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Large Cardinals - IMJ- pinto/Slides/Laura2juin2010.pdf Strongly Compact Cardinals Strongly Compact Cardinals Definition is strongly compact if for all set S;every -complete filter

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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

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