Forcing axioms and cardinal arithmetic I PDF
Preview Forcing axioms and cardinal arithmetic I
Forcing axioms and cardinal arithmetic I Boban Velickovic EquipedeLogique Universite´ deParis7 Logic Colloquium 2006, Nijmegen, July 27- August 2 2006 TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms Outline 1 The Go¨del program 2 Forcing axioms 3 Stationary set reflection 4 Bounded Forcing Axioms TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms Outline 1 The Go¨del program 2 Forcing axioms 3 Stationary set reflection 4 Bounded Forcing Axioms TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms The Go¨del program ZFC (Zermelo Fraenkel axioms with Choice) serves as a good axiomatization for mathematics. It describes the self evident properties of the cumulative hierarchy of sets and the set theoretic universe V. Cumulative hierarchy V = ∅ 0 V = P(V ) α+1 α S V = V , for α limit α ξ<α ξ V = S V . α∈ORD α TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms The Go¨del program ZFC (Zermelo Fraenkel axioms with Choice) serves as a good axiomatization for mathematics. It describes the self evident properties of the cumulative hierarchy of sets and the set theoretic universe V. Cumulative hierarchy V = ∅ 0 V = P(V ) α+1 α S V = V , for α limit α ξ<α ξ V = S V . α∈ORD α TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms However, ZFC does not decide some basic questions such as: Cantor’sContinuumHypothesis(CH),infactsaysverylittle about cardinal arithmetic in general Souslin’s Hypothesis (SH) and related problems in general topology regularity properties (i.e. Lebesgue measurability, the property of Baire, etc) of projective sets of reals Whitehead’s problem in Homological Algebra Kaplansky’s conjecture in Banach algebras many many more.... TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms However, ZFC does not decide some basic questions such as: Cantor’sContinuumHypothesis(CH),infactsaysverylittle about cardinal arithmetic in general Souslin’s Hypothesis (SH) and related problems in general topology regularity properties (i.e. Lebesgue measurability, the property of Baire, etc) of projective sets of reals Whitehead’s problem in Homological Algebra Kaplansky’s conjecture in Banach algebras many many more.... TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms However, ZFC does not decide some basic questions such as: Cantor’sContinuumHypothesis(CH),infactsaysverylittle about cardinal arithmetic in general Souslin’s Hypothesis (SH) and related problems in general topology regularity properties (i.e. Lebesgue measurability, the property of Baire, etc) of projective sets of reals Whitehead’s problem in Homological Algebra Kaplansky’s conjecture in Banach algebras many many more.... TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms However, ZFC does not decide some basic questions such as: Cantor’sContinuumHypothesis(CH),infactsaysverylittle about cardinal arithmetic in general Souslin’s Hypothesis (SH) and related problems in general topology regularity properties (i.e. Lebesgue measurability, the property of Baire, etc) of projective sets of reals Whitehead’s problem in Homological Algebra Kaplansky’s conjecture in Banach algebras many many more.... TheGo¨delprogram Forcingaxioms Stationarysetreflection BoundedForcingAxioms However, ZFC does not decide some basic questions such as: Cantor’sContinuumHypothesis(CH),infactsaysverylittle about cardinal arithmetic in general Souslin’s Hypothesis (SH) and related problems in general topology regularity properties (i.e. Lebesgue measurability, the property of Baire, etc) of projective sets of reals Whitehead’s problem in Homological Algebra Kaplansky’s conjecture in Banach algebras many many more....
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