The axiom of choice is a global forcing axiom. Sketch of proof. I show just the direction I want to bring forward: Assume F : X <ω → P ( X ) is a function. Let T be the subtree of X <ω given by finite sequences s ∈ X <ω such that s ( i ) ∈ F ( s ↾ i ) for all i < | s | . Consider the family given by the dense sets D n = { s ∈ T : | s | > n } . If G is a filter on T meeting the dense sets of this family, ∪ G works. 6 / 51
The axiom of choice is a global forcing axiom. More generally: Definition A partial order P is < λ -closed if all chains in P of length less than λ have a lower bound. Let AC ↾ λ abbreviate DC γ holds for all γ < λ and Γ λ be the class of < λ -closed posets. Fact DC λ is equivalent to FA λ (Γ λ ) over the theory ZF + AC ↾ λ . 7 / 51
The axiom of choice is a global forcing axiom. More generally: Definition A partial order P is < λ -closed if all chains in P of length less than λ have a lower bound. Let AC ↾ λ abbreviate DC γ holds for all γ < λ and Γ λ be the class of < λ -closed posets. Fact DC λ is equivalent to FA λ (Γ λ ) over the theory ZF + AC ↾ λ . 7 / 51
The axiom of choice is a global forcing axiom. More generally: Definition A partial order P is < λ -closed if all chains in P of length less than λ have a lower bound. Let AC ↾ λ abbreviate DC γ holds for all γ < λ and Γ λ be the class of < λ -closed posets. Fact DC λ is equivalent to FA λ (Γ λ ) over the theory ZF + AC ↾ λ . 7 / 51
The axiom of choice is a global forcing axiom. Conclusion: Fact The axiom of choice is equivalent over the theory ZF to the assertion that FA λ (Γ λ ) holds for all λ . The usual forcing axioms such as Martin’s maximum or the proper forcing axiom are natural strenghtenings of the axiom of choice. They aim to isolate a maximal strengthening of AC ↾ ω 2 enlarging the family Γ for which FA ℵ 1 (Γ) holds. 8 / 51
The axiom of choice is a global forcing axiom. Conclusion: Fact The axiom of choice is equivalent over the theory ZF to the assertion that FA λ (Γ λ ) holds for all λ . The usual forcing axioms such as Martin’s maximum or the proper forcing axiom are natural strenghtenings of the axiom of choice. They aim to isolate a maximal strengthening of AC ↾ ω 2 enlarging the family Γ for which FA ℵ 1 (Γ) holds. 8 / 51
Baire’s category theorem is a forcing axiom Theorem (BCT) Assume ( X , τ ) is compact and Hausdorff. Let { D n : n ∈ ω } be a family of dense open subsets of X. Then � n ∈ ω D n is non-empty. Fact FA ℵ 0 ( P ) for all forcing P entails BCT . 9 / 51
Baire’s category theorem is a forcing axiom Theorem (BCT) Assume ( X , τ ) is compact and Hausdorff. Let { D n : n ∈ ω } be a family of dense open subsets of X. Then � n ∈ ω D n is non-empty. Fact FA ℵ 0 ( P ) for all forcing P entails BCT . 9 / 51
Proof. Let ( X , τ ) compact Hausdorff and { D n : n ∈ ω } a family of dense open subsets of X . Let ( P , ≤ P ) = ( τ \ {∅} , ⊆ ) and E n = { A ∈ τ : A ⊆ D n } . Each E n is a dense subset of P . Let G be a filter on P with G ∩ E n � ∅ for all n . By compactness of X � � { Cl ( A ) : A ∈ G } ⊆ ∅ � D n . n ∈ ω � 10 / 51
Proof. Let ( X , τ ) compact Hausdorff and { D n : n ∈ ω } a family of dense open subsets of X . Let ( P , ≤ P ) = ( τ \ {∅} , ⊆ ) and E n = { A ∈ τ : A ⊆ D n } . Each E n is a dense subset of P . Let G be a filter on P with G ∩ E n � ∅ for all n . By compactness of X � � { Cl ( A ) : A ∈ G } ⊆ ∅ � D n . n ∈ ω � 10 / 51
More general forcing axioms Fact Let G be a filter on a poset P and X ⊆ P. Then G ∩ X is non-empty iff G ∩ ↓ X is non-empty. Hence G meets a predense set A iff it meets the dense open set ↓ A . Definition Given a poset P and a property φ , FA φ ( P ) holds if For all D collection of predense subsets of P such that φ ( D ) holds, there exists G filter on P such that G ∩ X � ∅ for all X ∈ D . FA κ ( P ) stands for FA φ ( P ) where φ ( D ) ≡ |D| = κ and each D ∈ D is predense . BFA ω 1 ( P ) stands for FA φ ( RO ( P )) where φ ( D ) ≡ |D| = ω 1 and each D ∈ D is a predense subset of RO ( P ) of size ω 1 . 11 / 51
More general forcing axioms Fact Let G be a filter on a poset P and X ⊆ P. Then G ∩ X is non-empty iff G ∩ ↓ X is non-empty. Hence G meets a predense set A iff it meets the dense open set ↓ A . Definition Given a poset P and a property φ , FA φ ( P ) holds if For all D collection of predense subsets of P such that φ ( D ) holds, there exists G filter on P such that G ∩ X � ∅ for all X ∈ D . FA κ ( P ) stands for FA φ ( P ) where φ ( D ) ≡ |D| = κ and each D ∈ D is predense . BFA ω 1 ( P ) stands for FA φ ( RO ( P )) where φ ( D ) ≡ |D| = ω 1 and each D ∈ D is a predense subset of RO ( P ) of size ω 1 . 11 / 51
More general forcing axioms Fact Let G be a filter on a poset P and X ⊆ P. Then G ∩ X is non-empty iff G ∩ ↓ X is non-empty. Hence G meets a predense set A iff it meets the dense open set ↓ A . Definition Given a poset P and a property φ , FA φ ( P ) holds if For all D collection of predense subsets of P such that φ ( D ) holds, there exists G filter on P such that G ∩ X � ∅ for all X ∈ D . FA κ ( P ) stands for FA φ ( P ) where φ ( D ) ≡ |D| = κ and each D ∈ D is predense . BFA ω 1 ( P ) stands for FA φ ( RO ( P )) where φ ( D ) ≡ |D| = ω 1 and each D ∈ D is a predense subset of RO ( P ) of size ω 1 . 11 / 51
More general forcing axioms Fact Let G be a filter on a poset P and X ⊆ P. Then G ∩ X is non-empty iff G ∩ ↓ X is non-empty. Hence G meets a predense set A iff it meets the dense open set ↓ A . Definition Given a poset P and a property φ , FA φ ( P ) holds if For all D collection of predense subsets of P such that φ ( D ) holds, there exists G filter on P such that G ∩ X � ∅ for all X ∈ D . FA κ ( P ) stands for FA φ ( P ) where φ ( D ) ≡ |D| = κ and each D ∈ D is predense . BFA ω 1 ( P ) stands for FA φ ( RO ( P )) where φ ( D ) ≡ |D| = ω 1 and each D ∈ D is a predense subset of RO ( P ) of size ω 1 . 11 / 51
More general forcing axioms Fact Let G be a filter on a poset P and X ⊆ P. Then G ∩ X is non-empty iff G ∩ ↓ X is non-empty. Hence G meets a predense set A iff it meets the dense open set ↓ A . Definition Given a poset P and a property φ , FA φ ( P ) holds if For all D collection of predense subsets of P such that φ ( D ) holds, there exists G filter on P such that G ∩ X � ∅ for all X ∈ D . FA κ ( P ) stands for FA φ ( P ) where φ ( D ) ≡ |D| = κ and each D ∈ D is predense . BFA ω 1 ( P ) stands for FA φ ( RO ( P )) where φ ( D ) ≡ |D| = ω 1 and each D ∈ D is a predense subset of RO ( P ) of size ω 1 . 11 / 51
Large cardinals as forcing axioms Given a cardinal κ , I κ is the ideal of bounded subsets of κ , A κ is the family of maximal antichains of size less than κ in P ( κ ) / I κ . Definition κ is measurable iff there is a ultrafilter G ∈ P ( κ ) / I κ such that G ∩ A � ∅ for all A ∈ A κ . I.e. κ is measurable if FA φ ( P ( κ ) / I κ ) , where φ ( D ) stands for D = A κ . Cofinally many large cardinal properties of κ can be formulated as forcing axiom of the type FA φ ( P ( P ( λ )) / J κ,λ ) , choosing φ and J κ,λ suitably. for example supercompact, huge, extendible, n -huge, I 1 , etc...... 12 / 51
Large cardinals as forcing axioms Given a cardinal κ , I κ is the ideal of bounded subsets of κ , A κ is the family of maximal antichains of size less than κ in P ( κ ) / I κ . Definition κ is measurable iff there is a ultrafilter G ∈ P ( κ ) / I κ such that G ∩ A � ∅ for all A ∈ A κ . I.e. κ is measurable if FA φ ( P ( κ ) / I κ ) , where φ ( D ) stands for D = A κ . Cofinally many large cardinal properties of κ can be formulated as forcing axiom of the type FA φ ( P ( P ( λ )) / J κ,λ ) , choosing φ and J κ,λ suitably. for example supercompact, huge, extendible, n -huge, I 1 , etc...... 12 / 51
Large cardinals as forcing axioms Given a cardinal κ , I κ is the ideal of bounded subsets of κ , A κ is the family of maximal antichains of size less than κ in P ( κ ) / I κ . Definition κ is measurable iff there is a ultrafilter G ∈ P ( κ ) / I κ such that G ∩ A � ∅ for all A ∈ A κ . I.e. κ is measurable if FA φ ( P ( κ ) / I κ ) , where φ ( D ) stands for D = A κ . Cofinally many large cardinal properties of κ can be formulated as forcing axiom of the type FA φ ( P ( P ( λ )) / J κ,λ ) , choosing φ and J κ,λ suitably. for example supercompact, huge, extendible, n -huge, I 1 , etc...... 12 / 51
Large cardinals as forcing axioms Given a cardinal κ , I κ is the ideal of bounded subsets of κ , A κ is the family of maximal antichains of size less than κ in P ( κ ) / I κ . Definition κ is measurable iff there is a ultrafilter G ∈ P ( κ ) / I κ such that G ∩ A � ∅ for all A ∈ A κ . I.e. κ is measurable if FA φ ( P ( κ ) / I κ ) , where φ ( D ) stands for D = A κ . Cofinally many large cardinal properties of κ can be formulated as forcing axiom of the type FA φ ( P ( P ( λ )) / J κ,λ ) , choosing φ and J κ,λ suitably. for example supercompact, huge, extendible, n -huge, I 1 , etc...... 12 / 51
Ło´ s theorem Theorem Let � M l = ( M l , R l ) : l ∈ L � be first oreder models for L = { R } . Let G ⊆ P ( L ) be a ultrafilter on L. Set [ f ] G = [ h ] G iff � l ∈ L : f ( l ) = h ( l )) � ∈ G, R ([ f 1 ] G , . . . , [ f n ] G ) iff � l ∈ L : R l ( f 1 ( l ) , . . . , f n ( l )) � ∈ G. ¯ Then: l ∈ L M l / G , ¯ For all φ ( x 1 , . . . , x n ) ( � R ) | = φ ([ f 1 ] G , . . . , [ f n ] G ) if and only if 1 � l ∈ L : M l | = φ ( f 1 ( l ) , . . . , f n ( l )) � ∈ G. If M l = M for all l ∈ L, M ≺ � l ∈ L M l / G as witnessed by the map 2 m �→ [ c m ] G (where c m : L → M is constant with value m). 13 / 51
Ło´ s theorem Theorem Let � M l = ( M l , R l ) : l ∈ L � be first oreder models for L = { R } . Let G ⊆ P ( L ) be a ultrafilter on L. Set [ f ] G = [ h ] G iff � l ∈ L : f ( l ) = h ( l )) � ∈ G, R ([ f 1 ] G , . . . , [ f n ] G ) iff � l ∈ L : R l ( f 1 ( l ) , . . . , f n ( l )) � ∈ G. ¯ Then: l ∈ L M l / G , ¯ For all φ ( x 1 , . . . , x n ) ( � R ) | = φ ([ f 1 ] G , . . . , [ f n ] G ) if and only if 1 � l ∈ L : M l | = φ ( f 1 ( l ) , . . . , f n ( l )) � ∈ G. If M l = M for all l ∈ L, M ≺ � l ∈ L M l / G as witnessed by the map 2 m �→ [ c m ] G (where c m : L → M is constant with value m). 13 / 51
Ło´ s theorem Theorem Let � M l = ( M l , R l ) : l ∈ L � be first oreder models for L = { R } . Let G ⊆ P ( L ) be a ultrafilter on L. Set [ f ] G = [ h ] G iff � l ∈ L : f ( l ) = h ( l )) � ∈ G, R ([ f 1 ] G , . . . , [ f n ] G ) iff � l ∈ L : R l ( f 1 ( l ) , . . . , f n ( l )) � ∈ G. ¯ Then: l ∈ L M l / G , ¯ For all φ ( x 1 , . . . , x n ) ( � R ) | = φ ([ f 1 ] G , . . . , [ f n ] G ) if and only if 1 � l ∈ L : M l | = φ ( f 1 ( l ) , . . . , f n ( l )) � ∈ G. If M l = M for all l ∈ L, M ≺ � l ∈ L M l / G as witnessed by the map 2 m �→ [ c m ] G (where c m : L → M is constant with value m). 13 / 51
Recall on boolean algebras and Stone spaces Given a boolean algebra B: St ( B ) is given by its ultrafilters G . St ( B ) is endowed with a compact, Hausdorff topology τ B whose clopens are N b = � G ∈ St ( B ) : b ∈ G � . The map b �→ N b defines a natural isomorphism of B with the boolean algebra CLOP ( St ( B )) of clopen subset of St ( B ) . B is complete if and only if CLOP ( St ( B )) = RO ( St ( B ) , τ B ) � B. Spaces X satisfying the property that its regular open sets are closed are extremally (or extremely) disconnected . P ( X ) is a complete boolean algebra, and β ( X ) = St ( P ( X )) is the Stone-Cech compactification of X with discrete topology and is extremally disconnected. 14 / 51
Recall on boolean algebras and Stone spaces Given a boolean algebra B: St ( B ) is given by its ultrafilters G . St ( B ) is endowed with a compact, Hausdorff topology τ B whose clopens are N b = � G ∈ St ( B ) : b ∈ G � . The map b �→ N b defines a natural isomorphism of B with the boolean algebra CLOP ( St ( B )) of clopen subset of St ( B ) . B is complete if and only if CLOP ( St ( B )) = RO ( St ( B ) , τ B ) � B. Spaces X satisfying the property that its regular open sets are closed are extremally (or extremely) disconnected . P ( X ) is a complete boolean algebra, and β ( X ) = St ( P ( X )) is the Stone-Cech compactification of X with discrete topology and is extremally disconnected. 14 / 51
Recall on boolean algebras and Stone spaces Given a boolean algebra B: St ( B ) is given by its ultrafilters G . St ( B ) is endowed with a compact, Hausdorff topology τ B whose clopens are N b = � G ∈ St ( B ) : b ∈ G � . The map b �→ N b defines a natural isomorphism of B with the boolean algebra CLOP ( St ( B )) of clopen subset of St ( B ) . B is complete if and only if CLOP ( St ( B )) = RO ( St ( B ) , τ B ) � B. Spaces X satisfying the property that its regular open sets are closed are extremally (or extremely) disconnected . P ( X ) is a complete boolean algebra, and β ( X ) = St ( P ( X )) is the Stone-Cech compactification of X with discrete topology and is extremally disconnected. 14 / 51
Recall on boolean algebras and Stone spaces Given a boolean algebra B: St ( B ) is given by its ultrafilters G . St ( B ) is endowed with a compact, Hausdorff topology τ B whose clopens are N b = � G ∈ St ( B ) : b ∈ G � . The map b �→ N b defines a natural isomorphism of B with the boolean algebra CLOP ( St ( B )) of clopen subset of St ( B ) . B is complete if and only if CLOP ( St ( B )) = RO ( St ( B ) , τ B ) � B. Spaces X satisfying the property that its regular open sets are closed are extremally (or extremely) disconnected . P ( X ) is a complete boolean algebra, and β ( X ) = St ( P ( X )) is the Stone-Cech compactification of X with discrete topology and is extremally disconnected. 14 / 51
Boolean valued models Definition Let B be a cba and a L be first order relational language. A B -valued model for L is a tuple M = � M , = M , R M : i ∈ I , c M : j ∈ J � with i j = M : M 2 → B ( τ, σ ) �→ � τ = σ � M B = � τ = σ � , R M : M n → B ( τ 1 , . . . , τ n ) �→ � R ( τ 1 , . . . , τ n ) � M B = � R ( τ 1 , . . . , τ n ) � for R ∈ L an n -ary relation symbol. 15 / 51
Boolean valued models Definition Let B be a cba and a L be first order relational language. A B -valued model for L is a tuple M = � M , = M , R M : i ∈ I , c M : j ∈ J � with i j = M : M 2 → B ( τ, σ ) �→ � τ = σ � M B = � τ = σ � , R M : M n → B ( τ 1 , . . . , τ n ) �→ � R ( τ 1 , . . . , τ n ) � M B = � R ( τ 1 , . . . , τ n ) � for R ∈ L an n -ary relation symbol. 15 / 51
Forcing relations on boolean valued models The constraints on R M and = M are the following: for τ, σ, χ ∈ M , � τ = τ � = 1 B ; 1 � τ = σ � = � σ = τ � ; 2 � τ = σ � ∧ � σ = χ � ≤ � τ = χ � ; 3 for R ∈ L with arity n , and ( τ 1 , . . . , τ n ) , ( σ 1 , . . . , σ n ) ∈ M n , � R ( τ 1 , . . . , τ n ) � ∧ � τ h = σ h � ≤ � R ( σ 1 , . . . , σ n ) � . � h ∈{ 1 ,..., n } 16 / 51
Forcing relations on boolean valued models The constraints on R M and = M are the following: for τ, σ, χ ∈ M , � τ = τ � = 1 B ; 1 � τ = σ � = � σ = τ � ; 2 � τ = σ � ∧ � σ = χ � ≤ � τ = χ � ; 3 for R ∈ L with arity n , and ( τ 1 , . . . , τ n ) , ( σ 1 , . . . , σ n ) ∈ M n , � R ( τ 1 , . . . , τ n ) � ∧ � τ h = σ h � ≤ � R ( σ 1 , . . . , σ n ) � . � h ∈{ 1 ,..., n } 16 / 51
Forcing relations on boolean valued models The constraints on R M and = M are the following: for τ, σ, χ ∈ M , � τ = τ � = 1 B ; 1 � τ = σ � = � σ = τ � ; 2 � τ = σ � ∧ � σ = χ � ≤ � τ = χ � ; 3 for R ∈ L with arity n , and ( τ 1 , . . . , τ n ) , ( σ 1 , . . . , σ n ) ∈ M n , � R ( τ 1 , . . . , τ n ) � ∧ � τ h = σ h � ≤ � R ( σ 1 , . . . , σ n ) � . � h ∈{ 1 ,..., n } 16 / 51
Boolean valued semantics Definition Let � M , = M , R M � be a B-valued model in the relational language L = { R } , φ ( x 1 , . . . , x n ) a L -formula with displayed free variables, ν : free variables → M . � φ � M ,ν = � φ � , the boolean value of φ with the assignment ν is defined by B recursion as follows: � t = s � = � ν ( t ) = ν ( s ) � , � R ( t 1 , . . . , t n ) � = � R ( ν ( t 1 ) , . . . , ν ( t n )) � ; � ¬ ψ � = ¬ � ψ � ; � ψ ∧ θ � = � ψ � ∧ � θ � ; � ∃ y ψ ( y ) � = � � ψ ( y /τ ) � . τ ∈ M 17 / 51
Boolean valued semantics Definition Let � M , = M , R M � be a B-valued model in the relational language L = { R } , φ ( x 1 , . . . , x n ) a L -formula with displayed free variables, ν : free variables → M . � φ � M ,ν = � φ � , the boolean value of φ with the assignment ν is defined by B recursion as follows: � t = s � = � ν ( t ) = ν ( s ) � , � R ( t 1 , . . . , t n ) � = � R ( ν ( t 1 ) , . . . , ν ( t n )) � ; � ¬ ψ � = ¬ � ψ � ; � ψ ∧ θ � = � ψ � ∧ � θ � ; � ∃ y ψ ( y ) � = � � ψ ( y /τ ) � . τ ∈ M 17 / 51
Boolean valued semantics Definition Let � M , = M , R M � be a B-valued model in the relational language L = { R } , φ ( x 1 , . . . , x n ) a L -formula with displayed free variables, ν : free variables → M . � φ � M ,ν = � φ � , the boolean value of φ with the assignment ν is defined by B recursion as follows: � t = s � = � ν ( t ) = ν ( s ) � , � R ( t 1 , . . . , t n ) � = � R ( ν ( t 1 ) , . . . , ν ( t n )) � ; � ¬ ψ � = ¬ � ψ � ; � ψ ∧ θ � = � ψ � ∧ � θ � ; � ∃ y ψ ( y ) � = � � ψ ( y /τ ) � . τ ∈ M 17 / 51
Soundness Theorem for B-valued semantics Theorem (Soundness Theorem) Assume L is a relational language and φ is a L -formula which is syntactically provable by a L -theory T. Assume each formula in T has boolean value at least b ∈ B in a B -valued model M with valuation ν . Then � φ � M ,ν ≥ b as well. B The completeness theorem is automatic given that 2 is a complete boolean algebra. 18 / 51
Soundness Theorem for B-valued semantics Theorem (Soundness Theorem) Assume L is a relational language and φ is a L -formula which is syntactically provable by a L -theory T. Assume each formula in T has boolean value at least b ∈ B in a B -valued model M with valuation ν . Then � φ � M ,ν ≥ b as well. B The completeness theorem is automatic given that 2 is a complete boolean algebra. 18 / 51
Tarski quotient of B-valued models Definition Let B be a cba , M a B-valued model for L , and G a ultrafilter over B. Consider the following equivalence relation on M : τ ≡ G σ ⇔ � τ = σ � ∈ G The first order (Tarski) model M / G = � M / G , R M / G : i ∈ I , c M / G : j ∈ J � is i j defined letting: For any n -ary relation symbol R in L R M / G = � ([ τ 1 ] G , . . . , [ τ n ] G ) ∈ ( M / G ) n : � R ( τ 1 , . . . , τ n ) � ∈ G � . For any constant symbol c in L c M / G = [ c M ] G . 19 / 51
Tarski quotient of B-valued models Definition Let B be a cba , M a B-valued model for L , and G a ultrafilter over B. Consider the following equivalence relation on M : τ ≡ G σ ⇔ � τ = σ � ∈ G The first order (Tarski) model M / G = � M / G , R M / G : i ∈ I , c M / G : j ∈ J � is i j defined letting: For any n -ary relation symbol R in L R M / G = � ([ τ 1 ] G , . . . , [ τ n ] G ) ∈ ( M / G ) n : � R ( τ 1 , . . . , τ n ) � ∈ G � . For any constant symbol c in L c M / G = [ c M ] G . 19 / 51
Full B-valued models Definition A B-valued model M for the language L is full if for every L -formula y | there is a σ ∈ M such that τ ∈ M | ¯ φ ( x , ¯ y ) and every ¯ τ ) � = � φ ( σ, ¯ � ∃ x φ ( x , ¯ τ ) � 20 / 51
Boolean valued Ło´ s Theorem — Forcing theorem Theorem (B-valued Ło´ s’s Theorem — Forcing theorem) Assume M is a full B -valued model for the relational language L . Then for every formula φ ( x 1 , . . . , x n ) in L and ( τ 1 , . . . , τ n ) ∈ M n : For all ultrafilters G over B , M / G | = φ ([ τ 1 ] G , . . . , [ τ n ] G ) if and only if 1 � φ ( τ 1 , . . . , τ n ) � ∈ G. For all a ∈ B the following are equivalent: 2 � φ ( f 1 , . . . , f n ) � ≥ a, 1 M / G | = φ ([ τ 1 ] G , . . . , [ τ n ] G ) for all G ∈ N a , 2 M / G | = φ ([ τ 1 ] G , . . . , [ τ n ] G ) for densely many G ∈ N a . 3 21 / 51
Ło´ s’s Theorem versus boolean valued Ło´ s’s Theorem Fact Let ( M x : x ∈ X ) be a family of Tarski-models in the first order relational language L . Then N = � x ∈ X M x is a full P ( X ) -model, letting for each n-ary relation symbol R ∈ L , � R ( f 1 , . . . , f n ) � P ( X ) = � x ∈ X : M x | = R ( f 1 ( x ) , . . . , f n ( x )) � . Let G be any non-principal ultrafilter on X . Then the Tarski quotient N / G is the familiar ultraproduct of the family ( M x : x ∈ X ) by G . The usual Ło´ s theorem for ultraproducts of Tarski models is the specialization to the case of the full P ( X ) -valued model N of the boolean valued Ło´ s theorem. If N is an ultrapower of a model M, the embedding a �→ [ c a ] G (where c a ( x ) = a for all x ∈ X and a ∈ M) is elementary. 22 / 51
Ło´ s’s Theorem versus boolean valued Ło´ s’s Theorem Fact Let ( M x : x ∈ X ) be a family of Tarski-models in the first order relational language L . Then N = � x ∈ X M x is a full P ( X ) -model, letting for each n-ary relation symbol R ∈ L , � R ( f 1 , . . . , f n ) � P ( X ) = � x ∈ X : M x | = R ( f 1 ( x ) , . . . , f n ( x )) � . Let G be any non-principal ultrafilter on X . Then the Tarski quotient N / G is the familiar ultraproduct of the family ( M x : x ∈ X ) by G . The usual Ło´ s theorem for ultraproducts of Tarski models is the specialization to the case of the full P ( X ) -valued model N of the boolean valued Ło´ s theorem. If N is an ultrapower of a model M, the embedding a �→ [ c a ] G (where c a ( x ) = a for all x ∈ X and a ∈ M) is elementary. 22 / 51
Ło´ s’s Theorem versus boolean valued Ło´ s’s Theorem Fact Let ( M x : x ∈ X ) be a family of Tarski-models in the first order relational language L . Then N = � x ∈ X M x is a full P ( X ) -model, letting for each n-ary relation symbol R ∈ L , � R ( f 1 , . . . , f n ) � P ( X ) = � x ∈ X : M x | = R ( f 1 ( x ) , . . . , f n ( x )) � . Let G be any non-principal ultrafilter on X . Then the Tarski quotient N / G is the familiar ultraproduct of the family ( M x : x ∈ X ) by G . The usual Ło´ s theorem for ultraproducts of Tarski models is the specialization to the case of the full P ( X ) -valued model N of the boolean valued Ło´ s theorem. If N is an ultrapower of a model M, the embedding a �→ [ c a ] G (where c a ( x ) = a for all x ∈ X and a ∈ M) is elementary. 22 / 51
Ło´ s’s Theorem versus boolean valued Ło´ s’s Theorem Fact Let ( M x : x ∈ X ) be a family of Tarski-models in the first order relational language L . Then N = � x ∈ X M x is a full P ( X ) -model, letting for each n-ary relation symbol R ∈ L , � R ( f 1 , . . . , f n ) � P ( X ) = � x ∈ X : M x | = R ( f 1 ( x ) , . . . , f n ( x )) � . Let G be any non-principal ultrafilter on X . Then the Tarski quotient N / G is the familiar ultraproduct of the family ( M x : x ∈ X ) by G . The usual Ło´ s theorem for ultraproducts of Tarski models is the specialization to the case of the full P ( X ) -valued model N of the boolean valued Ło´ s theorem. If N is an ultrapower of a model M, the embedding a �→ [ c a ] G (where c a ( x ) = a for all x ∈ X and a ∈ M) is elementary. 22 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of compact Hausdorff spaces Let X be a set with the discrete topology. For a ∈ X , G a ∈ St ( P ( X )) is the principal ultrafilter of supersets of { a } . The map a �→ G a embeds X as an open, dense, discrete subspace of St ( P ( X )) . For any space ( Y , τ ) , any f : X → Y is continuous. (since X has the discrete topology) Moreover if Y is compact Hausdorff: f : X → Y induces a unique continuous extension ¯ f : St ( P ( X )) → Y . (St ( P ( X )) is also the Stone-Cech compactification of X ). C ( X , Y ) = Y X is canonically isomorphic to C ( St ( P ( X )) , Y ) . C ( St ( P ( X )) , Y ) � Y X can be endowed of the structure of a P ( X ) -valued elementary extension of Y for any first order structure on Y . What if we replace P ( X ) with an arbitrary (complete) boolean algebra? 23 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary complete boolean algebra, and set M = C ( St ( B ) , 2 ω ) . Fix R a Borel (Universally Baire) relation on ( 2 ω ) n . The continuity of an n -tuple f 1 , . . . , f n ∈ M grants that { G : R ( f 1 ( G ) . . . , f n ( G )) } = ( f 1 × · · · × f n ) − 1 [ R ] has the Baire property in St ( B ) , where f 1 × · · · × f n ( G ) = ( f 1 ( G ) , . . . , f n ( G )) . Define: R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) where Reg ( A ) = Int ( Cl ( A )) . Also, since the diagonal is closed in ( 2 ω ) 2 , = M ( f , g ) = Reg ( � G : f ( G ) = g ( G ) � ) is well defined. 24 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary complete boolean algebra, and set M = C ( St ( B ) , 2 ω ) . Fix R a Borel (Universally Baire) relation on ( 2 ω ) n . The continuity of an n -tuple f 1 , . . . , f n ∈ M grants that { G : R ( f 1 ( G ) . . . , f n ( G )) } = ( f 1 × · · · × f n ) − 1 [ R ] has the Baire property in St ( B ) , where f 1 × · · · × f n ( G ) = ( f 1 ( G ) , . . . , f n ( G )) . Define: R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) where Reg ( A ) = Int ( Cl ( A )) . Also, since the diagonal is closed in ( 2 ω ) 2 , = M ( f , g ) = Reg ( � G : f ( G ) = g ( G ) � ) is well defined. 24 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary complete boolean algebra, and set M = C ( St ( B ) , 2 ω ) . Fix R a Borel (Universally Baire) relation on ( 2 ω ) n . The continuity of an n -tuple f 1 , . . . , f n ∈ M grants that { G : R ( f 1 ( G ) . . . , f n ( G )) } = ( f 1 × · · · × f n ) − 1 [ R ] has the Baire property in St ( B ) , where f 1 × · · · × f n ( G ) = ( f 1 ( G ) , . . . , f n ( G )) . Define: R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) where Reg ( A ) = Int ( Cl ( A )) . Also, since the diagonal is closed in ( 2 ω ) 2 , = M ( f , g ) = Reg ( � G : f ( G ) = g ( G ) � ) is well defined. 24 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary complete boolean algebra, and set M = C ( St ( B ) , 2 ω ) . Fix R a Borel (Universally Baire) relation on ( 2 ω ) n . The continuity of an n -tuple f 1 , . . . , f n ∈ M grants that { G : R ( f 1 ( G ) . . . , f n ( G )) } = ( f 1 × · · · × f n ) − 1 [ R ] has the Baire property in St ( B ) , where f 1 × · · · × f n ( G ) = ( f 1 ( G ) , . . . , f n ( G )) . Define: R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) where Reg ( A ) = Int ( Cl ( A )) . Also, since the diagonal is closed in ( 2 ω ) 2 , = M ( f , g ) = Reg ( � G : f ( G ) = g ( G ) � ) is well defined. 24 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary (even atomless) complete boolean algebra. The following holds: For any Borel (universally Baire) relation R on ( 2 ω ) n , the structure ( M , = M , R M ) is a full B-valued model. For G ∈ St ( B ) , i G : 2 ω → M / G x �→ [ c x ] G ( c x is the constant function with value x ) defines an injective morphism ( 2 ω , R ) into ( M / G , R M / G ) . It is not clear whether this morphism is an elementary map or not: This is the case for B = P ( X ) , since in this case we are analyzing the standard embedding of the first order structure ( 2 ω , R ) in its ultrapowers induced by ultrafilters on P ( X ) . What are the properties of this map if B is some other complete (atomless) boolean algebra? 25 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary (even atomless) complete boolean algebra. The following holds: For any Borel (universally Baire) relation R on ( 2 ω ) n , the structure ( M , = M , R M ) is a full B-valued model. For G ∈ St ( B ) , i G : 2 ω → M / G x �→ [ c x ] G ( c x is the constant function with value x ) defines an injective morphism ( 2 ω , R ) into ( M / G , R M / G ) . It is not clear whether this morphism is an elementary map or not: This is the case for B = P ( X ) , since in this case we are analyzing the standard embedding of the first order structure ( 2 ω , R ) in its ultrapowers induced by ultrafilters on P ( X ) . What are the properties of this map if B is some other complete (atomless) boolean algebra? 25 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary (even atomless) complete boolean algebra. The following holds: For any Borel (universally Baire) relation R on ( 2 ω ) n , the structure ( M , = M , R M ) is a full B-valued model. For G ∈ St ( B ) , i G : 2 ω → M / G x �→ [ c x ] G ( c x is the constant function with value x ) defines an injective morphism ( 2 ω , R ) into ( M / G , R M / G ) . It is not clear whether this morphism is an elementary map or not: This is the case for B = P ( X ) , since in this case we are analyzing the standard embedding of the first order structure ( 2 ω , R ) in its ultrapowers induced by ultrafilters on P ( X ) . What are the properties of this map if B is some other complete (atomless) boolean algebra? 25 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary (even atomless) complete boolean algebra. The following holds: For any Borel (universally Baire) relation R on ( 2 ω ) n , the structure ( M , = M , R M ) is a full B-valued model. For G ∈ St ( B ) , i G : 2 ω → M / G x �→ [ c x ] G ( c x is the constant function with value x ) defines an injective morphism ( 2 ω , R ) into ( M / G , R M / G ) . It is not clear whether this morphism is an elementary map or not: This is the case for B = P ( X ) , since in this case we are analyzing the standard embedding of the first order structure ( 2 ω , R ) in its ultrapowers induced by ultrafilters on P ( X ) . What are the properties of this map if B is some other complete (atomless) boolean algebra? 25 / 51
Boolean ultrapowers of 2 ω Let B be an arbitrary (even atomless) complete boolean algebra. The following holds: For any Borel (universally Baire) relation R on ( 2 ω ) n , the structure ( M , = M , R M ) is a full B-valued model. For G ∈ St ( B ) , i G : 2 ω → M / G x �→ [ c x ] G ( c x is the constant function with value x ) defines an injective morphism ( 2 ω , R ) into ( M / G , R M / G ) . It is not clear whether this morphism is an elementary map or not: This is the case for B = P ( X ) , since in this case we are analyzing the standard embedding of the first order structure ( 2 ω , R ) in its ultrapowers induced by ultrafilters on P ( X ) . What are the properties of this map if B is some other complete (atomless) boolean algebra? 25 / 51
Shoenfield’s absoluteness rephrased Theorem (Cohen’s absoluteness) Assume B is a complete boolean algebra and R ⊆ ( 2 ω ) n is a Borel (Universally Baire) relation. Let M = C ( St ( B ) , 2 ω ) and G ∈ St ( B ) . Then ( 2 ω , = , R ) ≺ Σ 2 ( M / G , = M / G , R M / G ) . If one assumes the existence of class many Woodin cardinals ( 2 ω , = , R ) ≺ ( M / G , = M / G , R M / G ) . Proof. C ( St ( B ) , 2 ω ) is isomorphic to the B-names in V B for elements of 2 ω (see next slide). Apply Shoenfield’s (or Woodin’s) absoluteness to V and V [ H ] (for H V -generic for B) to infer the desired conclusion. � 26 / 51
Shoenfield’s absoluteness rephrased Theorem (Cohen’s absoluteness) Assume B is a complete boolean algebra and R ⊆ ( 2 ω ) n is a Borel (Universally Baire) relation. Let M = C ( St ( B ) , 2 ω ) and G ∈ St ( B ) . Then ( 2 ω , = , R ) ≺ Σ 2 ( M / G , = M / G , R M / G ) . If one assumes the existence of class many Woodin cardinals ( 2 ω , = , R ) ≺ ( M / G , = M / G , R M / G ) . Proof. C ( St ( B ) , 2 ω ) is isomorphic to the B-names in V B for elements of 2 ω (see next slide). Apply Shoenfield’s (or Woodin’s) absoluteness to V and V [ H ] (for H V -generic for B) to infer the desired conclusion. � 26 / 51
Shoenfield’s absoluteness rephrased Theorem (Cohen’s absoluteness) Assume B is a complete boolean algebra and R ⊆ ( 2 ω ) n is a Borel (Universally Baire) relation. Let M = C ( St ( B ) , 2 ω ) and G ∈ St ( B ) . Then ( 2 ω , = , R ) ≺ Σ 2 ( M / G , = M / G , R M / G ) . If one assumes the existence of class many Woodin cardinals ( 2 ω , = , R ) ≺ ( M / G , = M / G , R M / G ) . Proof. C ( St ( B ) , 2 ω ) is isomorphic to the B-names in V B for elements of 2 ω (see next slide). Apply Shoenfield’s (or Woodin’s) absoluteness to V and V [ H ] (for H V -generic for B) to infer the desired conclusion. � 26 / 51
C ( St ( B ) , 2 ω ) and V B Given f ∈ C ( St ( B ) , 2 ω ) = M , σ ∈ V B with � σ ∈ 2 ω � = 1 B define: � � �� n , i � , f − 1 [ N n , i ] � : n < ω, i < 2 ∈ V B , τ f = g σ ∈ M by g σ ( G )( n ) = i iff � σ ( n ) = i � ∈ G . Then g τ f = f , � � τ g σ = σ = 1 B . These identities allow to translate forcing relations from both sides. The lift of a Universally Baire relation R to V B is translated as the forcing relation (on M ) R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) . Universal Baireness grants that the lift R M behaves as desired. 27 / 51
C ( St ( B ) , 2 ω ) and V B Given f ∈ C ( St ( B ) , 2 ω ) = M , σ ∈ V B with � σ ∈ 2 ω � = 1 B define: � � �� n , i � , f − 1 [ N n , i ] � : n < ω, i < 2 ∈ V B , τ f = g σ ∈ M by g σ ( G )( n ) = i iff � σ ( n ) = i � ∈ G . Then g τ f = f , � � τ g σ = σ = 1 B . These identities allow to translate forcing relations from both sides. The lift of a Universally Baire relation R to V B is translated as the forcing relation (on M ) R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) . Universal Baireness grants that the lift R M behaves as desired. 27 / 51
C ( St ( B ) , 2 ω ) and V B Given f ∈ C ( St ( B ) , 2 ω ) = M , σ ∈ V B with � σ ∈ 2 ω � = 1 B define: � � �� n , i � , f − 1 [ N n , i ] � : n < ω, i < 2 ∈ V B , τ f = g σ ∈ M by g σ ( G )( n ) = i iff � σ ( n ) = i � ∈ G . Then g τ f = f , � � τ g σ = σ = 1 B . These identities allow to translate forcing relations from both sides. The lift of a Universally Baire relation R to V B is translated as the forcing relation (on M ) R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) . Universal Baireness grants that the lift R M behaves as desired. 27 / 51
C ( St ( B ) , 2 ω ) and V B Given f ∈ C ( St ( B ) , 2 ω ) = M , σ ∈ V B with � σ ∈ 2 ω � = 1 B define: � � �� n , i � , f − 1 [ N n , i ] � : n < ω, i < 2 ∈ V B , τ f = g σ ∈ M by g σ ( G )( n ) = i iff � σ ( n ) = i � ∈ G . Then g τ f = f , � � τ g σ = σ = 1 B . These identities allow to translate forcing relations from both sides. The lift of a Universally Baire relation R to V B is translated as the forcing relation (on M ) R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) . Universal Baireness grants that the lift R M behaves as desired. 27 / 51
C ( St ( B ) , 2 ω ) and V B Given f ∈ C ( St ( B ) , 2 ω ) = M , σ ∈ V B with � σ ∈ 2 ω � = 1 B define: � � �� n , i � , f − 1 [ N n , i ] � : n < ω, i < 2 ∈ V B , τ f = g σ ∈ M by g σ ( G )( n ) = i iff � σ ( n ) = i � ∈ G . Then g τ f = f , � � τ g σ = σ = 1 B . These identities allow to translate forcing relations from both sides. The lift of a Universally Baire relation R to V B is translated as the forcing relation (on M ) R M : M n → B ( f 1 , . . . , f n ) �→ Reg ( � G : R ( f 1 ( G ) , . . . , f n ( G ) � ) . Universal Baireness grants that the lift R M behaves as desired. 27 / 51
Two questions Where are forcing axioms playing a role in the above proof (and 1 rephrasing) of Shoenfield’s absoluteness? What if Y � 2 ω is some other compact Hausdorff space? 2 Time not permitting I won’t give a proof of the above rephrasing of 1 Shoenfield’s absoluteness, which can be based on a Baire category argument and on Cohen’s forcing theorem. We will now inquire on the second question, which leads us to other 2 stronger formulation of forcing axioms in categorial terms. 28 / 51
Two questions Where are forcing axioms playing a role in the above proof (and 1 rephrasing) of Shoenfield’s absoluteness? What if Y � 2 ω is some other compact Hausdorff space? 2 Time not permitting I won’t give a proof of the above rephrasing of 1 Shoenfield’s absoluteness, which can be based on a Baire category argument and on Cohen’s forcing theorem. We will now inquire on the second question, which leads us to other 2 stronger formulation of forcing axioms in categorial terms. 28 / 51
Two questions Where are forcing axioms playing a role in the above proof (and 1 rephrasing) of Shoenfield’s absoluteness? What if Y � 2 ω is some other compact Hausdorff space? 2 Time not permitting I won’t give a proof of the above rephrasing of 1 Shoenfield’s absoluteness, which can be based on a Baire category argument and on Cohen’s forcing theorem. We will now inquire on the second question, which leads us to other 2 stronger formulation of forcing axioms in categorial terms. 28 / 51
Two questions Where are forcing axioms playing a role in the above proof (and 1 rephrasing) of Shoenfield’s absoluteness? What if Y � 2 ω is some other compact Hausdorff space? 2 Time not permitting I won’t give a proof of the above rephrasing of 1 Shoenfield’s absoluteness, which can be based on a Baire category argument and on Cohen’s forcing theorem. We will now inquire on the second question, which leads us to other 2 stronger formulation of forcing axioms in categorial terms. 28 / 51
Two questions Where are forcing axioms playing a role in the above proof (and 1 rephrasing) of Shoenfield’s absoluteness? What if Y � 2 ω is some other compact Hausdorff space? 2 Time not permitting I won’t give a proof of the above rephrasing of 1 Shoenfield’s absoluteness, which can be based on a Baire category argument and on Cohen’s forcing theorem. We will now inquire on the second question, which leads us to other 2 stronger formulation of forcing axioms in categorial terms. 28 / 51
Looking at 2 ω is the same as looking at H ω 1 There exists a natural correspondence between the theory of projective subsets of 2 ω and the first order theory of H ω 1 . Any Σ 1 2 -property of 2 ω corresponds to a Σ 1 -property on H ω 1 . Moreover 2 ω is a definable class in H ω 1 , hence the first order theory of H ω 1 interprets that of 2 ω with projective predicates. The converse holds as well. Hence it is essentially the same to look at the first order theory of 2 ω or at the first order theory of H ω 1 . 29 / 51
Looking at 2 ω is the same as looking at H ω 1 There exists a natural correspondence between the theory of projective subsets of 2 ω and the first order theory of H ω 1 . Any Σ 1 2 -property of 2 ω corresponds to a Σ 1 -property on H ω 1 . Moreover 2 ω is a definable class in H ω 1 , hence the first order theory of H ω 1 interprets that of 2 ω with projective predicates. The converse holds as well. Hence it is essentially the same to look at the first order theory of 2 ω or at the first order theory of H ω 1 . 29 / 51
Looking at 2 ω is the same as looking at H ω 1 There exists a natural correspondence between the theory of projective subsets of 2 ω and the first order theory of H ω 1 . Any Σ 1 2 -property of 2 ω corresponds to a Σ 1 -property on H ω 1 . Moreover 2 ω is a definable class in H ω 1 , hence the first order theory of H ω 1 interprets that of 2 ω with projective predicates. The converse holds as well. Hence it is essentially the same to look at the first order theory of 2 ω or at the first order theory of H ω 1 . 29 / 51
Boolean ultrapowers of H κ To analyze how to use forcing for the analysis of compact spaces other than 2 ω it is more convenient to move from an analysis of a compact space X to the analysis of the H κ in which X is definable for κ large enough. If we can define elementary boolean ultrapowers of H κ , we can naturally define elementary boolean ultrapowers of any compact Hausdorff Y (or more generally any mathematical structure) definable in H κ . Let us address now the question of how to use generic absoluteness results as a template to formulate stronger and stronger forcing axioms. 30 / 51
Boolean ultrapowers of H κ To analyze how to use forcing for the analysis of compact spaces other than 2 ω it is more convenient to move from an analysis of a compact space X to the analysis of the H κ in which X is definable for κ large enough. If we can define elementary boolean ultrapowers of H κ , we can naturally define elementary boolean ultrapowers of any compact Hausdorff Y (or more generally any mathematical structure) definable in H κ . Let us address now the question of how to use generic absoluteness results as a template to formulate stronger and stronger forcing axioms. 30 / 51
Boolean ultrapowers of H κ To analyze how to use forcing for the analysis of compact spaces other than 2 ω it is more convenient to move from an analysis of a compact space X to the analysis of the H κ in which X is definable for κ large enough. If we can define elementary boolean ultrapowers of H κ , we can naturally define elementary boolean ultrapowers of any compact Hausdorff Y (or more generally any mathematical structure) definable in H κ . Let us address now the question of how to use generic absoluteness results as a template to formulate stronger and stronger forcing axioms. 30 / 51
Forcing axioms as density properties of class posets. Definition Let Γ be a class of complete boolean algebras and Θ be a class of complete homomorphisms between elements of Γ and closed under composition and identity maps. B ≥ Θ Q if there is a complete homomorphism i : B → Q in Θ . B ≥ ∗ Θ Q if there is a complete and injective homomorphism i : B → Q in Θ . With these definitions (Γ , ≤ Θ ) and (Γ , ≤ ∗ Θ ) are class partial orders. 31 / 51
Forcing axioms as density properties of class posets. Definition Let Γ be a class of complete boolean algebras and Θ be a class of complete homomorphisms between elements of Γ and closed under composition and identity maps. B ≥ Θ Q if there is a complete homomorphism i : B → Q in Θ . B ≥ ∗ Θ Q if there is a complete and injective homomorphism i : B → Q in Θ . With these definitions (Γ , ≤ Θ ) and (Γ , ≤ ∗ Θ ) are class partial orders. 31 / 51
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