Forcing axioms and cardinal arithmetic I

76 Pages·2006·0.6 MB·English

by  Boban Velickovic

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....