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Topic: Axiom of pairs

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axiom of choice

Separation axioms

Peano axioms

probability axioms

axiom of extensionality

Zermelo Fraenkel axioms

Axiom of regularity

Kuratowski closure axioms

Axiom schema of specification

axiom of infinity

axiom of empty set

Axiom of pairs

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	Axiom of pairing - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory.
		The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		Nevertheless, in the standard formulation of the Zermelo-Fraenkel set theory, the axiom of pairing follows from the axiom of power set and the schema of replacement, thus it is sometimes omitted.
	en.wikipedia.org /wiki/Axiom_of_pairing (531 words)

	Zermelo set theory - Wikipedia, the free encyclopedia
		AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa...
		Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is definite for all elements of a set M, M possesses a subset M' containing as elements precisely those elements x of M for which –(x) is true".
		AXIOM V. Axiom of the union (Axiom der Vereinigung) "To every set T there corresponds a set ∪T, the union of T, that contains as elements precisely all elements of the elements of T".
	en.wikipedia.org /wiki/Zermelo_set_theory (986 words)

	Ordered pair (Site not responding. Last check: 2007-10-08)
		An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element.
		The set of all ordered pairs whose first element is in some set X and second element in some set Y is called the Cartesian product of X and Y, and written X × Y.
		In the usual ZF formulation of set theory including the axiom of regularity, ordered pairs (a, b) can also be defined as the set {a, {a, b}}.
	pedia.newsfilter.co.uk /wikipedia/o/or/ordered_pair.html (381 words)

	Talk:Ordered pair - Wikipedia, the free encyclopedia
		The ordered pair (x,x) is symmetric -- switching the elements has no effect, and that's exactly what is reflected by the set notation.
		I replaced the text In the usual Zermelo-Fraenkel formulation of set theory including the axiom of regularity, ordered pairs (a, b) can also be defined as the set {a, {a, b}}.
		However, the axiom of regularity is required, since without it, it is possible to consider sets x and z such that x = {z}, z = {x}, and x ≠ z.
	www.wikipedia.org /wiki/Talk:Ordered_pair (917 words)

	Zermelo set theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		AXIOM I. (Click link for more info and facts about Axiom of extensionality) Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa...
		(Click link for more info and facts about Axiom of separation) Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is definite for all elements of a set M, M possesses a subset M' containing as elements precisely those elements x of M for which –(x) is true".
		Axiom of the power set (Axiom der Potenzmenge) "To every set T there corresponds a set T', the (Click link for more info and facts about power set) power set of T, that contains as elements precisely all subsets of T".
	www.absoluteastronomy.com /encyclopedia/Z/Ze/Zermelo_set_theory.htm (1000 words)

	axiom of choice
		An axiom in set theory that is one of the most controversial axioms in mathematics; it was formulated in 1904 by the German mathematician Ernst Zermelo (1871-1953) and, at first, seems obvious and trivial.
		His point is that the two socks in a pair are identical in appearance, so, to pick one of them, we have to make an arbitrary choice.
		Thus there are no contradictions in choosing to reject it; among the alternatives are to adopt a contradictory axiom or to use a completely different framework for mathematics, such as category theory.
	www.daviddarling.info /encyclopedia/A/axiom_of_choice.html (567 words)

	Ordered pair (Site not responding. Last check: 2007-10-08)
		An ordered pair is a collection of two objects such that one can be distinguished as the first elementand the other as the second element.
		The set of all ordered pairs whose first element is in some set X and second element in some set Y is calledthe Cartesian product of said sets.
		In the usual ZF formulation of set theoryincluding the axiom of regularity, ordered pairs (a,b) can also be defined as the set {a, {a, b}}.
	www.therfcc.org /ordered-pair-62909.html (351 words)

	Ordered pair: Definition and Links by Encyclopedian.com - All about Ordered pair (Site not responding. Last check: 2007-10-08)
		The set of all ordered pairs whose first element is in some set X and second element in some set Y is called the Cartesian product of said sets.
		Ordered triples and n-tuples (ordered lists of n terms) are defined recursively from this definition: an ordered triple (a,b,c) can be defined as (a, (b,c)): two nested pairs.
		In pure set theory, where there are only sets, ordered pairs (a, b) can be defined as the set { {a}, {a, b} }.
	www.encyclopedian.com /or/Ordered-pair.html (393 words)

	The Annotated Axioms
		Axiom 4 should read something like "Space is a separation of existences, beingnesses and identities held apart with non-existences, non-beingnesses and non-identities." The original space at the highest level I've spotted is separation between selves.
		So this axiom is partially true, but ignores the motion by/through/via postulated fields (which still isn't correct but I have no idea how to describe the actuality in words).
		In Axiom 19 he is describing his observation of the fact that if you view as-is the alteration (significance or meaning, which is on a scale of values, read Dennis Stephens on value and importance) you will obtain a partial relief.
	www.censorthis.com /ouran/GD52.html (2330 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The axiom that expresses this is as follows: Total-Order() --> (A t1,t2)[Instant(t1) & Instant(t2) --> [before(t1,t2) v t1 = t2 v before(t2,t1)]] This eliminates models of time with branching futures and other conflations of time and possibility or limited knowledge.
		The statements of the axioms have been complicated modestly in order to localize the difference between the two approaches to the choice between two pairs of simple existence axioms, which are themselves conditioned on 0-argument propositions indicating the choice of that option.
		The axiom for this is as follows, where the 0-argument predicate indicating the exercising of this option is "Dense()": Dense() --> (A t1,t2)[Instant(t1) & Instant(t2) & before(t1,t2) --> (E t)[Instant(t) & before(t1,t) & before(t,t2)]] This is weaker than the mathematical property of continuity, which we will not axiomatize here.
	www.isi.edu /~pan/time/owl-time-july04.txt (10792 words)

	Computer Music Generation via Decision Tree Learning (Site not responding. Last check: 2007-10-08)
		Axiom 11: A subscale of the 12-note chromatic scale is the 7-note major scale, formed by steps of {2, 2, 1, 2, 2, 2, 1} from the chromatic scale.
		This axiom is largely arbitrary, but it is important to note that the major scale contains all of the frequencies of the chromatic scale that are very close to small whole-number ratios, except for the 7:4 ratio, which was the furthest away from the ideal ratio anyway.
		By Axiom C1, all the notes of a chord are in the major scale with respect to the key.
	pace.pace.net /rmg/RMG_thesis.htm (8610 words)

	RMG_5 (Site not responding. Last check: 2007-10-08)
		Axiom S1: Pairs of waveforms whose frequencies are smaller whole-number ratios are generally preferred to those with larger whole-number ratios.
		Axiom S2: Notes in a scale are generally preferred to notes not in that scale.
		Axiom S4: Scales with notes whose frequencies are too close together are less preferred.
	pace.pace.net /rmg/presentation/long/RMG_5.htm (63 words)

	Math 410
		Axiom of Identity -- If A and B are sets such that every element of A is an element of B and every element of B is an element of A, then A and B are the same sets.
		Axiom of Specification -- If A is a set and Q(x) is a condition, then there is a subset B of A containing exactly those elements of A for which the statement Q(x) is true.
		Axiom of Powers -- If A is a set, then there is a set P(A) (called the power set of A) consisting of all possible subsets of the set A.
	mathserv.monmouth.edu /coursenotes/kuntz/math410/m41002.htm (1590 words)

	untitled (Site not responding. Last check: 2007-10-08)
		Axiom 4 tells us that two distinct lines are on at least one point.
		By Axiom 2, line L must have a point in common with each of the three other lines.
		Suppoae there were a fourth point, say P, on the line L. Then by Axiom 3, P must also be on one of the other tree lines.
	pegasus.cc.ucf.edu /~xli/three-pt.htm (372 words)

	MathAction and Axiom AxiomDevelopment
		One tasks of the Axiom developers is to analyze known problems and to propose patches.
		If you are new to Axiom or computer algebra in general, you might find it useful to browse through "A Critique of the Mathematical Abilities of CA Systems" by Michael Wester,
		Axiom is written in boot and common lisp.
	www.axiom-developer.org /zope/mathaction/AxiomDevelopment (1495 words)

	Relevance of the Axiom of Choice
		The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate.
		The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid's five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid's Parallel Postulate:
		For many sets, including any finite set, the first six axioms of set theory (abbreviated ZF) are enough to guarantee the existence of a choice function but there do exist sets for which AC is required to show the existence of a choice function.
	db.uwaterloo.ca /~alopez-o/math-faq/node69.html (1384 words)

	Axiom of Choice (Site not responding. Last check: 2007-10-08)
		The Axiom of Choice (AC) was formulated about a century ago, and it was controversial for a few of decades after that; it may be considered the last great controversy of mathematics.
		In effect, when we accept the Axiom of Choice, this means we are agreeing to the convention that we shall permit ourselves to use a choice function f in proofs, as though it "exists" in some sense, even though we cannot give an explicit example of it or an explicit algorithm for it.
		Consequences of the Axiom of Choice is a book by Paul Howard and Jean E. Rubin that was published by the American Mathematical Society in 1998.
	math.vanderbilt.edu /~schectex/ccc/choice.html (3751 words)

	Heine continuity implies Cauchy continuity without the Axiom of Choice - Apronus.com
		The Axiom of Choice is not needed to prove that if f is Cauchy continuous at x then f is Heine continuous at x.
		Imagine those pairs as a set of points on the plane and write a simple computer program which spirals outward from the origin visiting each point.
		This time you will not need the Axiom of Choice to obtain the sequence whose existence contradicts Heine continuity because you will be able to define this sequence by applying Fact 1.
	www.apronus.com /math/cauchyheine.htm (683 words)

	Reference-Manual for CREATE-AXIOM (Site not responding. Last check: 2007-10-08)
		Most axioms can be associated with a definition, and belong there.
		Some axioms involve several basic terms and are central to the theory being specified by the ontology.
		The tree can be either a string, or a list of strings, or a list of ( ) pairs, where is a string or symbol that names the type of the comment.
	www-ksl.stanford.edu /pub/knowledge-sharing/ontologies/html/reference-manual/CREATE-AXIOM.html (331 words)

	ZERMELO SET THEORY (Site not responding. Last check: 2007-10-08)
		A set theory with the following set of axioms: Extensionality: two sets are equal if and only if they have the same elements.
		Axiom of elementary sets (Axiom der Elementarmengen) "There exists a (fictitious) a set, the null set, phi, that contains no element at all.
		Let Mo be the subset of M for which, by AXIOM III, is separated out by the notion "x ∉ x".
	www.websters-online-dictionary.org /Ze/Zermelo+set+theory.html (1216 words)

	Axiom of the empty set (Site not responding. Last check: 2007-10-08)
		The axiom of the empty set uses the existential quantifier (
		The axiom of the empty set is as follows.
		The definition of the integers requires two axioms for constructing finite sets.
	www.mtnmath.com /whatrh/node42.html (53 words)

	AXIOM Global Trading (Site not responding. Last check: 2007-10-08)
		AXIOM Global Trading Launches Joint Venture With Schubert Group International New York, NY June 01, 2005: AXIOM Global Trading, a leading integrator and developer of institutional centric Direct Market Access and advanced algorithmic technology, and SGI LLC, Schubert Group International, a member firm of the NYSE, have announced today thelaunch of AXIOM SGI.
		AXIOM SGI will be headed by Wall Street veteran Scott F. Carotenuto, founder of AXIOM Global Trading, a technology company providing solutions for global equity order management, execution, and other market access demands of both domestic and foreignmoney managers, hedge funds and broker dealers.
		AXIOM Global Trading is focused on providing buy-side institutions with the most advanced technology.
	axiomtrading.com /html/news.php?nID=14 (546 words)

	Simple Axiom Systems for Boolean Algebra
		In particular, Stephen Wolfram proposed a study of 27 candidate axiom systems consisting of 25 single equations and 2 pairs of equations.
		A single axiom of length 105 is presented in [6].
		A number of single axioms in terms of operators other than the Sheffer stroke are presented in [6].
	www.cs.unm.edu /~veroff/BA (804 words)

	MathAction and Axiom #8 (Savannah bug #9297) output misses some parenthesis
		The text mode output of Axiom appears to be incorrect, although internally everything is correct, as can be seen by the LaTeX formatted output.
		Expressions (1) and (2) above appear to be indistinquishable in the native Axiom text mode output.
		In expressions (3) and (4) the scope of the summation is ambiguous in both the text mode and formatted outputs.
	page.axiom-developer.org /zope/mathaction/8SavannahBug9297OutputMissesSomeParenthesis (1074 words)

	Bibliography
		Review of Arata Ishimoto, A Set of Axioms of the modal propositional calculus equivalent to S3, Ibid., p.
		Review of Arata Ishimoto, 'A note on the paper "A Set of Axioms of the modal propositional calculus equivalent to S3" ', and 'A formulation of the modal propositional calculus equivalent to S4', ibid., p.
		Review of B. Sobocinski, 'On the Single Axioms of Protothetic', The Journal of Symbolic Logic, vol.
	www.kommunikation.aau.dk /prior/biblio/bibliogr.htm (2479 words)

	Candidate Axiom Systems (Site not responding. Last check: 2007-10-08)
		The candidate set consists of 25 single equations and 2 pairs of equations, all in terms of the Sheffer stroke operator .
		Here is the candidate set in clause form, suitable as input to Otter.
		% candidate single axioms f(f(x, f(x, f(x, y))), f(y, f(x, z))) = y # label("Equation 1").
	www.cs.unm.edu /~veroff/BA/candidates.html (410 words)

	Some set theoretical aspects.
		The axioms of the Peano's arithmetic can be shown to be theorems under any strong enough set theory, for example the
		The axiom does actually show you the form of this set.
		7 : Axiom of the power set : For any set A there is a set B that includes every subset of A. This set is called the power set of A and is written P(A).
	linas.org /mirrors/www.torget.se/2001.03.23/users/m/mauritz/math/num/set.htm (398 words)

	Composite Computing Methods Integrating Symbolic, Numeric and Graphical Packages for Research Engineers
		The trade-off for the speed of the evaluation process using Fortran is in the cost of the Fortran generation and compilation(The Fortran generation process in Axiom is the costliest in practice although the compilation itself is at least of the order n*n on the size of the code).
		The Fortran generation utilities included with recent implementations of Axiom are used to both create the Fortran code and also to verify that the full ANSI 1978 standards are adhered to with respect to variable names, types and constructs.
		Axiom also provides the link to enable the calling of routines from the NAG Numerical Library, the output graphical facilities and the hypertext model, HyperDoc, for the user interface (tutorial package) [Tapia et.
	www.bath.ac.uk /~masjhd/jtap/www/report.htm (6396 words)

	Integers.
		Using the Axiom of replacement will this allow us to create the set of all non negative integers.
		and each pair (p,q)={p,{p,q}}, where p and q are naturals in the form decried on the previous page.
		Z will thus be a infinite set of infinite sets of ordered pairs of integers.
	hemsidor.torget.se /users/m/mauritz/math/num/setint.htm (397 words)

	Natural Numbers
		Using the Axiom of the empty set we can assure that there exist a set containing no members, the empty set.
		We now use the Axiom of the power set, 'For any set A there is a set B that includes every subset of A', to assure the
		The 3:rd axiom is a bit harder to show to be true, and we will leave that to be done later.
	hemsidor.torget.se /users/m/mauritz/math/num/setnat.htm (692 words)

	Abstract for "Logic from Computer Science" (Site not responding. Last check: 2007-10-08)
		This language has been used as an example programming language in past theoretical studies and, with the addition of numerals and a few basic operations, it is sufficient to define all partial recursive functions.
		The natural equational axioms include eta-equivalence and the so-called "surjective pairing" axiom for pairs.
		The equational axioms for PCF (including eta and surjective pairing) are sound for observational equivalence with respect to R-.
	www.seas.upenn.edu /~sweirich/types/archive/1989/msg00095.html (309 words)

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