Large Cardinals Laura Fontanella University of Paris 7 2 nd June 2010 Laura Fontanella (ICIS) Sminaire des thsards 18/05/10 1 / 18
Introduction Introduction Cohen (1963) CH is independent from ZFC . G¨ odel’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¨ odel’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¨ odel’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¨ odel’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 V κ is a model of set theory. We can’t prove the existence of an inaccessible cardinal (G¨ odel). 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 V κ is a model of set theory. We can’t prove the existence of an inaccessible cardinal (G¨ odel). 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 { X α ; α < γ } with γ < κ, [ X α ∈ U ⇒ ∃ α < γ ( X α ∈ U ) . α<γ 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 { X α ; α < γ } with γ < κ, [ X α ∈ U ⇒ ∃ α < γ ( X α ∈ U ) . α<γ 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 { X α ; α < γ } with γ < κ, [ X α ∈ U ⇒ ∃ α < γ ( X α ∈ U ) . α<γ 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¨ odel’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¨ odel’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¨ odel’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 ( x 1 , ..., x n ) is a formula of set theory, then Ult ( U , V ) | = ϕ ([ f 1 ] , ..., [ f n ]) ⇐ ⇒ { x ∈ S ; ϕ ( f 1 ( x ) , ..., f n ( 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 . j π � Ult � M V 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 ( x 1 , ..., x n ) is a formula of set theory, then Ult ( U , V ) | = ϕ ([ f 1 ] , ..., [ f n ]) ⇐ ⇒ { x ∈ S ; ϕ ( f 1 ( x ) , ..., f n ( 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 . j π � Ult � M V 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 ( x 1 , ..., x n ) is a formula of set theory, then Ult ( U , V ) | = ϕ ([ f 1 ] , ..., [ f n ]) ⇐ ⇒ { x ∈ S ; ϕ ( f 1 ( x ) , ..., f n ( 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 . j π � Ult � M V 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
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