Large Cardinals Laura Fontanella University of Paris 7 2 nd June - - PowerPoint PPT Presentation

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. G¨

• del’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. G¨

• del’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. G¨

• del’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. G¨

• del’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 (G¨

• del). 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 (G¨

• del). 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<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 by µ(X) = 1 ⇐ ⇒ X ∈ U is a measure over κ. Proposition Every measurable cardinal is inaccessible.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 5 / 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<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 by µ(X) = 1 ⇐ ⇒ X ∈ U is a measure over κ. Proposition Every measurable cardinal is inaccessible.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 5 / 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<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 by µ(X) = 1 ⇐ ⇒ X ∈ U is a measure over κ. Proposition Every measurable cardinal is inaccessible.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 5 / 18

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: G¨

• del’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: G¨

• del’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: G¨

• del’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). 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). 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 If there is a measurable cardinal, then V = L. Proof. Assume V = L and κ is the least measurable cardinal. Let U be a κ-complete ultrafilter

• n κ and j : V → M the corresponding elementary embedding. Then j(κ) > κ. The

universe 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.

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

Strongly Compact Cardinals

Strongly Compact Cardinals

Definition κ is strongly compact if for all set S, every κ-complete filter on S can be extended to a κ-complete ultrafilter on S. Theorem κ is strongly compact if, and only if, the language Lκ,κ satisfies the Strong Compactness Theorem. Theorem κ is weakly compact if, and only if, the language Lκ,κ satisfies the Weak Compactness Theorem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 11 / 18

Strongly Compact Cardinals

Strongly Compact Cardinals

Definition κ is strongly compact if for all set S, every κ-complete filter on S can be extended to a κ-complete ultrafilter on S. Theorem κ is strongly compact if, and only if, the language Lκ,κ satisfies the Strong Compactness Theorem. Theorem κ is weakly compact if, and only if, the language Lκ,κ satisfies the Weak Compactness Theorem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 11 / 18

Strongly Compact Cardinals

Strongly Compact Cardinals

Definition κ is strongly compact if for all set S, every κ-complete filter on S can be extended to a κ-complete ultrafilter on S. Theorem κ is strongly compact if, and only if, the language Lκ,κ satisfies the Strong Compactness Theorem. Theorem κ is weakly compact if, and only if, the language Lκ,κ satisfies the Weak Compactness Theorem.

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 11 / 18

Strongly Compact Cardinals

Strongly Compact

• Measurable
• Weakly Compact
• Inaccessible

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 12 / 18

Supercompact Cardinals

Supercompact Cardinals

Definition κ is supercompact if for all S such that |S| ≥ κ there is a normal and κ-complete ultrafilter 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).

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 13 / 18

Supercompact Cardinals

Supercompact Cardinals

Definition κ is supercompact if for all S such that |S| ≥ κ there is a normal and κ-complete ultrafilter 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).

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 13 / 18

Supercompact Cardinals

Supercompact Cardinals

Definition κ is supercompact if for all S such that |S| ≥ κ there is a normal and κ-complete ultrafilter 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).

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 13 / 18

The Hierarchy of Large Cardinals

The Hierarchy

Supercompact

• Strongly Compact
• Measurable
• Weakly Compact
• Mahlo
• Inaccessible

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 14 / 18

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 of cardinality less than κ, has a branch of cardinality κ. Theorem κ est weakly compact if, and only if, κ → (κ)2

2.

Definition κ is Ramsey if κ → (κ)<ω

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 15 / 18

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 of cardinality less than κ, has a branch of cardinality κ. Theorem κ est weakly compact if, and only if, κ → (κ)2

2.

Definition κ is Ramsey if κ → (κ)<ω

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 15 / 18

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 of cardinality less than κ, has a branch of cardinality κ. Theorem κ est weakly compact if, and only if, κ → (κ)2

2.

Definition κ is Ramsey if κ → (κ)<ω

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 15 / 18

The Hierarchy of Large Cardinals

The Hierarchy

Supercompact

• Strongly Compact
• Measurable
• Ramsey
• Weakly Compact
• Inaccessible

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 16 / 18

The Hierarchy of Large Cardinals

0=1

• I0-I3
• n-huge
• superhuge
• huge
• almost huge
• Vopenka’s Principle
• Extendible
• Supercompact
• Superstrong
• Strongly compact
• Woodin
• Strong
• 0† exists
• Measurable
• Ramsey

Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 17 / 18

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 = L.

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

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 = L.

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