Multiverse views and the Ω Conjecture

W. Hugh Woodin

Ziwet LecturesUniversity of Michigan

November 4, 2010

Does the Continuum Hypothesis have an answer?

(Is Set Theory meaningful?)

The skeptical view

The Continuum Hypothesis is neither true nor false because theentire conception of the universe of sets is a complete fiction.

I All the theorems of set theory are merely finitistic truths, areflection of the mathematician and not of any genuinemathematical “reality”.

In search of V ... modeling multiverse views

Start with a countable transitive set M such that M |= ZFCI think of M as V but from the perspective of a “super” V .

Specify the multiverse M generates

I as a collection of transitive sets or just as a collection ofmodels of ZFC.

1. The multiverse is M.I This is Platonism.

2. The multiverse is set of all (countable) models of ZFC.I In this case the multiverse does not depend on M but just on

the integers.

I This is formalism.I this view reduces truth in Set Theory to truth in Number

Theory.

I Cohen’s method of forcing suggests an intermediate possibility.

The generic-multiverse of sets

Suppose that M is a countable transitive set and that

M |= ZFC.

Let VM be the smallest set of countable transitive sets such thatM ∈ VM and such that for all pairs, (M1,M2), of countabletransitive sets, if

1. M1 |= ZFC,

2. M2 is a generic extension of M1,

3. M1 ∈ VM or M2 ∈ VM ,

then both M1 and M2 are in VM .

Definition

VM is the generic-multiverse generated from M.

Evaluating truth in the generic-multiverse...

Theorem

For each sentence ϕ there is a sentence ϕ∗ such that for allcountable transitive sets M if

M |= ZFC

then the following are equivalent.

1. M |= ϕ∗,

2. For all N ∈ VM , N |= ϕ.

I The same theorem holds where the generic-multiverse, VM , isreplaced by any refinement which is based on a definablerestriction on the forcing notions allowed, for example:cardinal preserving, proper, semi-proper, etc.

The generic-multiverse view of truth

A Π2-sentence, ϕ, is a generic-multiverse truth if ϕ holds in eachuniverse of the generic-multiverse generated by V .

Theorem

Suppose there is a proper class of strongly inaccessible cardinals.Then the following are equivalent in the generic-multiverse view oftruth (each if true implies the truth of the other).

1. L(R) |= AD.

2. L(R) 6|= Axiom of Choice.

Question

Is the generic-multiverse view of truth a reasonable one?

Question

Does it realize the goal of finding a conception of truth in SetTheory which is comparable to our conception of truth in NumberTheory?

Potential advantage

Cohen’s method of forcing cannot be used to show independenceof a sentence in this view of truth:

I but the resolution may be that the sentence is meaningless.

Ω-logic(The logic of the generic-multiverse)

Definition

Suppose ϕ is a Π2-sentence. Then

|=Ω ϕ

if ϕ holds in all generic extensions of V .

Theorem

Suppose there is a proper class of Woodin cardinals and that ϕ is aΠ2-sentence.

Then ϕ is a generic-multiverse truth if and only if |=Ω ϕ.

Universally Baire sets and strong closure

Definition

Suppose that A ⊆ R is universally Baire and suppose that M is acountable transitive model of ZFC.

Then M is strongly A-closed if for all countable transitive sets Nsuch that N is a generic extension of M,

A ∩ N ∈ N.

Examples where M is necessarily strongly A-closed.

1. A is finite.

2. A is ∆11.

3. (Shoenfield Absoluteness) A is Π11.

The definition of `Ω ϕ

Definition

Suppose there is a proper class of Woodin cardinals. Suppose thatϕ is a Π2-sentence.

Then `Ω ϕ if there exists a set A ⊂ R such that:

1. A is universally Baire,

2. for all countable transitive models, M, if M is stronglyA-closed then

M |= “|=Ω ϕ”.

I “`Ω ϕ” is invariant across the generic-multiverse.

The Ω Conjecture

Theorem (Ω Soundness)

Suppose that there exists a proper class of Woodin cardinals andsuppose that ϕ is Π2-sentence.

If `Ω ϕ then |=Ω ϕ

Definition (Ω Conjecture)

Suppose that there exists a proper class of Woodin cardinals andsuppose that ϕ is a Π2-sentence.

Then |=Ω ϕ if and only if `Ω ϕ.

The Ω Conjecture and generic-multiverse view of truth

Definition

1. T0 is the set of sentences ϕ such that |=Ω “H(ω2) |= ϕ”.

2. T is the set of Π2-sentences ϕ such that |= ϕ.

I T0 is trivially (recursively) reducible to T .

Theorem

Suppose that there is a proper class of Woodin cardinals and thatthe Ω Conjecture holds.

Then T is (recursively) reducible to T0.

Claim

If there is a proper class of Woodin cardinals and the Ω Conjectureholds then the generic-multiverse view of truth is not viable.

The generic-multiverse view of truth is simply a form offormalism; it reduces truth to Third Order Number Theory.

Claim

If there is a proper class of Woodin cardinals and the Ω Conjectureholds then there is no (mathematical) evidence that theContinuum Hypothesis has no answer.

HOD and the Ω Conjecture

Question

But is the Ω Conjecture true?

Theorem

Suppose that there is a proper class of Woodin cardinals and thatfor all sets A ⊂ R, if A is OD then A is universally Baire.

Then HOD |= “ The Ω Conjecture ”.

Theorem

The Ω Conjecture is invariant across the generic-multiverse.

I The generic invariance of the Ω Conjecture suggests that ifthe Ω Conjecture is false it must be refuted from some largecardinal hypothesis.

Large cardinal axioms

I There is a proper class of inaccessible cardinals.

I There is a proper class of measurable cardinals.

I There is a proper class of Woodin cardinals.

I There is a proper class of superstrong cardinals.

· · · · · · · · · · · ·

I There is a proper class of supercompact cardinals.

I There is a proper class of extendible cardinals.

I There is a proper class of huge cardinals.

I There is a proper class of ω-huge cardinals.

The hierarchy of large cardinal axioms has emerged as thefundamental core of Set Theory.

I It is (empirically) a wellordered hierarchy and provides acalibration of the unsolvability of problems in Set Theory.

Theorem

Suppose that there is a proper class of Woodin cardinals and thatthere is no transitive set M such that

M |= ZFC + “There is a proper class of Woodin cardinals”.

Then the Ω Conjecture holds.

The Inner Model Program

The inner model problem

For a specified large cardinal axiom, produce a generalization ofGodel’s construction of L which is compatible with the given largecardinal axiom.

I Fundamental issue: how to make this problem precise.

Goals

I Understand the hierarchy of large cardinals.I Use this to understand the Universe of Sets.

I perhaps even to find an ultimate version of L.

The building blocks for inner models: Extenders

Suppose thatj : V → M

is an elementary embedding with critical point κ, κ < γ, and that

Vγ+ω ⊂ M.

The extender E of length γ derived from j

The extender E of length γ defined from j is the function:

E : P(γ)→ P(γ)

where E (A) = j(A) ∩ γ.

Two ordinals associated to the extender E :

I CRT(E ) = minα E (α) 6= α = κ.I LTH(E ) = γ where dom(E ) = P(γ).

Three fundamental large cardinal axioms formulated interms of extenders

Definition

1. δ is a strong cardinal if for each γ > δ there is an extender Esuch that

(a) LTH(E ) ≥ γ,(b) CRT(E ) = δ.

2. δ is a supercompact cardinal if for each γ > δ there is anextender E such that

(a) LTH(E ) ≥ γ,(b) E (κ) = δ where κ = CRT(E ).

3. δ is a extendible cardinal if for each γ > δ there is an extenderE such that

(a) CRT(E ) = δ and γ < LTH(E ),(b) γ < E (δ) and E (γ) < LTH(E ). ut

Martin-Steel Extender Models

Theorem (Martin, Steel)

Assume V = HOD. Then there is an extender sequenceE = 〈Eα : α ∈ Ord〉 such that the following hold.

1 E is Σ2-definable.

2 if δ is a strong cardinal then 〈Eα : α ∈ Ord〉 witnesses that δis a strong cardinal.

3 L[E ] 6= V . ut

I (1) and (2) imply that every strong cardinal of V is a strongcardinal of L[E ].

Extender models and weak extender models forsupercompactness

Definition

Suppose E = 〈Eα : α ∈ Ord〉 is a sequence of extenders whichwitnesses that δ is supercompact.

Then L[E ] is an extender model for δ is supercompact. ut

Definition

A transitive class N |= ZFC is a weak extender model for δ issupercompact if there is a class E of extenders such that

1. E witnesses δ is supercompact,

2. E |N ∈ N for each E ∈ E . ut

I If N is an extender model for δ is supercompact then N is aweak extender model for δ is supercompact.

The Universality Theorem

Theorem

Suppose that N is a weak extender model for δ is supercompact.Suppose E is an extender of length γ such that

1 for all a ⊂ γ, if a ∈ N then E (a) ∈ N,

2 CRT(E ) ≥ δ.

Then E |N ∈ N. ut

Ramifications

Any generalization of L to a weak extender model for δ issupercompact is necessarily an ultimate version of L.

Definition

Suppose that E = Eα : α ∈ Ord is an extender sequence.

Then E is weakly Σ2-definable if there is a formula ϕ(x) such thatfor all β ∈ Ord,

I for all β < η1 < η2 < η3, if

(E )Vη1 |β = (E )Vη3 |β

then (E )Vη1 |β = (E )Vη2 |β = (E )Vη3 |β.

where (E )Vγ = a ∈ Vα Vγ |= ϕ[a]. ut

I Similarly define when a transitive class N is weaklyΣ2-definable

I set N|β = N ∩ Vβ .

Starting from large cardinals one can force to obtain:

1. V = HOD.

2. There is a proper class of supercompact cardinals.

3. Suppose E = 〈Eα : α ∈ Ord〉 is any extender sequence andthat δ is any supercompact cardinal such that

(a) E is weakly Σ2-definable,(b) 〈Eα : α ∈ Ord〉 witnesses that δ is supercompact.

Then V ⊆ L[E ].

Ramifications

1. Rules out generalizing the Inner Model Program to thelevel of extender models for δ is supercompact.

2. Significantly constrains the possibilities of generalizingthe Inner Model Program to the level of weak extendermodels for δ is supercompact.

Martin-Steel Inner Models revisited

Theorem (Martin, Steel)

Assume there is a proper class of Woodin cardinals and V = HOD.

Then there is an extender sequence E = 〈Eα : α ∈ Ord〉 such that

1) E is weakly Σ2-definable.

2) if δ is a strong cardinal then 〈Eα : α ∈ Ord〉 witnesses that δis a strong cardinal.

3) Suppose A ∈ L(R)[E ] ∩ P(R). Then A is universally Baire. ut

I Illustrates the connection between the Inner Model Programand the universally Baire sets.

Ultimate L

The axiom for ultimate L

There is a proper class of Woodin cardinals. Further for each Σ3

sentence ϕ, if ϕ holds in V then there is a proper initial segment Γof the universally Baire sets such that

HODL(Γ,R) ∩ VΘ |= ϕ

where Θ = ΘL(Γ,R).

I A natural strengthening would be as a scheme with norestriction on ϕ.

(ULC): The Ultimate L Conjecture

Suppose that there is a proper class of supercompact cardinals.Then for all sufficiently large supercompact cardinals δ, there is atransitive class N ⊂ V such that the following hold.

1. N is Σ2-definable from δ.

2. N is a weak extender model for δ is supercompact.

3. N |= “V = ultimate L”.

Consequences of ULC

Theorem

Assume ZFC ` ULC and that δ is an extendible cardinal. Thenthe following hold.

(1) HOD is a weak extender model for δ is supercompact.

(2) Suppose γ > δ is a singular cardinal. Then γ is a singularcardinal in HOD and

γ+ = (γ+)HOD. ut

More consequences of ULC

Theorem (ZF)

Assume ZFC ` ULC and that δ is an extendible cardinal. Thenthere is a transitive class N such that the following hold.

(1) N |= ZFC.

(2) N is Σ2-definable from N ∩ Vδ.

(3) For all γ > δ, γ is a cardinal if and only if γ is a cardinal inN. ut

Conjecture (ZF)

Suppose that δ is an extendible cardinal and that

G ⊂ Coll(ω,Vδ)

is V -generic. Then V [G ] |= Axiom of Choice. ut

An (infinitely?) optimistic view

ZFC ` ULC.

The axiom, V = ultimate L, reduces all questions in Set Theory tolarge cardinal axioms.

The axiom, V = ultimate L, will be validated on the basis of largecardinal axioms.