster.com /encyclopedia/ZFC-set-theory (546 words)

	[No title]
		This is an arrangement that focuses on the paradoxes contained in the naïve set theory, the iterative conception of set.
		On the other hand, it appears that an ontology cannot be satisfied with the restrictions established by the ZFC axiomatic.
		j(a, b, c) means that the set of the free variables of j is a part of {a, b, c}.
	membres.lycos.fr /laurentduboislaurent/zfset.htm (456 words)

	epsilon and omega (Site not responding. Last check: 2007-10-15)
		For all sets x consisting of pairwise disjoint non-empty sets there is a set y which has exactly one element in common with every element of x.
		The study of natural extensions of ZFC is one of the main areas of modern set theory.
		Regardless of this, the study of extensions of ZFC is fruitful: Surprising interactions with other fields of mathematics have emerged (the area of left distributivity is an example), and a canonical axiom system for second order number theory has been identified, which might be regarded as the second order counterpart of the Peano Axioms.
	www.mathematik.uni-muenchen.de /~deiser/set.html (1423 words)

	Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks
		set theory, axioms, Zermelo, Fraenkel, Frankel, infinity, Cantor, Frege, Russell, paradox, formal, axiomatic, Russell paradox, axiom, axiomatic set theory, comprehension, axiom of infinity, ZF, ZFC
		The set theory adopting the axiom of extensionality (C1), the axiom C1', the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF.
		The set theory ZF+AC is denoted traditionally by ZFC.
	www.ltn.lv /~podnieks/gt2.html (8336 words)

	Axiom of Choice
		A mathematically complete statement of the above requires a definition in the language of set theory of function.
		A function is a set of ordered pairs where the first element is in the domain of the function and the second element is in the range of the function.
		Figure 6.4: The axioms of ZFC set theory
	www.mtnmath.com /whatrh/node57.html (328 words)

	03E: Set theory
		Fuzzy set theory replaces the two-valued set-membership function with a real-valued function, that is, membership is treated as a probability, or as a degree of truthfulness.
		The theory of finite sets is, arguably, a definition of Combinatorics.
		Since Axiomatic Set Theory is often used to construct the natural numbers (satisfying the Peano axioms, say) it is possible to translate statements about Number Theory to Set Theory.
	www.math.niu.edu /~rusin/known-math/index/03EXX.html (1585 words)

	Math Forum Discussions
		> theory, though this fact is somewhat mysterious.
		Re: ordinal strength of a theory - and large cardinals
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/message.jspa?messageID=4278401&tstart=390 (446 words)

	Indispensability Arguments in the Philosophy of Mathematics
		Once one rejects the picture of scientific theories as homogeneous units, the question arises whether the mathematical portions of theories fall within the true elements of the confirmed theories or within the idealized elements.
		Her reason for this is that scientists themselves do not seem to take the indispensable application of a mathematical theory to be an indication of the truth of the mathematics in question.
		That is, set theorists should be assessing the new axiom candidates with one eye on the latest developments in physics.
	setis.library.usyd.edu.au /stanford/entries/mathphil-indis (4408 words)

	Hilbert Space Explorer Home Page
		(the set of complex numbers cc) is defined as a specific set of set theory.
		The set of closed subspaces of Hilbert space obey the laws of a simple equational algebra called "orthomodular lattice algebra." This algebra is sometimes called "quantum logic," and it can be used as the basis for a propositional calculus that resembles but is somewhat weaker than standard (classical) propositional calculus.
		The orthocomplement of a subspace is the set of vectors orthogonal to all vectors in the subpace.
	us.metamath.org /mpegif/mmhil.html (2104 words)

	Zermelo-Fraenkel set theory : ZFC (Site not responding. Last check: 2007-10-15)
		The Zermelo-Fraenkel axioms of set theory, denoted ZF, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based.
		An equivalent finite alternative system is given by the von Neumann-Bernays-Gödel axioms[?] (NBG), which distinguish between classes and sets.
		Axiom of extension: Two sets are the same if and only if they have the same elements.
	www.factbase.info /zf/zfc.html (403 words)

	Inaccessible Set Axioms May Have Little Consistency Strength (ResearchIndex) (Site not responding. Last check: 2007-10-15)
		The paper investigates inaccessible set axioms and their consistency strength in constructive set theory.
		In ZFC inaccessible sets are of the form V where is a strongly inaccessible cardinal and V denotes the - th level of the von Neumann hierarchy.
		Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory.
	citeseer.ist.psu.edu /384300.html (357 words)

	Kids.net.au - Encyclopedia Set theory - (Site not responding. Last check: 2007-10-15)
		Naive set theory is the original set theory developed by mathematicians.
		Axiomatic set theory is a rigorous axiomatic set theory developed in response to the discovery of serious flaws (such as Russell's Paradox) in naive set theory.
		This is a disambiguation page; that is, one that just points to other pages that might otherwise have the same name.
	www.kids.net.au /encyclopedia-wiki/se/Set_theory (142 words)

	Reverse mathematics (Site not responding. Last check: 2007-10-15)
		It turns out that over a weak base theory, many mathematical statements are equivalent to the particular additional axiom needed to prove them.
		Most relevant sets of real numbers, including all Borel sets, can be coded by real numbers with the membership relation expressible in second order arithmetic.
		The primary difference between doing classical mathematics in set theory (ZFC) and doing it in second order arithmetic is that in second order arithmetic one deals with codes for sets rather than sets themselves (except sets of integers).
	www.sciencedaily.com /encyclopedia/reverse_mathematics (294 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		These will include von Neumann-G\"odel-Bernays (predicative) class theory, Kelley-Morse (impredicative) class theory, the set theory of Ackermann with sets and classes and powerful extensions of Jensen's NFU (New Foundations with urelements) which may not seem at first glance to be superstructures on ZFC.
		Positive set theory may also be discussed, if time permits.
		The idea of viewing set theories with universal set such as NFU or positive set theory as alternative superstructures to the usual theory of classes has been discussed earlier by Roland Hinnion.
	math.boisestate.edu /~best/best7/talks/randall.html (119 words)

	Descriptive Set Theory. By K.Podnieks (Site not responding. Last check: 2007-10-15)
		In the descriptive set theory the meaning of "simple", "definable" sets (of real numbers) is defined explicitly by introducing the so-called Borel sets and projective sets.
		The class of projective sets is closed is closed under finite unions, finite intersections, and inverse continuous images, yet (unlike the class of Borel sets) it is not closed under countable unions and countable intersections.
		be the set of all sets of pairs of natural numbers.
	www.ltn.lv /~podnieks/gtaa.html (1770 words)

	PlanetMath: Zermelo-Fraenkel axioms
		Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a foundation for what is now called Zermelo-Fraenkel set theory, or ZF.
		If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC.
		Union over a set: If is a set, then there exists a set that contains every element of each
	planetmath.org /encyclopedia/ZermeloFraenkelAxioms.html (215 words)

	Abstract of Consistency paper (Site not responding. Last check: 2007-10-15)
		A kappa-denotational semantics for Map Theory in ZFC+SI Chantal Berline and Klaus Grue
		It is based on lambda-calculus instead of logic and sets, and it fulfills Church's original aim of introducing lambda-calculus.
		The lower bound is proved in \cite{grue92} by means of a syntactical translation of ZFC (including classical propositional and predicate calculus) into map theory, and the upper bound by building an (exceedingly complex) model of map theory within ZFC+SI.
	www.diku.dk /~grue/papers/kappa/kappa.html (325 words)

	Hajnal & Hamburger's Set Theory Book Site
		This is a classical introduction to set theory in three parts.
		The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required.
		An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part.
	www.ipfw.edu /math/Hamburger/book.html (191 words)

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