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Topic: Axiom of constructibility

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	Axiom of constructibility - Wikipedia, the free encyclopedia
		The axiom of constructibility is a possible axiom for set theory in mathematics.
		It asserts that V equals L where V is the universe of sets and L is the constructible universe.
		The axiom of constructibility implies the generalized continuum hypothesis and also the axiom of choice.
	en.wikipedia.org /wiki/Axiom_of_constructibility (243 words)

	Axiom of constructibility: Encyclopedia topic (Site not responding. Last check: 2007-10-09)
		The axiom of constructibility is a possible axiom (axiom: (logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident) for set theory (set theory: The branch of pure mathematics that deals with the nature and relations of sets) in mathematics.
		It asserts that V equals L where V is the universe of sets and L is the constructible universe (constructible universe: in mathematics, the constructible universe (or gödels constructible universe)...
		The axiom of constructibility implies the generalized continuum hypothesis (generalized continuum hypothesis: in mathematics, the continuum hypothesis is a hypothesis about the possible sizes...
	www.absoluteastronomy.com /reference/axiom_of_constructibility (286 words)

	sci.math FAQ: The Axiom of Choice
		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: 1.
		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.
		Since AC gives no method for constructing a choice set constructivists belong to school C. Formalism A formalist believes that mathematics is strictly symbol manipulation and any consistent theory is reasonable to study.
	www.cs.uu.nl /wais/html/na-dir/sci-math-faq/axiomchoice.html (1586 words)

	[No title]
		The negation of this axiom asserts that all ordinals are finite.
		This is similar to the first formulation of the axiom of choice, except it is restricted to well-founded relations.
		In order to construct some of the more important admissible sets, those in which Barwise develops his definability theory, he invokes Gödel's hierarchy of sets to build a class L of constructible sets in KPU via a list of operations.
	www.afn.org /~afn07474/kppaper.html (1632 words)

	Morasses, by Daniel J. Velleman (Site not responding. Last check: 2007-10-09)
		One common way to define an infinite structure is to construct it as a union of an increasing chain of substructures.
		One way to carry out such a construction is to use forcing, with the forcing conditions being countable substructures of the final structure.
		The definition of simplified morasses is based directly on the kinds of constructions for which morasses are usually used, and is not closely tied to the structure of L.
	at.yorku.ca /z/a/a/b/17.htm (538 words)

	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:
		A constructivist believes that the only acceptable mathematical objects are ones that can be constructed by the human mind, and the only acceptable proofs are constructive proofs.
	www.cs.uwaterloo.ca /~alopez-o/math-faq/mathtext/node35.html (1455 words)

	Diary for Bram
		The consensus reality interpretation of PA is inconsistent with the axiom of inconsistency of PA. I believe that there is an interpretation of large cardinal axioms which both has meaning and implies their existence.
		It is conceivable that there is also an interpretation of an axiom which contradicts the large cardinal axioms, such as the axiom of constructibility, which also has meaning.
		We then wish to add an axiom stating that that axiom is consistent, then generalize to another, and another, etc. until we generalize to aleph null.
	www.advogato.org /person/Bram/diary.html?start=7 (3318 words)

	Continuum Hypothesis. Axiom of Constructibility. Axiom of Determinateness. Ackermann's Set Theory. By K.Podnieks
		The consistency of the axiom of choice and of the generalized continuum hypothesis.
		Cohen's method of forcing allows proving also that the axiom of choice (of course!) cannot be derived from the (normal!) axioms of ZF.
		Thus, both scenarios (axiom of constructibility and axiom of determinacy) have produced already a plentiful collection of beautiful and interesting results.
	www.ltn.lv /~podnieks/gt2a.html (5349 words)

	Sy Friedman (Site not responding. Last check: 2007-10-09)
		One attractive way is to adjoin the axiom V=L, asserting that every set is constructible.
		This axiom has many desirable consequences, such as the generalised continuum hypothesis, the existence of a definable wellordering of the class of all sets, as well as strong combinatorial principles, such as Diamond, Square and Morass.
		As many interesting set-theoretic statements have consistency strength beyond ZFC, it is now common in set theory to assume the existence of ``large'' inner models of the set-theoretic universe, i.e., inner models containing large cardinals.
	www.math.cas.cz /~krajicek/sy05.html (244 words)

	Reference.com/Encyclopedia/List of axioms
		In epistemology, the word axiom is understood differently; see axiom and self-evidence.
		Individual axioms are almost always part of a larger axiomatic system.
		These are the de facto standard axioms for contemporary mathematics
	www.reference.com /browse/wiki/List_of_axioms (110 words)

	Continuum Hypothesis. Alternative Set Theories
		Cohen's method of forcing allows proving also that the axiom of choice (of course!) can not be derived from the (normal!) axioms of ZF.
		The axiom of determinateness contradicts V=L and the axiom of choice.
		Thus, both scenarios (axiom of constructibility and axiom of determinateness) have produced already a plentiful collection of beautiful and interesting results.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt6_2.html (5306 words)

	The Strength of Mac Lane Set Theory - Mathias (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		We begin by reviewing the axioms of the two...
		21 The fine structure of the constructible hierarchy (context) - Jensen - 1972
		3 The Consistency of the Axiom of Choice and of the Generalise..
	citeseer.ist.psu.edu /331036.html (621 words)

	Find in a Library: The axiom of constructibility : a guide for the mathematician
		Find in a Library: The axiom of constructibility : a guide for the mathematician
		The axiom of constructibility : a guide for the mathematician
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	www.worldcatlibraries.org /wcpa/ow/1f4a478daa14d111.html (67 words)

	R. G\"obel, B. Goldsmith "{This work was written under contract SC/014/88 from Eolas, the Irish Science and Technology ... (Site not responding. Last check: 2007-10-09)*
		Abstract:The discrete algebras $A$ over a commutative ring $R$ which can be realized as the full endomorphism algebra of a torsion-free $R$-module have been investigated by Dugas and G\"obel under the additional set-theoretic axiom of constructibility, $V=L$.
		Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra.
		The results obtained are independent of the usual Zermelo-Fraenkel set theory ZFC.
	www.ii.uj.edu.pl /EMIS/journals/CMUC/cmuc9301/abs/gobel.htm (157 words)

	Presentations
		This presentation was for a seminar I took on contemporary set theory.
		I had to present chapter 2 of Keith Devlin's The Joy of Sets to the class which covers the language of set theory (simply first-order logic plus the '∈' symbol), the cumulative hierarchy of sets (the V hierarchy), an introduction to class theory, and finally the Zermelo-Fraenkel axioms themselves (minus the Axiom of Choice).
		It covers constructibility, the constructible hierarchy, and how to use the axiom of constructibility to prove that IF Zermelo-Fraenkel set theory is consistent THEN so is Zermelo-Fraenkel set theory plus the axiom of choice.
	www.ryanflannery.net /presentations.php (662 words)

	toc1 (Site not responding. Last check: 2007-10-09)
		5.5 The class L[I]: sets constructible from internal sets
		5.5b Proof of the theorem on I-constructible sets
		5.5d Transfinite constructions in L[I] Historical and other notes to Chapter 5
	www.math.uni-wuppertal.de /~reeken/toc5.html (45 words)

	Relevance of the Axiom of Choice
		There are many equivalent statements of the Axiom of Choice.
		The following version gave rise to its name:
		Accept an axiom that implies AC such as the Axiom of Constructibility (V=L) or the Generalized Continuum Hypothesis (GCH).
	db.uwaterloo.ca /~alopez-o/math-faq/node69.html (1384 words)

	Keith J. Devlin Discussion
		All the Math that's Fit to Print : Articles from The Guardian (Spectrum)
		The axiom of constructibility: A guide for the mathematician (Lecture notes in mathematics ; 617)
		Becoming a Better Reasoner.(Logic Made Easy: How to Know When Language Deceives You)(Book Review) : An article from: American Scientist
	www.gnooks.com /discussion/keith+j-2e+devlin.html (132 words)

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