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Topic: Axiom of projective determinacy

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	Axiom of projective determinacy - Wikipedia, the free encyclopedia
		In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.
		The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect information of length ω in which the players play natural numbers, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then one player or the other has a winning strategy.
		PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfect set property and the property of Baire.
	en.wikipedia.org /wiki/Axiom_of_projective_determinacy (199 words)

	Encyclopedia: Axiom of determinacy (Site not responding. Last check: 2007-11-05)
		The axiom of determinacy is inconsistent with the axiom of choice (AC); however, it has been shown that it implies that all sets of reals are Lebesgue measurable and have the Baire property.
		The axiom of determinacy has not been proved consistent with ZF and cannot even be proved to be independent of ZF (assuming that ZF is consistent) without further axioms.
		It is possible that the axiom of determinacy can be proved false without the use of the axiom of choice.
	www.nationmaster.com /encyclopedia/Axiom-of-determinacy (742 words)

	Determinacy Maximum (Site not responding. Last check: 2007-11-05)
		Many natural propositions that are undecidable in ZFC can be resolved by determinacy hypotheses. For example, projective determinacy provides a reasonably complete theory of second order arithmetic. This paper introduces a strong determinacy hypothesis, which we hope resolves more of the natural undecidable propositions.
		Projective determinacy only claims determinacy for games where positions can be coded by integers and payoff sets definable in a simple way from real numbers.
		The consequences of determinacy maximum are yet to be explored. One hopes that they lead to a canonical theory of the larger fragments of the set theoretical universe, but first one must investigate the consistency of determinacy maximum.
	web.mit.edu /dmytro/www/DeterminacyMaximum.htm (1325 words)

	info: AXIOM (Site not responding. Last check: 2007-11-05)
		Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups).
		They are a set of axioms strong enough to prove many important facts about number theory and they allowed G? to establish his famous second incompleteness theorem.
		The axioms are referred to as '4 + 1' because for nearly two millennia the fifth (parallel) postulate ('through a point outside a line there is exactly one parallel') was suspected of being derivable from the first four.
	www.info-macedonia.com /Axiom (1896 words)

	List of axioms -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		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 (972 words)

	Category:Set theory - Wikipedia, the free encyclopedia
		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.
		It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms.
		Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal elements within the real numbers.
	en.wikipedia.org /wiki/Category:Set_theory (163 words)

	Transactions of the American Mathematical Society
		This, together with a construction of W. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey.
		And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey.
		W. Woodin, On the consistency strength of projective uniformization, in Proceedings of Herbrand Symposium.
	www.ams.org /tran/2002-354-04/S0002-9947-01-02926-9/home.html (485 words)

	Set Theory
		The Axiom of Choice states that for every set of mutually disjoint nonempty sets there exists a set that has exactly one member common with each of these sets.
		The Axiom of Choice, which postulates the existence of a certain set (the choice set) without giving specific instructions how to construct such a set, is of different nature than the other axioms, which all formulate certain construction principles for sets.
		The legitimate question is whether the Axiom of Choice is consistent, that is whether it cannot be refuted from the other axioms.
	setis.library.usyd.edu.au /stanford/entries/set-theory (3302 words)

	Infinity debate
		There are weaker axioms that seem to work: = = = = AD + countable choice = = = = AC + projective determinacy = = = = I will note, however, that the large cardinal characteristics suggest = = that these distinctions are actually manifestations of geometric = = representations of small numbers.
		There are weaker axioms that seem to work: = = AD + countable choice = = AC + projective determinacy = = I will note, however, that the large cardinal characteristics suggest = that these distinctions are actually manifestations of geometric = representations of small numbers.
		There are weaker axioms that seem to work: AD + countable choice AC + projective determinacy I will note, however, that the large cardinal characteristics suggest that these distinctions are actually manifestations of geometric representations of small numbers.
	www.pych-one.com /new-748901-4765.html (10533 words)

	Dictionary of Meaning www.mauspfeil.net (Site not responding. Last check: 2007-11-05)
		The axiom of determinacy has not been proved consistent with ZF and cannot even be proved to be independent (Set theory) independent of ZF (assuming that ZF is consistent) without further axioms.
		This is just because the axiom of determinacy may be inconsistent.
		Assuming consistency of ZF the axiom of determinacy cannot be proved to be independent of ZF (in ZF).
	www.mauspfeil.net /Axiom_of_determinacy.html (1241 words)

	Chaitin: Irreducible Complexity in Pure Mathematics (Site not responding. Last check: 2007-11-05)
		The concept of "axiom" is closely related to the idea of logical irreducibility.
		Axioms are mathematical facts that we take as self-evident and do not attempt to prove from simpler principles.
		All formal mathematical theories start with axioms, and then deduce the consequences of these axioms, which are called its theorems.
	www.umcs.maine.edu /~chaitin/xxx.html (5054 words)

	Antimeta: Eight Views of Mathematics (Site not responding. Last check: 2007-11-05)
		Thus, the axiom makes many important and divergent explanatory predictions, and thus should be accepted as true.
		Similarly, if there is enough evidence in the theory of ZFC, it seems that projective determinacy must be true in the fiction as well, even though it was never explicitly stipulated.
		If mathematical truth is identified with provability from the axioms, then negative answers to the second and third question follow immediately from Gödel's incompleteness results.
	www.antimeta.org /blog/archives/2005/05/eight_views_of.html (1445 words)

	Encyclopedia: List of axioms (Site not responding. Last check: 2007-11-05)
		These are the de facto standard axioms for contemporary mathematics
		With the Zermelo-Frankel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable
		Alternates incompatible with AC Freiling's axiom of symmetry
	www.nationmaster.com /encyclopedia/List-of-axioms (137 words)

	Citations: A proof of projective determinacy - Martin, Steel (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Donald A. Martin, John R. Steel, A proof of projective determinacy, Journal of the American Mathematical Society 2 (1989), p.
		Martin and J. Steel, A proof of projective determinacy, Journal of the American Mathematical Society 2 (1989), 71--125.
		In the final section of this paper I formulate an analogous axiom for games of imperfect information, and explore some of the consequences of this axiom.
	citeseer.ist.psu.edu /context/390987/0 (1643 words)

	List of axioms - Wikipedia, the free encyclopedia
		In epistemology, the word axiom is understood differently; see axiom and self-evidence.
		Individual axioms are almost always part of a larger axiomatic system.
		Alternates incompatible with AC Axiom of real determinacy
	en.wikipedia.org /wiki/List_of_axioms (109 words)

	Logic Colloquium
		In a core model induction argument, one produces canonical inner models which are correct for statements at a given level of complexity, using core model theory together with the existence of models which are correct at lower levels.
		The principle of "collecting the universe" justifies axioms asserting the existence of large cardinals which can be "built up" by iterating Mahlo's operation.
		The pattern of connections between determinacy and large cardinals suggests that there should be games which capture the theory of indiscernible Woodin cardinals.
	www.math.ucla.edu /~hbe/logic.html (2686 words)

	Antimeta: Fictionalism Archives (Site not responding. Last check: 2007-11-05)
		A group theorist thinks that what she is doing is just writing down results that hold for whatever things happen to satisfy the group axioms, as a topologist is coming up with results that apply for anything that happens to satisfy the topological axioms assumed.
		Conservativity is the (testable) claim that there are no observational or purely concrete claims in, say, physics that are decided by the addition of mathematical axioms making reference to a separate class of non-observable entities.
		The Copenhagen Interpretation takes this sort of instrumentalist approach (which Steel refers to in "Does Mathematics Need New Axioms?" with regards to large cardinal axioms and their consistency), but Field tries to give a purely concretely stateable theory over which mathematics is conservative.
	www.antimeta.org /blog/archives/fictionalism (4042 words)

	Colloquium Logicum 2004, Abstracts
		Abstract: A characteristic feature of infinitary combinatorics under the Axiom of Determinacy is the existence of sequences of partition cardinals, called Kleinberg sequence.
		Ralf Schindler (Universität Münster): Cardinal arithmetic and determinacy
		We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness.
	math.uni-heidelberg.de /logic/CL_2004/abstracts.html (2552 words)

	Infinite Ink: The Continuum Hypothesis by Nancy McGough
		Like the Axiom of Choice (AC), Gödel showed that CH is consistent with standard set theory and Cohen showed that ~CH is consistent with standard set theory (and thus CH is independent of standard set theory).
		Many of these axioms were proposed because they shed light on the continuum and CH.
		The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe.
	www.ii.com /math/ch (4563 words)

	Techniques For Approaching The Dual Ramsey Property In The Projective Hierarchy (ResearchIndex)
		Abstract: This paper can be understood as a catalogue of some of the similarities; in fact, one could see parts of this paper as an attempt to reach the obvious dualization of Theorem 1.2: (Update)
		2 Martin: The axiom of determinateness and reduction principle..
		1 Moschovakis: Some consequences of the axiom of de- nable det..
	citeseer.ist.psu.edu /311506.html (513 words)

	GedankenTravelExperiment
		We need some intuitive grounds for choosing the axioms that set theory uses, as they are the other part of the problem.
		The more axioms we have, the more robust our mathematics is. But adding an axiom because it is somewhat useful is not the same as adding something that fundamental that it is indispensable to the discipline.
		Maddy is using it to mean that an axiom can be said to have verifiable consequences if we show that the axiom that the theorem relies on can be proved without it.
	philosophicalkarl.blogspot.com /2004_11_01_philosophicalkarl_archive.html (6693 words)

	The usual problem for a philosophy of mathematics is applicability – the unreasonable effectiveness of mathematics (Site not responding. Last check: 2007-11-05)
		A "new" set-theoretic axiom may be valued for the light it sheds not on concreta but on mathematical objects already in play.
		So it is, e.g., with the axiom of projective determinacy and the sets of reals studied in descriptive set theory.
		The theory might be a collection of axioms; it might be that plus some informal depiction of the kind of object the axioms attempt to characterize; or it might be an informal depiction pure and simple.
	www.mit.edu /~yablo/mgk.html (12359 words)

	Dictionary of Meaning www.mauspfeil.net
		to be the set of all subsets which are projections of Borel set Borel subsets of ''R''
		The '''axiom of projective determinacy''' states that for any axiom of determinacy ω-game, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then the game has a winning strategy.
		There you find a list of all editors and the possibility to edit the original text of the article Axiom of projective determinacy.
	www.mauspfeil.net /Axiom_of_projective_determinacy.html (215 words)

	Chaitin, Meta Math! The Quest for Omega
		And part of this package is that a finite set of mathematical axioms or postulates are explicitly given, plus you use symbolic logic to deduce all the possible consequences of the axioms.
		The axioms are the starting point for any mathematical theory; they are taken as self-evident, without need for proof.
		The consequences of these axioms, and the consequences of the consequences, and the consequences of that, and so forth and so on ad infinitum, are called the "theorems" of the FAS.
	ibiwan.com /omega.html (20804 words)

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