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Topic: Axiom of extensionality

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	Axiom of extensionality - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.
		The axiom given above assumes that equality is a primitive symbol in predicate logic.
		In this case, the usual axiom of extensionality would imply that every ur-element is equal to the empty set.
	en.wikipedia.org /wiki/Axiom_of_extensionality (537 words)

	Encyclopedia: Axiomatic set theory (Site not responding. Last check: 2007-10-07)
		Axiom of extensionality: Two sets are the same if and only if they have the same elements.
		Axiom of separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds.
		Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.
	www.nationmaster.com /encyclopedia/Axiomatic-set-theory (6130 words)

	PlanetMath: axiom of extensionality (Site not responding. Last check: 2007-10-07)
		The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory.
		Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.
		This is version 2 of axiom of extensionality, born on 2003-06-24, modified 2003-06-24.
	www.planetmath.org /encyclopedia/AxiomOfExtensionality.html (70 words)

	Axiom of union - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x.
		The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		A as {C in B : for all D in A, C is in D} using the axiom schema of specification.
	en.wikipedia.org /wiki/Axiom_of_union (310 words)

	Zermelo Fraenkel Set Theory (Site not responding. Last check: 2007-10-07)
		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.
		The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 (Zermelo 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.
	www.wikiverse.org /zermelo-fraenkel-set-theory (469 words)

	Axiom of extensionality (Site not responding. Last check: 2007-10-07)
		The axiom of extensionality is generally considered and it or an equivalent appears in about any alternative axiomatisation of set theory.
		Then it's necessary to the usual axioms of equality from predicate as axioms about this defined symbol and becomes these axioms that are referred to as axioms of extensionality.
		In this case usual axiom of extensionality would imply that ur-element is equal to the empty set.
	www.freeglossary.com /Axiom_of_extension (755 words)

	The Varied Sorrows of Logical Abstraction
		He suggested this might be achieved by translating language asserting mutual subordination into statements of the form 'the extension of the concept X is the same as the extension of the concept Y' in which the descriptions would then be regarded as proper names as indicated by the presence of the definite article.
		This specially designed axiom would be "equivalent to the assumption that 'any combination or disjunction of predicates is equivalent to a single predicate"' (Russell 1973, 250; 1927, 58-59), and would provide a way of dealing with any function of a particular argument by means of some formally equivalent function of a particular type.
		Extensionality, Marcus explains, has acquired the undeserved reputation of being a clear, unambiguous concept, and as such well-suited to the needs of mathematics and the empirical sciences where, it is claimed, there is no need to traffic in fuzzy, troublesome non-extensional notions.
	perso.wanadoo.fr /rancho.pancho/Varied.htm (10304 words)

	Axiomatic set theory - Wikipedia, the free encyclopedia
		For instance this method can be used to demonstrate the existence of large cardinals is not provable in ZFC (but it is essentially impossible to shown they are consistent).
		From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function, etc.
		For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists.
	en.wikipedia.org /wiki/Axiomatic_set_theory (2518 words)

	Set Theory:Axioms - Wikibooks
		Axiom of separation is sometimes called schema of separation, since it comprises infinitely many axioms - one for each condition P.
		The following axiom is somewhat of a convention but various models of set theory have been defined without it, or even using axiom stating things close to the opposite.
		It remains the most controversial axiom among mathematicians, and for that reason, when using ZF with the Axiom of Choice, it is often specified as ZFC.
	en.wikibooks.org /wiki/Set_Theory:Axioms (703 words)

	4Reference \|\| Axiom schema of specification (Site not responding. Last check: 2007-10-07)
		Thus, what the axiom schema is really saying is that, given a set A and a predicate P, we can find a subset B of A whose members are precisely the members of A that satisfy P.
		The axiom schema of specification is generally considered uncontroversial as far as it goes, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		Most of the other Zermelo-Fraenkel axioms (but not the axiom of extensionality or the axiom of regularity) then became necessary to serve as an additional replacement for the axiom schema of comprehension; each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy.
	www.4reference.net /encyclopedias/wikipedia/Axiom_schema_of_specification.html (1060 words)

	Positive set theory - Wikipedia, the free encyclopedia
		The axiom of infinity: the von Neumann ordinal ω exists.
		The axiom of closure: for every set x, a set exists which is the intersection of all sets containing x; this is called the closure of x and is written {x}.
		The axiom of empty set: there exists a set
	en.wikipedia.org /wiki/Positive_set_theory (178 words)

	Reference and Paradox
		There was nothing, he found, to stop one from transforming an equality holding between two concepts into an equality of extensions in conformity with the first part of his law, but from the fact that concepts are equal in extension one cannot infer that whatever falls under one falls under the other.
		They should be replaced with this axiom which seemed to him "to be the essence of the usual assumption of classes" and to retain "as much of classes as we have any use for, and little enough to avoid the contradictions" (Russell 1927,166-67; 1956, 82; 1919, 191).
		She called a principle extensional if it either "(a) directly, or indirectly imposes restrictions on the possible values of the functional variables such that some intensional functions are prohibited or (b) it has the consequence of equating identity with a weaker form of equivalence" (Marcus 1960, 46).
	perso.wanadoo.fr /rancho.pancho/Refer.htm (8818 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		The axiom of extensionality is just a statement: sets X and Y are identical if and only if they have the same members.
		This however is not a problem with the axiom of extensionality.
		We just can't use the axiom of extensionality to decide whether there are many sets a such that a={a}, but must decide this on other grounds.
	www.math.niu.edu /~rusin/known-math/00_incoming/extensionality (436 words)

	PlanetMath: axiom of pairing (Site not responding. Last check: 2007-10-07)
		The Axiom of Pairing is one of the axioms of Zermelo-Fraenkel set theory.
		Using the Axiom of Extensionality, we see that the set
		This is version 4 of axiom of pairing, born on 2003-06-24, modified 2003-06-24.
	www.planetmath.org /encyclopedia/AxiomOfPairing.html (94 words)

	Citations: Proofs in Higher Order Logics - Miller (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		Indeed, the axiom of choice is not provable in type theory [4] but its skolemized form is. In Miller proposes a Skolem like theorem for type theory: the symbol f is a Skolem s symbol of arity n.
		Whenever a Skolem s symbol f of arity n occurs in a term it must occur in a subterm of the form (f a 1 : an) and the free variables of the a i s cannot be bound higher in the term.
		It is possible to prove an instance of the axiom of choice that is known to be independent of HOL using naive Skolemization.
	citeseer.ist.psu.edu /context/93985/0 (3774 words)

	List of axioms -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		In (The philosophical theory of knowledge) epistemology, the word axiom is understood differently; see ((logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident) axiom and (Click link for more info and facts about self-evidence) self-evidence.
		Individual axioms are almost always part of a larger (Click link for more info and facts about axiomatic system) axiomatic system.
		Other axioms of (Any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity) mathematical logic
	www.absoluteastronomy.com /encyclopedia/L/Li/List_of_axioms.htm (956 words)

	Mereology
		A second familiar objection is familiar from the literature on material constitution, where the principle of mereological extensionality is sometimes taken to contradict the possibility that an object may be distinct from the matter constituting it.
		But their independent motivation also bears witness to the fact that the controversies about extensionality, and particularly about (40), stem from genuine and fundamental philosophical conundrums and cannot be assessed by appealing to our intuitions about the meaning of ‘part’.
		The intuitive idea behind these two axioms is best appreciated in the presence of extensionality, for in that case the entities whose conditional existence is asserted by (P.6) and (P.7) must be unique.
	setis.library.usyd.edu.au /stanford/entries/mereology (9661 words)

	Extension %28semantics%29 (Site not responding. Last check: 2007-10-07)
		UCI Extension is the continuing education branch of the renowned University of California, Irvine.
		A proposal for an extension of I-83 as a bypass of Washington, DC.
		Extension service offering educational programs delivered by county extension educators, and focus on the strengthening and sustaining families within the community.
	www.omniknow.com /common/wiki.php?in=en&term=Extension_%28semantics%29 (1130 words)

	Extensionality (Site not responding. Last check: 2007-10-07)
		In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them.
		In axiomatic set theory, extensionality is expressed in the axiom of extension, which states that two sets are equal if and only if they contain the same elements.
		In lambda calculus, extensionality is expressed by the eta-conversion rule, which allows conversion between any two expressions that denote the same function.
	www.wikiverse.org /extensionality (96 words)

	ZermeloFraenkel set theory - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-10-07)
		, are the standard axioms of axiomatic_set_theory on which, together with the axiom_of_choice, all of ordinary mathematics is based.
		The axioms are the result of work by Thoralf_Skolem in 1922, based on earlier work by Adolf_Fraenkel in the same year, which was based on the axiom system put forth by Ernst_Zermelo in 1908 (Zermelo_set_theory).
		The axiom system has an infinite number of axioms because an axiom_schema is used.
	www.indexsuche.com /Zermelo-Fraenkel_set_theory.html (432 words)

	Olivier Esser's homepage (Site not responding. Last check: 2007-10-07)
		KM is the Kelley-Morse class theory, ''On has the tree property'' is the natural translation to the class of ordinals of the corresponding property for cardinals in ZF.
		Abstract: This is the study of the relative interpretability of the axiom of extensionality in the positive set theory.
		This work has to be considered in the line of works of R.~O.~Gandy, D.~Scott and R.~Hinnion who have studied the relative interpretability of the axiom of extensionality in set theories of Zermelo and Zermelo-Fraenkel.
	homepages.vub.ac.be /~oesser/publications.html (983 words)

	axiom of extensionality - OneLook Dictionary Search (Site not responding. Last check: 2007-10-07)
		We found 3 dictionaries with English definitions that include the word axiom of extensionality:
		Tip: Click on the first link on a line below to go directly to a page where "axiom of extensionality" is defined.
		Axiom of Extensionality : Eric Weisstein's World of Mathematics [home, info]
	www.onelook.com /?w=axiom+of+extensionality&ls=a (88 words)

	why Bill-semantics is not DS
		So Bill said that transfinite induction (the axiom of choice, the "C" in ZFC) is not a separate axiom, and is not independent of the others, but is a consequence of the basic ZFC axioms.
		I've known about the ZFC axiom of extensionality for years: 2 sets are equal iff they have the same elements.
		The ZF subset/comprehension axiom says you can define subsets by functions, and these functions are formulas in the language of set theory, so in particular they can involve the quantifiers \forall & \exists.
	www.codecomments.com /showthread.php?threadid=235806&perpage=10&pagenumber=5 (3638 words)

	Metamath Proof Explorer - ax-ext (Site not responding. Last check: 2007-10-07)
		I suppose this is a philosophical issue: Under traditional predicate calculus, the equality-free version of Extensionality moves the properties of equality into set theory.
		But proper substitution has an intimate tie-in with equality that is implicitly "hidden" in the traditional axioms of predicate calculus.
		The axiomatics of a Metamath-style system with equality-free Extensionality, including devising simple, elegant axioms, have apparently never been studied nor even considered.
	metamath.planetmirror.com /mpegif/ax-ext.html (217 words)

	Metamath Proof Explorer - df-cleq (Site not responding. Last check: 2007-10-07)
		This is an example of a somewhat "risky" definition, because it extends the use of the existing equality symbol rather than introducing a new symbol, allowing us to make statements in the original language that may not be true.
		which is not a theorem of logic but rather presupposes the Axiom of Extensionality, which we must therefore include as a hypothesis.
		We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play.
	metamath.planetmirror.com /mpegif/df-cleq.html (178 words)

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