Forcing axioms and cardinal arithmetic Iboban/pdf/LC2006_PART1.pdf · 2006-07-21 · Large cardinal...

Forcing axiomsand

cardinal arithmeticI

Boban Velickovic

Equipe de LogiqueUniversite de Paris 7

Logic Colloquium 2006, Nijmegen,July 27- August 2 2006

The Godel program Forcing axioms Stationary set reflection Bounded Forcing Axioms

Outline

1 The Godel program

2 Forcing axioms

3 Stationary set reflection

4 Bounded Forcing Axioms


Outline

1 The Godel program

2 Forcing axioms




The Godel program

ZFC (Zermelo Fraenkel axioms with Choice) serves as a goodaxiomatization for mathematics. It describes the self evidentproperties of the cumulative hierarchy of sets and the settheoretic universe V.

Cumulative hierarchy

V0 = ∅Vα+1 = P(Vα)

Vα =⋃

ξ<α Vξ, for α limit

V =⋃

α∈ORD Vα.


The Godel program

ZFC (Zermelo Fraenkel axioms with Choice) serves as a goodaxiomatization for mathematics. It describes the self evidentproperties of the cumulative hierarchy of sets and the settheoretic universe V.

Cumulative hierarchy

V0 = ∅Vα+1 = P(Vα)

Vα =⋃

ξ<α Vξ, for α limit

V =⋃

α∈ORD Vα.


However, ZFC does not decide some basic questions such as:

Cantor’s Continuum Hypothesis (CH), in fact says very littleabout cardinal arithmetic in general

Souslin’s Hypothesis (SH) and related problems in generaltopology

regularity properties (i.e. Lebesgue measurability, theproperty of Baire, etc) of projective sets of reals

Whitehead’s problem in Homological Algebra

Kaplansky’s conjecture in Banach algebras

many many more....








many many more....








many many more....








many many more....








many many more....








many many more....








many many more....


Godel [1947]

... if the meaning of the primitive terms of set theory... areaccepted as sound, it follows that the set-theoretical conceptsand theorems describe some well-determined reality, in whichCantor’s conjecture must be either true or false and itsundecidability from the axioms as known today can only meanthat these axioms do not contain a complete description of thisreality.

Godel’s program

Search for new axioms and rules of inference for set theorywhich would decide the value of the continuum and otherproblems undecidable in ZFC alone.


Godel [1947]

... if the meaning of the primitive terms of set theory... areaccepted as sound, it follows that the set-theoretical conceptsand theorems describe some well-determined reality, in whichCantor’s conjecture must be either true or false and itsundecidability from the axioms as known today can only meanthat these axioms do not contain a complete description of thisreality.

Godel’s program

Search for new axioms and rules of inference for set theorywhich would decide the value of the continuum and otherproblems undecidable in ZFC alone.


New axioms typically assert the richness of the set theoreticuniverse.

Large cardinal axioms - assert that the set theoreticuniverse is ’tall’. Provide a linear hierarchy of consistencystrength. Have impact on the low levels of the cumulativehierarchy. Do not decide cardinal arithmetic, e.g. theContinuum Hypothesis.Forcing axioms - Assert a kind of ’saturation’ of theuniverse of sets, i.e. if a set satisfying certain propertiescan be found in a suitable generic extension of theuniverse then such a set already exists. Decidecombinatorial questions about uncountable cardinals leftopen by ZFC. Have strong influence on cardinal arithmetic.

These two types of axioms are very closely intertwined.Typically one needs large cardinals to prove the consistency ofstrong forcing axioms.














Outline

1 The Godel program

2 Forcing axioms




Forcing axioms

(Martin and Solovay) Martin’s Axiom (MAℵ1). Resolvesmany questions about sets of reals, uncountable trees, etc.However, does not decide many problems, e.g. does notsettle the value of the continuum.

(Baumgartner and Shelah) Proper Forcing Axiom (PFA)Very successful in resolving questions left open by MAℵ1 .

(Foreman, Magidor, Shelah) Martin’s Maximum (MM) - theprovably maximal forcing axiom.


Forcing axioms





Forcing axioms





General form of forcing axioms. Let K be a class of forcingnotions and κ an uncountable cardinal.

FAκ(K)

For every P ∈ K and a family D of κ dense subsets of P thereis a filter G in P such that G ∩ D 6= ∅, for all D ∈ D.

MAℵ1 ≡ FAℵ1(ccc)

PFA ≡ FAℵ1(proper)

SPFA ≡ FAℵ1(semi-proper)

MM ≡ FAℵ1(stationary preserving) ≡ SPFA.

RemarkK cannot be the class of all posets or even all posetspreserving ℵ1.



FAκ(K)









FAκ(K)









FAκ(K)








Stationary sets in P(I).

An algebra on I is just a function F : I<ω → I.C ⊆ P(I) is club if there is an algebra F such that

C = {X ⊆ I : clF (X ) = X}.

S ⊆ P(I) is stationary if S ∩ C 6= ∅, for all club C ⊆ P(I).

Example

Given an infinite cardinal κ ≤ |I| the following is stationary

[I]κ = {X ⊆ I : |X | = κ}.

We will be interested in stationary subsets of [I]ℵ0 . This is ageneralization of the usual notion of stationary subsets of aregular cardinal λ > ℵ0.




C = {X ⊆ I : clF (X ) = X}.


Example


[I]κ = {X ⊆ I : |X | = κ}.





C = {X ⊆ I : clF (X ) = X}.


Example


[I]κ = {X ⊆ I : |X | = κ}.





C = {X ⊆ I : clF (X ) = X}.


Example


[I]κ = {X ⊆ I : |X | = κ}.



Definition

A forcing notion P is proper if for every λ > ℵ0 and S ⊆ [λ]ℵ0

stationary P S is stationary in [λ]ℵ0 .

This is stronger than preserving ℵ1.

Equivalently, P is proper if for every θ large enough and everycountable N ≺ Hθ, P ∈ N, p ∈ P ∩ N there is q ≤ p such that

q P N[G] ∩ORD = N ∩ORD.

Such q is called (N,P)-generic.This is the working definition of properness.


Definition

A forcing notion P is proper if for every λ > ℵ0 and S ⊆ [λ]ℵ0

stationary P S is stationary in [λ]ℵ0 .

This is stronger than preserving ℵ1.

Equivalently, P is proper if for every θ large enough and everycountable N ≺ Hθ, P ∈ N, p ∈ P ∩ N there is q ≤ p such that

q P N[G] ∩ORD = N ∩ORD.

Such q is called (N,P)-generic.This is the working definition of properness.


Key feature of proper forcing [Shelah]

A countable support iteration of proper forcing notions is proper.

Semi proper forcing notions. Same as above, but require only

q P N[G] ∩ ω1 = N ∩ ω1.

Such q is called (N,P)-semi-generic.

Shelah defined revised countable support and showed that arevised countable support iteration of semi-proper forcingnotions is semi-proper.





q P N[G] ∩ ω1 = N ∩ ω1.







q P N[G] ∩ ω1 = N ∩ ω1.




Martin’s Maximum (MM) is FAℵ1(K) for K the class of all forcingnotions preserving stationary subsets of ω1. Such forcingnotions cannot be iterated, but MM is equivalent toFAℵ1(semi-proper), i.e. SPFA and semi-proper forcing notionscan be iterated.


The consistency proof of PFA and MM uses a supercompactcardinal. These axioms resolve many questions left open byMAℵ1 .

PFA implies

every two ℵ1-dense sets of reals without endpoints areisomorphic

no Kurepa trees on ω1

no ℵ2-Aronszajn trees

�(κ) fails, for all κ > ℵ1

2ℵ0 = 2ℵ1 = ℵ2, etc.


The consistency proof of PFA and MM uses a supercompactcardinal. These axioms resolve many questions left open byMAℵ1 .

PFA implies

every two ℵ1-dense sets of reals without endpoints areisomorphic

no Kurepa trees on ω1

no ℵ2-Aronszajn trees

�(κ) fails, for all κ > ℵ1

2ℵ0 = 2ℵ1 = ℵ2, etc.


The consistency proof of PFA and MM uses a supercompactcardinal. PFA and MM resolve many questions left open byMAℵ1 .

MM implies

NSω1 is ℵ2-saturated

Chang’s conjecture (ℵ2,ℵ1) → (ℵ1,ℵ0)

κℵ1 = κ, for all regular κ > ℵ1, etc...


The consistency proof of PFA and MM uses a supercompactcardinal. PFA and MM resolve many questions left open byMAℵ1 .

MM implies

NSω1 is ℵ2-saturated

Chang’s conjecture (ℵ2,ℵ1) → (ℵ1,ℵ0)

κℵ1 = κ, for all regular κ > ℵ1, etc...


MAℵ1 can be applied even without knowing forcing. This is nottrue for PFA and MM. In applying, e.g. PFA one often has touse finite iterations of proper forcing notions.

IdeaFormulate combinatorial principles which follow from strongforcing axioms, but can be applied directly. Increase theusefulness of these axioms.

One very useful principle which follows from MM but not PFA isstationary set reflection.










Outline

1 The Godel program

2 Forcing axioms




Stationary set reflection

Definition

Suppose S ⊆ [κ]ℵ0 is stationary and X ⊆ κ. S reflects to X ifS ∩ [X ]ℵ0 is stationary in [X ]ℵ0

Stationary set reflection (SSR)

For all κ ≥ ℵ1 and S ⊆ [κ]ℵ0 there is X ⊆ κ of size ℵ1 such thatω1 ⊆ X and S reflects to X .

One can strengthen SSR by requiring X to be the trace on κ ofsome internally approachable N ≺ Hθ (θ regular, large enough)of size ℵ1. N is called internally approachable if it is the unionof a continuous ∈-chain of elementary submodels. Thisstronger principle is SSR∗.



Definition







Definition






There is an even stronger reflection principle.

Suppose S ⊆ [κ]ℵ0 is stationary and E ⊆ ω1 is stationary. Then

SE = {X ∈ S : X ∩ ω1 ∈ E}.

Definition

S ⊆ [κ]ℵ0 is projective stationary if SE is stationary, for everystationary E ⊆ ω1.

Example

Suppose κ = ℵ2. Let T ⊆ {α < ω2 : cof(α) = ω} be stationary.Let

S = {X ∈ [ω2]ℵ0 : sup(X ) ∈ T}.

Then S is projective stationary.




SE = {X ∈ S : X ∩ ω1 ∈ E}.

Definition


Example


S = {X ∈ [ω2]ℵ0 : sup(X ) ∈ T}.





SE = {X ∈ S : X ∩ ω1 ∈ E}.

Definition


Example


S = {X ∈ [ω2]ℵ0 : sup(X ) ∈ T}.





SE = {X ∈ S : X ∩ ω1 ∈ E}.

Definition


Example


S = {X ∈ [ω2]ℵ0 : sup(X ) ∈ T}.



Strong Reflection Principle ( SRP)[Todorcevic]

Let κ > ℵ1 and θ > κ regular. Suppose S ⊆ [κ]ℵ0 is projectivestationary. Let θ be regular, large enough. Then there is acontinuous, increasing, ∈-chain 〈Nξ : ξ < ω1〉 of countableelementary submodels of Hθ such that Nξ ∩ κ ∈ S, for all ξ.

It is not obvious that SRP implies SSR. Given a stationaryS ⊆ [κ]ℵ0 one first uses SRP to show that there is a stationaryE ⊆ ω1 such that

T = {X ∈ [κ]ℵ0 : X ∈ S or X /∈ E}

is stationary and then applies SRP to T .


Strong Reflection Principle ( SRP)[Todorcevic]

Let κ > ℵ1 and θ > κ regular. Suppose S ⊆ [κ]ℵ0 is projectivestationary. Let θ be regular, large enough. Then there is acontinuous, increasing, ∈-chain 〈Nξ : ξ < ω1〉 of countableelementary submodels of Hθ such that Nξ ∩ κ ∈ S, for all ξ.

It is not obvious that SRP implies SSR. Given a stationaryS ⊆ [κ]ℵ0 one first uses SRP to show that there is a stationaryE ⊆ ω1 such that

T = {X ∈ [κ]ℵ0 : X ∈ S or X /∈ E}

is stationary and then applies SRP to T .


PFA does not imply SSR, but there is a technical strengtheningPFA+ which holds in the standard model for PFA and whichimplies SSR∗.

FactSSR∗ implies �(κ) fails, for all regular κ > ω1. Thus SSRhas considerable large cardinals.

SSR implies κℵ0 = κ, for all regular κ > ℵ1.


PFA does not imply SSR, but there is a technical strengtheningPFA+ which holds in the standard model for PFA and whichimplies SSR∗.

FactSSR∗ implies �(κ) fails, for all regular κ > ω1. Thus SSRhas considerable large cardinals.

SSR implies κℵ0 = κ, for all regular κ > ℵ1.


The stronger principle SRP is a consequence of MM, but not ofPFA (or PFA+). In fact, it is used in many applications of MMwhich do not follow from PFA. For instance,

FactSRP implies Chang’s conjecture

SRP implies NSω1 is saturated, etc.


The stronger principle SRP is a consequence of MM, but not ofPFA (or PFA+). In fact, it is used in many applications of MMwhich do not follow from PFA. For instance,

FactSRP implies Chang’s conjecture

SRP implies NSω1 is saturated, etc.


We sketch the proof of the following.

Theorem (Foreman, Magidor, Shelah)

Assume SRP. Then κℵ1 = κ, for every regular κ > ω1.

Proof.Let S = {α < κ : cof(α) = ω}. Partition S into κ disjointstationary sets Sξ, ξ < κ. Fix a partition ω1 =

⋃{Eη : η < ω1}

into disjoint stationary sets. Let f : ω1 → κ be 1− 1. Let

Uf = {X ∈ [κ]ℵ0 : X ∩ ω1 ∈ Eξ ↔ sup(X ) ∈ Sf (ξ)}.

One can show Uf is projective stationary.


We sketch the proof of the following.

Theorem (Foreman, Magidor, Shelah)

Assume SRP. Then κℵ1 = κ, for every regular κ > ω1.

Proof.Let S = {α < κ : cof(α) = ω}. Partition S into κ disjointstationary sets Sξ, ξ < κ. Fix a partition ω1 =

⋃{Eη : η < ω1}

into disjoint stationary sets. Let f : ω1 → κ be 1− 1. Let

Uf = {X ∈ [κ]ℵ0 : X ∩ ω1 ∈ Eξ ↔ sup(X ) ∈ Sf (ξ)}.

One can show Uf is projective stationary.


Now get a continuous increasing ∈-chain 〈Nξ : ξ < ω1〉 ofcountable elementary submodels of some Hθ (θ regular, largeenough) such that Nξ ∩ κ ∈ Uf , for all ξ.

Let δξ = sup(Nξ ∩ κ) and δ = supξ δξ. Then δ codes f in thesense that f is the unique function g : ω1 → κ such that there isa continuous increasing ϕ : ω1 → δ cofinal in δ such that forevery η

η ∈ Eξ ↔ ϕ(η) ∈ Sg(ξ).

Every injective f : ω1 → κ is coded by some δ < κ in this way.So, κω1 = κ.




η ∈ Eξ ↔ ϕ(η) ∈ Sg(ξ).





η ∈ Eξ ↔ ϕ(η) ∈ Sg(ξ).



Corollary

SRP implies1 2ℵ0 = 2ℵ1 = ℵ2

2 The Singular Cardinal Hypothesis, i.e. if κ is a singularstrong limit then 2κ = κ+.

In 1. one needs to show also that SRP implies that CH fails.

2. is proved by induction on cof(κ). If cof(κ) ≤ ω1 apply theabove theorem to κ+. The general case follows by Silver’stheorem.


Corollary






Corollary






Corollary






We will present another combinatorial principle which is similarin spirit to stationary set reflection, yet follows from PFA.

This principle, the Mapping Reflection Principle (MRP) wasintroduced by Justin Moore in 2003 and used by him and otherto resolve a number of important open problems.

Before we do that we need to make a detour and discussbounded forcing axioms.


Outline

1 The Godel program

2 Forcing axioms




Bounded Forcing Axioms

Introduced by Goldstern and Shelah in the mid 1990s. Havemany of the same consequences, yet require much smallerlarge cardinal assumptions than PFA and MM. The followingformulation is due to Bagaria. Let K be a class of forcingnotions.

DefinitionBFA(K) For every P ∈ K

(Hℵ2 ,∈) ≺Σ1 (VP ,∈).

So, BFA(K) states that for every Σ0 formula ψ(x ,a) withparameter a ∈ Hℵ2 , if some forcing from K introduces a witnessx for ψ(x ,a), then such an x already exists.





(Hℵ2 ,∈) ≺Σ1 (VP ,∈).






(Hℵ2 ,∈) ≺Σ1 (VP ,∈).



Many consequences of strong forcing axioms about subsets ofω1 already follow from their bounded versions. Yet, boundedforcing axioms have much smaller consistency strength.

DefinitionA cardinal κ is called Σ1-reflecting if it is inaccessible and forevery a ∈ Vκ and formula ϕ(a) if there is α such that Vα |= ϕ(a)then there is β < κ such that Vβ |= ϕ(a).

Reflecting cardinals are rather weak. If λ is Mahlo then

S = {κ < λ : κ inaccessible and Vκ ≺ Vλ}

is stationary. For every κ ∈ S Vλ |= κ is reflecting. So, we cancut the universe to Vλ.














TheoremGoldstern, Shelah

CON(BPFA) ↔ CON(there is a reflecting cardinal)

CON(BSPFA) ↔ CON(there is a reflecting cardinal)

REMARK While SPFA is equivalent to MM the same is not truefor the bounded versions. In fact, Schindler has shown thatBMM has much higher consistency strength than a reflectingcardinal, i.e. at least a strong cardinal.


TheoremGoldstern, Shelah

CON(BPFA) ↔ CON(there is a reflecting cardinal)

CON(BSPFA) ↔ CON(there is a reflecting cardinal)

REMARK While SPFA is equivalent to MM the same is not truefor the bounded versions. In fact, Schindler has shown thatBMM has much higher consistency strength than a reflectingcardinal, i.e. at least a strong cardinal.

Date post:	02-Aug-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Forcing axioms and cardinal arithmetic Iboban/pdf/LC2006_PART1.pdf · 2006-07-21 · Large cardinal...

Documents