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Large Cardinals

Laura Fontanella

University of Paris 7

2nd June 2010

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 1 / 18

Introduction

Introduction

Cohen (1963)

CH is independent from ZFC.

Godel’s Program

Let’s find new axioms!

Forcing Axioms

They imply ¬CH.

Large Cardinal Axioms

They don’t decide the Continuum Problem.


Introduction

Introduction

Cohen (1963)


Godel’s Program


Forcing Axioms

They imply ¬CH.




Introduction

Introduction

Cohen (1963)


Godel’s Program


Forcing Axioms

They imply ¬CH.




Introduction

Introduction

Cohen (1963)


Godel’s Program


Forcing Axioms

They imply ¬CH.




Inaccessible cardinals

Inaccessible Cardinals

Definition

An weakly inaccessible cardinal is a limit and regular cardinal.A strongly inaccessible cardinal (or just inaccessible) is a strong limit and regularcardinal.

Theorem

If there is an inaccessible cardinal κ, then Vκ is a model of set theory.

We can’t prove the existence of an inaccessible cardinal (Godel). So the first largecardinal axiom is:

let’s assume such a large cardinal exists!



Inaccessible Cardinals

Definition

An weakly inaccessible cardinal is a limit and regular cardinal.A strongly inaccessible cardinal (or just inaccessible) is a strong limit and regularcardinal.

Theorem

If there is an inaccessible cardinal κ, then Vκ is a model of set theory.

We can’t prove the existence of an inaccessible cardinal (Godel). So the first largecardinal axiom is:

let’s assume such a large cardinal exists!



Mahlo Cardinals

Why don’t we assume there are ”a lot” of inaccessible cardinals?

Definition

A Mahlo cardinal is an inaccessible cardinal κ such that{λ < κ;λ is an inaccessible cardinal } is stationary in κ.

Mahlo

��Inaccessible



Mahlo Cardinals

Why don’t we assume there are ”a lot” of inaccessible cardinals?

Definition

A Mahlo cardinal is an inaccessible cardinal κ such that{λ < κ;λ is an inaccessible cardinal } is stationary in κ.

Mahlo

��Inaccessible


Measurable Cardinals


Definition

κ is a measurable cardinal if there exists a κ-complete (non principal) ultrafilter over κ.

An ultrafilter U is κ-complete if for all family {Xα;α < γ} with γ < κ,[Xα

α<γ

∈ U ⇒ ∃α < γ(Xα ∈ U).

Proposition

If U is a κ-complete ultrafilter over κ, then the function µ : P(κ)→ {0, 1} defined byµ(X ) = 1 ⇐⇒ X ∈ U is a measure over κ.

Proposition

Every measurable cardinal is inaccessible.




Definition



α<γ

∈ U ⇒ ∃α < γ(Xα ∈ U).

Proposition


Proposition





Definition



α<γ

∈ U ⇒ ∃α < γ(Xα ∈ U).

Proposition


Proposition




Measurable

��Mahlo

��Inaccessible



Embeddings

Definition

Let M ⊆ V , we say M is an inner model if (M,∈) is a transitif model of ZFC withOrd ⊆ M.

Example: Godel’s univers L is an inner model.

Theorem

If there is a measurable cardinal, then there is an inner model M and an elementaryembedding j : V → M

Let U be a ultrafilter on a set S, and let f , g be functions with domain S, we define:

f =∗ g ⇐⇒ {x ∈ S; f (x) = g(x)} ∈ U

f ∈∗ g ⇐⇒ {x ∈ S; f (x) ∈ g(x)} ∈ U

For each f , we denote [f ] the equivalence class of f (w.r.t. =∗) and Ult(U,V ) is theclass of all [f ], where f is a function on S.



Embeddings

Definition



Theorem



f =∗ g ⇐⇒ {x ∈ S; f (x) = g(x)} ∈ U

f ∈∗ g ⇐⇒ {x ∈ S; f (x) ∈ g(x)} ∈ U




Embeddings

Definition



Theorem



f =∗ g ⇐⇒ {x ∈ S; f (x) = g(x)} ∈ U

f ∈∗ g ⇐⇒ {x ∈ S; f (x) ∈ g(x)} ∈ U




Ult(U,V ) is an ultrapower of the univers.

If ϕ(x1, ..., xn) is a formula of set theory, then

Ult(U,V ) |= ϕ([f1], ..., [fn]) ⇐⇒ {x ∈ S;ϕ(f1(x), ..., fn(x))} ∈ U.

There is, then, an elementary embedding j : V → Ult(U,V ), defined by j(x) = [x ].

Theorem

If U is a κ-complete ultrafilter, then Ult(U,V ) is a well founded model of ZFC.

Corollary

If U is a κ-complete ultrafilter, then Ult(U,V ) is isomorphic to a transitive model of ZFC.

Vj // Ult

π // M

We will denote [f ] the set π([f ]) to simplify notation.







Theorem


Corollary


Vj // Ult

π // M








Theorem


Corollary


Vj // Ult

π // M




Theorem

M is an inner model (Ord ⊆ M).

Some properties:

j(α) = α, for all α < κ;

j(κ) > κ.

We say that κ is the critical point (and we write cr(j) = κ).

Theorem

A cardinal κ is measurable if, and only if there exists an inner model M and anelementary embedding j : V → M such that cr(j) = κ.



Theorem


Some properties:


j(κ) > κ.


Theorem




Theorem


Some properties:


j(κ) > κ.


Theorem




Theorem

If there is a measurable cardinal, then V 6= L.

Proof.

Assume V = L and κ is the least measurable cardinal. Let U be a κ-complete ultrafilteron κ and j : V → M the corresponding elementary embedding. Then j(κ) > κ. Theuniverse is the only inner model, that is V = M = L. By elementarity,M |= j(κ) is the least measurable cardinal, hence j(κ) is the least measurable cardinal.But j(κ) > κ. Contradiction.


Strongly Compact Cardinals


Definition

κ is strongly compact if for all set S, every κ-complete filter on S can be extended to aκ-complete ultrafilter on S.

Theorem

κ is strongly compact if, and only if, the language Lκ,κ satisfies the StrongCompactness Theorem.

Theorem

κ is weakly compact if, and only if, the language Lκ,κ satisfies the Weak CompactnessTheorem.




Definition


Theorem


Theorem





Definition


Theorem


Theorem




Strongly Compact

��Measurable

��Weakly Compact

��Inaccessible


Supercompact Cardinals


Definition

κ is supercompact if for all S such that |S| ≥ κ there is a normal and κ-completeultrafilter on S.

Definition

κ is λ-supercompact if there exists an elementary embedding j : V → M such that:

cr(j) = κ;

j(κ) > λ;

Mλ ⊆ M.

Theorem

If there is a supercompact cardinal, then Cons(PFA).




Definition


Definition


cr(j) = κ;

j(κ) > λ;

Mλ ⊆ M.

Theorem





Definition


Definition


cr(j) = κ;

j(κ) > λ;

Mλ ⊆ M.

Theorem



The Hierarchy of Large Cardinals

The Hierarchy

Supercompact

��Strongly Compact

��Measurable


��Mahlo

��Inaccessible


Ramsey Cardinals

Ramsey Cardinals

Theorem

κ is weakly compact if, and only if, κ is inaccessible and has the Tree Property for κ

Definition

We say that κ has the Tree Property if every tree of height κ and each level ofcardinality less than κ, has a branch of cardinality κ.

Theorem

κ est weakly compact if, and only if, κ→ (κ)22.

Definition

κ is Ramsey if κ→ (κ)<ω


Ramsey Cardinals

Ramsey Cardinals

Theorem


Definition


Theorem


Definition



Ramsey Cardinals

Ramsey Cardinals

Theorem


Definition


Theorem


Definition




The Hierarchy

Supercompact

��Strongly Compact

��Measurable

��Ramsey


��Inaccessible



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Vopenka’s Principle��

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Superstrong

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Conclusion

Conclusion

What about CH?

Large cardinal axioms imply Consistency of Forcing Axioms;

Forcing axioms imply ¬CH.

So, why should we be interested in Large Cardinal Axioms?

Large cardinal axioms imply V 6= L.


Conclusion

Conclusion

What about CH?

Large cardinal axioms imply Consistency of Forcing Axioms;

Forcing axioms imply ¬CH.

So, why should we be interested in Large Cardinal Axioms?

Large cardinal axioms imply V 6= L.
