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Topic: Axiom of regularity

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	Axiom of regularity - Wikipedia, the free encyclopedia
		The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory.
		The axiom of regularity is arguably the least useful ingredient of Zermelo-Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity.
		The axiom of regularity is irrelevant to the resolution of Russell's paradox.
	en.wikipedia.org /wiki/Axiom_of_regularity (719 words)

	Axiom schema of specification - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory.
		The axiom schema of specification is characteristic of systems of axiomatic set theory related to the usual set theory ZFC, but does not usually appear in radically different systems of alternative 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.
	en.wikipedia.org /wiki/Axiom_schema_of_specification (1005 words)

	Zermelo-Fraenkel set theory : ZFC
		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.
		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.factspider.com /zf/zfc.html (628 words)

	PlanetMath: axiom of foundation
		The axiom of foundation (also called the axiom of regularity) is an axiom of ZF set theory prohibiting circular sets and sets with infinite levels of containment.
		It is known that, if ZF without this axiom is consistent, then this axiom does not add any inconsistencies.
		This is version 4 of axiom of foundation, born on 2002-09-28, modified 2005-11-18.
	planetmath.org /encyclopedia/AxiomOfRegularity.html (113 words)

	Axiom of regularity (Site not responding. Last check: 2007-10-06)
		''Axiom of regularity implies that no set is an element of itself''Let ''A'' be a set such that ''A'' is an element of itself and define ''B'' = {''A''}, which is a set by the [[axiom of pairing]].
		Thus ''B'' does not satisfy the axiom of regularity and we have a contradiction, proving that ''A'' cannot exist.''Axiom of regularity implies that no infinite descending sequence of sets exists''Let ''f'' be a [[function (mathematics)function]] of the natural numbers with ''f''(''n''+1) an element of ''f''(''n'') for each ''n''.
		This is a contradiction, hence no such ''f'' exists.''Assuming the axiom of choice, no infinite descending sequence of sets implies the axiom of regularity''Let the non-empty set ''S'' be a counter-example to the axiom of regularity; that is, every non-empty element ''s'' of ''S'' has a non-empty intersection with ''S''.
	axiomofregularity.quickseek.com (398 words)

	Paradox Solutions (Site not responding. Last check: 2007-10-06)
		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 replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.
		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.
	students.odl.qmul.ac.uk /~roger10/ODL122/page4.html (515 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.
		This is version 13 of Zermelo-Fraenkel axioms, born on 2001-10-18, modified 2004-02-18.
	planetmath.org /encyclopedia/ZermeloFraenkelAxioms.html (216 words)

	Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter
		The Axiom of Choice is the most well-known nonconstructive assertion of existence; it has important consequences for many branches of mathematics.
		The Axiom of Foundation (also known as the Axiom of Regularity) is also nonconstructive, but it has few applications in ``ordinary'' mathematics (i.e., outside of set theory).
		A couple of very weak consequences of the Axiom of Choice are the existence of (i) subsets of R which are not Lebesgue measurable, and (ii) subsets of R which lack the Baire property.
	at.yorku.ca /z/a/a/b/18.htm (848 words)

	Introduction
		In conventional terms, this axiom states a very strong form of the so-called 'axiom of choice': arb chooses a first element from each nonempty set, 'first' in the sense that there exists no other element of s which is also an element of arb(s).
		As with the axioms, this rule is to be understood as a template, covering all of its substituted instances.
		Note that axioms (xviii-xx) ensure that the equivalence relator '*eq' has the same transitivity, symmetry, and reflexivity properties as equality, while (xi-xiii) allow us to replace any subexpression of an expression formed using only the three operators and, or, not by any equivalent subexpression.
	www.settheory.com /intro.html (18848 words)

	[No title]
		On page 269, Moore notes that Zermelo's new axiom system of 1930 contained the Axiom of Foundation, "probably adopted from von Neumann but perhaps stated independently", in two formulations, (1) "there is no infinite descending epsilon-sequence" (A contains B contains C etc.) and (2) every nonempty set A contains an epsilon-minimal element, i.e.
		The proof appeared in his article "The axiom of Fundierung and the axiom of choice", Archiv für mathematische Logik und Grundlagenforschung, volume 4, pp.
		In the index of Moore's book, under Axiom of Foundation, page 266 should have been cited since it mentions von Neumann's introduction of this axiom, although the axiom is not named but just called "an axiom which prohibited infinite descending epsilon-sequences".
	www.rpi.edu /dept/cogsci/regularity.axiom (519 words)

	Wikinfo \| Set (Site not responding. Last check: 2007-10-06)
		Cantor's theorem states that the cardinality of the set of all subsets of a set A is strictly greater than the cardinality of A itself.
		For a discussion of the properties and axioms concerning the construction of sets, see naive set theory and axiomatic set theory.
		In ZFC without the axiom of regularity, the possibility of non-well-founded sets arises.
	www.wikinfo.org /wiki.php?title=Set (1227 words)

	ZFC and Russell's Paradox
		What it does have in its stead is an axiom that every predicate can define a subset of an existing set, plus a few other axioms to make up for the lost functionality.
		A model of the group axioms is a group, but the elements in it are group elements.
		A model of the axioms of vector space is a vector space and the elements in it are vectors.
	www.physicsforums.com /showthread.php?t=51980 (1896 words)

	[No title] (Site not responding. Last check: 2007-10-06)
		Here are all the axioms in ZFC, one of the most popular modern mathematical theories.
		Axiom of Exstensionality: Two sets are equal iff they have the same elements.
		EN: 0 e N & (x e N => x U {x} e N) Axiom of Regularity: Given any nonempty set, there is an element of the set whose intersection with the set is empty.
	www.math.niu.edu /~rusin/known-math/98/zfc (300 words)

	[No title] (Site not responding. Last check: 2007-10-06)
		The axiom asserts that for any given set x, there is a set y which has as members all of the members of all of the members of x.
		Alternatively, if the axiom is defined as a biconditional, then it is assuming the theory is extending a predicate logic with equality defined as a predicate constant The Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.
		Axiom: (x(y(u (u(y) ((v(v(u (v(x)) The Infinity Axiom states the existence of at least one infinite set x, from which other infinite sets can be formed.
	www.utdallas.edu /~kcooper/teaching/2305/set/Aset.doc (1104 words)

	Metamath Proof Explorer Home Page (Site not responding. Last check: 2007-10-06)
		When an axiom or theorem with a distinct variable condition is referenced in a proof, the distinct variable conditions attached the theorem being proved must satisfy those of the referenced axiom or theorem after substitutions are made into the referenced axiom or theorem.
		The first three are the axiom and rule schemes for traditional predicate calculus, and the last two are the axiom schemes for the traditional theory of equality.
		Although in some sense the traditional axiom schemes are more compact than Metamath's ax-4 through ax-16, their goal is simply to provide recipes for generating actual axioms, from which we then prove actual theorems.
	metamath.planetmirror.com /mpegif/mmset.html (9063 words)

	PlanetMath: Russell's paradox
		The regularity axiom, one of the Zermelo-Fraenkel axioms, was devised to avoid this paradox by prohibiting self-swallowing sets.
		An interpretation of Russell paradox without any formal language of set theory could be stated like ``If the barber shaves all those who do not themselves shave, does he shave himself?''.
		If you answer someone else that is also false because he shaves all those who do not themselves shave and in this case he is part of that set since he does not shave himself.
	planetmath.org /encyclopedia/RussellsParadox.html (158 words)

	paradoxes of set theory (Site not responding. Last check: 2007-10-06)
		Every instance of the axiom of specification specifies a different specification set because otherwise z would be the set of all sets.
		Therefore the axiom scheme of specification contradicts the axiom of regularity.
		The axiom of specification fares no better than the axiom of comprehension and we will be proposing that the latter be reinstated bec ause it is a less ad hoc axiom than specification.
	alex.edfac.usyd.edu.au /chatrooms/Maths/267726716.html (518 words)

	Undecidable Questions (Site not responding. Last check: 2007-10-06)
		Axioms of Replacement: Let F(x,y) be any wff, and let F(x,z) be the result of substitution z for y in all the latter’s free occurrences in F(x,y).
		A(t) is an axiom, where x is a variable, A(x) a wff, and t is a term free for x in A(x).
		A’)) is an axiom, where x and y are variables, the wff A’ is obtained by substituting y for one or more free occurrences of x in the wff A, and y is free for x in the occurrences of x it replaces.
	www.arches.uga.edu /~mathclub/godel_notes.html (3186 words)

	Metamath Proof Explorer - mmtheorems27 (Site not responding. Last check: 2007-10-06)
		A member of an ordinal class is not equal to it.
		This lemma is needed for ordon 2640 in order to eliminate the need for the Axiom of Regularity.
		The class of all ordinal numbers is ordinal.
	metamath.planetmirror.com /mpegif/mmtheorems27.html (1017 words)

	How do you see the world?: December 2004
		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.
		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}.
		Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y1) and P(x,y2) implies y1 = y2, there is a set containing precisely the images of the original set's elements.
	noctos.blogspot.com /2004_12_01_noctos_archive.html (5230 words)

	Zermelo-Frankel Set Theory
		The Axiom of Extensionality remains the same, but the Comprehension Axiom is replaced by something of a hodgepodge of axioms that are not as intuitively obvious.
		It may seem at first that the Axiom of Regularity does not rule out all sets that contain themselves as members.
		So these two axioms together show that there is no set that is a member of itself.
	www.trinity.edu /cbrown/topics_in_logic/sets/node4.html (514 words)

	Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks (Site not responding. Last check: 2007-10-06)
		The axioms C1, C1' and C2[F] (for all formulas F that do not contain x) and the axiom of choice define a formal set theory C which corresponds almost 100% to Cantor's intuitive set theory (of the "pre-paradox" period of 1873-94).
		The axiom of infinity completes the list of comprehension axioms, which are necessary for reconstruction of common mathematics, i.e.
		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.
	www.ltn.lv /~podnieks/gt2.html#BM2 (8336 words)

	Comments on 4823 \| Ask MetaFilter (Site not responding. Last check: 2007-10-06)
		One of Euclid's geometry axioms was the "parallel postulate", but it seemed for a long time that it was not such an obvious and intuitive thing to assume.
		you can't prove that we will never need more axioms (at least, i have no idea how to), but people generally believe that the axioms of zfc provide a basis for most of mathematics (and there are 7 of them).
		It's also worth noting that the Axiom of Regularity in ZFC is basically designed to outlaw an entire category of contradictions and paradoxes that doomed Russell and Whitehead's Principia Mathematica.
	ask.metafilter.com /mefi/4823 (3314 words)

	Russell's paradox - Psychology Central (Site not responding. Last check: 2007-10-06)
		While Zermelo was working on his version of set theory, he also noticed the paradox, but thought it too obvious and never published anything about it.
		Zermelo's system avoids the difficulty through the famous Axiom of separation (Aussonderung).
		The most common version of axiomatic set theory in use today is Zermelo-Fraenkel set theory including the axiom of choice, ZFC, which avoids the notion of types.
	psychcentral.com /psypsych/Russell's_paradox (1861 words)

	Guide to the proof summary files (Site not responding. Last check: 2007-10-06)
		The presence of this flag identifies theorems which require the axiom of regularity.
		The presence of the flag identifies theorems that use this axiom.
		It is proved that the axiom of regularity is equivalent to the assertion that equal(REGULAR,V).
	www.math.gatech.edu /~belinfan/research/autoreas/otter/sum (1884 words)

	Logical foundations and formal verification - Philosophy and Ontology
		Formalism, a doctrine and a programme due to Hilbert, is characterised by the view that classical mathematics may be established by formal derivation from plausible axioms, provided that the consistency of the formal axiomatisation is established by "finitary" or "constructive" means.
		The logicist position failed to be established primarily because two of the principles (axioms) necessary for the development of classical mathematics are difficult to establish as principles of logic.
		Neither the axiom of infinity nor the axiom of choice can be convincingly shown to be logically necessary propositions.
	www.rbjones.com /rbjpub/rbjcv/papers/dtc112.htm (1729 words)

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