Forcing consequences of PFA together with the continuum large David - PowerPoint PPT Presentation

Forcing consequences of PFA together with the continuum large David Asper´ o (joint work with Miguel ´ Angel Mota) ICREA at U. Barcelona RIMS Workshop, Kyoto, 2009 Nov. 16–19

PFA implies 2 ℵ 0 = ℵ 2 . All known proofs of this implication use forcing notions that collapse ω 2 . Question: Does FA ( { P : P proper and cardinal–preserving } ) imply 2 ℵ 0 = ℵ 2 ? Does even FA ( { P : P proper , | P | = ℵ 1 } ) imply 2 ℵ 0 = ℵ 2 ?

In the first part of the talk I will isolate a certain subclass Γ of { P : P proper , | P | = ℵ 1 } and will sketch a proof that FA (Γ) + 2 ℵ 0 > ℵ 2 is consistent. FA (Γ) will be strong enough to imply for example the negation of Justin Moore’s ℧ and other strong forms of the negation of Club Guessing.

In the first part of the talk I will isolate a certain subclass Γ of { P : P proper , | P | = ℵ 1 } and will sketch a proof that FA (Γ) + 2 ℵ 0 > ℵ 2 is consistent. FA (Γ) will be strong enough to imply for example the negation of Justin Moore’s ℧ and other strong forms of the negation of Club Guessing.

Notation If N is a set such that N ∩ ω 1 ∈ ω 1 , set δ N = N ∩ ω 1 . Let X be a set. If W ⊆ [ X ] ℵ 0 and N is a set, W is an N–unbounded subset of [ X ] ℵ 0 if for every x ∈ N ∩ X there is some M ∈ W ∩ N with x ∈ M . If P is a partial order, P is nice if (a) conditions in P are functions with domain included in ω 1 , and (b) if p , q ∈ P are compatible, then the greatest lower bound r of p and q exists, dom ( r ) = dom ( p ) ∪ dom ( q ) , and r ( ν ) = p ( ν ) ∪ q ( ν ) for all ν ∈ dom ( r ) (where f ( ν ) = ∅ if ν / ∈ dom ( f ) ). Exercise: Every set–forcing for which glb ( p , q ) exists whenever p and q are compatible conditions is isomorphic to a nice forcing.

Notation If N is a set such that N ∩ ω 1 ∈ ω 1 , set δ N = N ∩ ω 1 . Let X be a set. If W ⊆ [ X ] ℵ 0 and N is a set, W is an N–unbounded subset of [ X ] ℵ 0 if for every x ∈ N ∩ X there is some M ∈ W ∩ N with x ∈ M . If P is a partial order, P is nice if (a) conditions in P are functions with domain included in ω 1 , and (b) if p , q ∈ P are compatible, then the greatest lower bound r of p and q exists, dom ( r ) = dom ( p ) ∪ dom ( q ) , and r ( ν ) = p ( ν ) ∪ q ( ν ) for all ν ∈ dom ( r ) (where f ( ν ) = ∅ if ν / ∈ dom ( f ) ). Exercise: Every set–forcing for which glb ( p , q ) exists whenever p and q are compatible conditions is isomorphic to a nice forcing.

More notation Given a nice partial order ( P , ≤ ) , a P –condition p and a set M such that δ M exists, we say that M is good for p iff p ↾ δ M ∈ P and, letting X = { s ∈ P ∩ M : s ≤ p ↾ δ M , s compatible with p } , (i) X � = ∅ , and (ii) for every s ∈ X there is some t ≤ s , t ∈ M , such that for all t ′ ≤ t , if t ′ ∈ M , then t ′ ∈ X .

A class of posets Let P be a nice poset and κ an infinite cardinal. P is κ –suitable if there are a binary relation R and a club C ⊆ ω 1 with the following properties. (1) If p R ( N , W ) , then the following conditions hold. (1.1) N is a countable subset of H ( κ ) , W is an N –unbounded subset of [ H ( κ )] ℵ 0 , and all members of W ∩ N are good for p . (1.2) If p ′ is a P –condition extending p , then there is some W ′ ⊆ W such that p ′ R ( N , W ′ ) . (1.3) If W ′ ⊆ W is N –unbounded, then p R ( N , W ′ ) . (1.4) p ↾ δ N ∈ N , and for all N ′ and all W ′ with δ N ′ < δ N , p R ( N ′ , W ′ ) if and only if p ↾ δ N R ( N ′ , W ′ )

A class of posets (2) For every p ∈ P and every finite set { ( N i , W i ) : i < m } such that ( ◦ ) each N i is a countable subset of H ( κ ) containing p , ω N i 1 = ω 1 , δ N i ∈ C , N i | = ZFC ∗ , and ( ◦ ) each W i is N i –unbounded there is a condition q ∈ P extending p and there are W ′ i ⊆ W i ( i < m ) such that q R ( N i , W ′ i ) for all i < m . We will say that a nice partial order is absolutely κ –suitable if it is κ –suitable in every ground model W containing it and such that ω W 1 = ω 1 .

A class of posets (2) For every p ∈ P and every finite set { ( N i , W i ) : i < m } such that ( ◦ ) each N i is a countable subset of H ( κ ) containing p , ω N i 1 = ω 1 , δ N i ∈ C , N i | = ZFC ∗ , and ( ◦ ) each W i is N i –unbounded there is a condition q ∈ P extending p and there are W ′ i ⊆ W i ( i < m ) such that q R ( N i , W ′ i ) for all i < m . We will say that a nice partial order is absolutely κ –suitable if it is κ –suitable in every ground model W containing it and such that ω W 1 = ω 1 .

A class of posets Let Γ κ denote the class of all absolutely κ –suitable posets consisting of finite functions included in ω 1 × [ ω 1 ] <ω . Easy: For all κ ≥ ω 2 , Γ κ ⊆ Proper . FA (Γ κ ) : For every P ∈ Γ κ and every collection D of size ℵ 1 consisting of dense subsets of P there is a filter G ⊆ P such that G ∩ D � = ∅ for all D ∈ D .

A class of posets Let Γ κ denote the class of all absolutely κ –suitable posets consisting of finite functions included in ω 1 × [ ω 1 ] <ω . Easy: For all κ ≥ ω 2 , Γ κ ⊆ Proper . FA (Γ κ ) : For every P ∈ Γ κ and every collection D of size ℵ 1 consisting of dense subsets of P there is a filter G ⊆ P such that G ∩ D � = ∅ for all D ∈ D .

A class of posets Let Γ κ denote the class of all absolutely κ –suitable posets consisting of finite functions included in ω 1 × [ ω 1 ] <ω . Easy: For all κ ≥ ω 2 , Γ κ ⊆ Proper . FA (Γ κ ) : For every P ∈ Γ κ and every collection D of size ℵ 1 consisting of dense subsets of P there is a filter G ⊆ P such that G ∩ D � = ∅ for all D ∈ D .

One application of FA (Γ κ ) : Ω ℧ : There is a sequence � g δ : δ < ω 1 � Definition (Moore) such that each g δ : δ − → ω is continuous with respect to the order topology and such that for every club C ⊆ ω 1 there is some δ ∈ C with g δ “ C = ω . ( ◦ ) Club Guessing implies ℧ . ( ◦ ) ℧ preserved by ccc forcing, and in fact by ω –proper forcing. ( ◦ ) Each of BPFA and MRP implies Ω := ¬ ℧ .

One application of FA (Γ κ ) : Ω ℧ : There is a sequence � g δ : δ < ω 1 � Definition (Moore) such that each g δ : δ − → ω is continuous with respect to the order topology and such that for every club C ⊆ ω 1 there is some δ ∈ C with g δ “ C = ω . ( ◦ ) Club Guessing implies ℧ . ( ◦ ) ℧ preserved by ccc forcing, and in fact by ω –proper forcing. ( ◦ ) Each of BPFA and MRP implies Ω := ¬ ℧ .

Theorem (Moore) ℧ implies the existence of an Aronszajn line which does not contain any Contryman suborder. Question (Moore): Does Ω imply 2 ℵ 0 ≤ ℵ 2 ?

Theorem (Moore) ℧ implies the existence of an Aronszajn line which does not contain any Contryman suborder. Question (Moore): Does Ω imply 2 ℵ 0 ≤ ℵ 2 ?

Proposition: For every κ ≥ ω 2 , FA (Γ κ ) implies Ω . Proof sketch: Notation: Given X , a set of ordinals, and δ , an ordinal, set ( ◦ ) rank ( X , δ ) = 0 iff δ is not a limit point of X , and ( ◦ ) rank ( X , δ ) > η if and only if δ is a limit of ordinals ǫ such that rank ( X , ǫ ) ≥ η . Given a sequence G = � g δ : δ < ω 1 � of continuous colourings, let P G be the following poset:

Proposition: For every κ ≥ ω 2 , FA (Γ κ ) implies Ω . Proof sketch: Notation: Given X , a set of ordinals, and δ , an ordinal, set ( ◦ ) rank ( X , δ ) = 0 iff δ is not a limit point of X , and ( ◦ ) rank ( X , δ ) > η if and only if δ is a limit of ordinals ǫ such that rank ( X , ǫ ) ≥ η . Given a sequence G = � g δ : δ < ω 1 � of continuous colourings, let P G be the following poset:

Proposition: For every κ ≥ ω 2 , FA (Γ κ ) implies Ω . Proof sketch: Notation: Given X , a set of ordinals, and δ , an ordinal, set ( ◦ ) rank ( X , δ ) = 0 iff δ is not a limit point of X , and ( ◦ ) rank ( X , δ ) > η if and only if δ is a limit of ordinals ǫ such that rank ( X , ǫ ) ≥ η . Given a sequence G = � g δ : δ < ω 1 � of continuous colourings, let P G be the following poset:

Conditions in P G are pairs p = ( f , � k ξ : ξ ∈ D � ) satisfying the following properties: (1) f is a finite function that can be extended to a normal function F : ω 1 − → ω 1 . (2) For every ξ ∈ dom ( f ) , rank ( f ( ξ ) , f ( ξ )) ≥ ξ . (3) D ⊆ dom ( f ) and for every ξ ∈ D , (3.1) k ξ < ω , (3.2) g f ( ξ ) “ range ( f ) ⊆ ω \{ k ξ } , and (3.3) rank ( { γ < f ( ξ ) : g f ( ξ ) ( γ ) � = k ξ } , f ( ξ )) = rank ( f ( ξ ) , f ( ξ )) .

Given conditions p ǫ = ( f ǫ , ( k ǫ ξ : ξ ∈ D ǫ )) ∈ P G for ǫ ∈ { 0 , 1 } , p 1 extends p 0 iff (i) f 0 ⊆ f 1 , (ii) D 0 ⊆ D 1 , and (iii) k 1 ξ = k 0 ξ for all ξ ∈ D 0 . Easy: If G is P G –generic and C = range ( � { f : ( ∃ � k )( � f ,� k � ∈ G ) } ) , then C is a club of ω V 1 and for every δ ∈ C there is k δ ∈ ω such that g δ “ C ⊆ ω \{ k δ } .