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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.

Gödel’s Program

Let’s find new axioms!

Forcing Axioms

They imply ¬CH.

Large Cardinal Axioms

They don’t decide the Continuum Problem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 2 / 18

Introduction

Introduction

Cohen (1963)

CH is independent from ZFC.

Gödel’s Program

Let’s find new axioms!

Forcing Axioms

They imply ¬CH.

Large Cardinal Axioms

They don’t decide the Continuum Problem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 2 / 18

Introduction

Introduction

Cohen (1963)

CH is independent from ZFC.

Gödel’s Program

Let’s find new axioms!

Forcing Axioms

They imply ¬CH.

Large Cardinal Axioms

They don’t decide the Continuum Problem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 2 / 18

Introduction

Introduction

Cohen (1963)

CH is independent from ZFC.

Gödel’s Program

Let’s find new axioms!

Forcing Axioms

They imply ¬CH.

Large Cardinal Axioms

They don’t decide the Continuum Problem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 2 / 18

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 regular cardinal.

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 (Gödel). So the first large cardinal axiom is:

let’s assume such a large cardinal exists!

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 3 / 18

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 regular cardinal.

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 (Gödel). So the first large cardinal axiom is:

let’s assume such a large cardinal exists!

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 3 / 18

Inaccessible cardinals

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

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 4 / 18

Inaccessible cardinals

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

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 4 / 18

Measurable Cardinals

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

Measurable Cardinals

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

Measurable Cardinals

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

Measurable Cardinals

Measurable

�� Mahlo

�� Inaccessible

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 6 / 18

Measurable Cardinals

Embeddings

Definition

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

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

Theorem

If there is a measurable cardinal, then there is an inner model M and an elementary embedding 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 the class of all [f ], where f is a function on S.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 7 / 18

Measurable Cardinals

Embeddings

Definition

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

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

Theorem

If there is a measurable cardinal, then there is an inner model M and an elementary embedding 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 the class of all [f ], where f is a function on S.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 7 / 18

Measurable Cardinals

Embeddings

Definition

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

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

Theorem

If there is a measurable cardinal, then there is an inner model M and an elementary embedding 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 the class of all [f ], where f is a function on S.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 7 / 18

Measurable Cardinals

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.

V j // Ult

π // M

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

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 8 / 18

Measurable Cardinals

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.

V j // Ult

π // M

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

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 8 / 18

Measurable Cardinals

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.

V j // Ult

π // M

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

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 8 / 18

Measurable Cardinals

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 an elementary embedding j : V → M such that cr(j) = κ.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 9 / 18

Measurable Cardinals

Theorem

M is an inner model (Ord ⊆ M).

Som