MAD families, splitting families and large continuum Vera Fischer - PowerPoint PPT Presentation

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions MAD families, splitting families and large continuum Vera Fischer Kurt G¨ odel Research Center University of Vienna April 2012 Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) General overview The forcing construction Matrix Iteration Open Questions ◮ con( s < b ) (Baumgartner, Dordal, 1984) ◮ con( b = ℵ 1 < s = a = ℵ 2 ) (Shelah, 1985) ◮ con( b = κ < a = κ + ) (Brendle, 1998) ◮ con( b = κ < s = κ + ) (F., Stepr¯ ans, 2008) Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) General overview The forcing construction Matrix Iteration Open Questions Theorem (Brendle, F., 2011) Let κ < λ be arbitrary regular uncountable cardinals. Then there is a ccc generic extension in which b = a = κ < s = λ . Theorem (Brenlde, F., 2011) Let µ be a measurable cardinal, κ < λ regular such that µ < κ . Then there is a ccc generic extension in which b = κ < s = a = λ . Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions For γ an ordinal, P γ is the poset of all finite partial functions p : γ × ω → 2 such that dom( p ) = F p × n p where F p ∈ [ γ ] <ω , n p ∈ ω . The order is given by q ≤ p if p ⊆ q and | q − 1 (1) ∩ F p × { i }| ≤ 1 for all i ∈ n q \ n p . Let G be a P γ -generic filter and for δ ∈ γ let A α = { i : ∃ p ∈ G ( p ( α, i ) = 1) } . Then ◮ { A α : α ∈ γ } is an a.d. family (maximal for γ ≥ ω 1 ), ◮ if p ∈ P γ then for all α ∈ F p ( p � ˙ A α ↾ n p = p ↾ { α } × n p ), ◮ for all α, β ∈ F p ( p � ˙ A α ∩ ˙ A β ⊆ n p ). Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions Let γ < δ , G a P γ -generic filter. In V [ G ], let P [ γ,δ ) consist of all ( p , H ) such that p ∈ P δ with F p ∈ [ δ \ γ ] <ω and H ∈ [ γ ] <ω . The order is given by ( q , K ) ≤ ( p , H ) if q ≤ P δ p , H ⊆ K and for all α ∈ F p , β ∈ H , i ∈ n q \ n p if i ∈ A β , then q ( α, i ) = 0. ◮ That is for all α ∈ F p , β ∈ H , p � ˙ A α ∩ ˇ A β ⊆ n p . ◮ P δ is forcing equivalent to P γ ∗ P [ γ,δ ) . Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions Property ⋆ Let M ⊆ N , B = { B α } α<γ ⊆ [ ω ] ω ∩ M , A ∈ N ∩ [ ω ] ω . Then B , A ) holds if for every h : ω × [ γ ] <ω → ω , h ∈ M and m ∈ ω ( ⋆ M , N there are n ≥ m , F ∈ [ γ ] <ω such that [ n , h ( n , F )) \ � α ∈ F B α ⊆ A . Lemma A If G γ +1 is P γ +1 -generic, G γ = G γ +1 ∩ P γ , A γ = { A α } α<γ , where A α = { i : ∃ p ∈ G ( p ( α, i ) = 1) } . Then ( ⋆ V [ G γ ] , V [ G γ +1 ] ) holds. A γ , A γ Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions Lemma B Let ( ⋆ M , N B , A ) hold, where B = { B α } α<γ , let I ( B ) be the ideal generated by B and the finite sets and let B ∈ M ∩ [ ω ] ω , B / ∈ I ( B ). Then | A ∩ B | = ℵ 0 . Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions Lemma C Let M ⊆ N , B = { B α } α<γ ⊆ M ∩ [ ω ] ω , A ∈ N ∩ [ ω ] ω such that ( ⋆ M , N B , A ). Let U be an ultrafilter in M . Then there is an ultrafilter V ⊇ U in N such that 1. every maximal antichain of M U which belongs to M is a maximal antichain of M V in N , 2. ( ⋆ M [ G ] , N [ G ] ) holds where G is M V -generic over N (and thus, B , A by (1), M U -generic over M ). Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions Lemma D Let M ⊆ N , P ∈ M a poset such that P ⊆ M , G a P -generic filter over M , N . Let B = { B α } α ∈ γ ⊆ M ∩ [ ω ] ω , A ∈ N ∩ [ ω ] ω such that B , A ) holds. Then ( ⋆ M [ G ] , N [ G ] ( ⋆ M , N ) holds. B , A Lemma E Let � P ℓ, n , ˙ Q ℓ, n : n ∈ ω � , ℓ ∈ { 0 , 1 } be finite support iterations such that P 0 , n is a complete suborder of P 1 , n for all n . Let V ℓ, n = V P ℓ, n . Let B = { A γ } γ<α ⊆ V 0 , 0 ∩ [ ω ] ω , A ∈ V 1 , 0 ∩ [ ω ] ω . If ( ⋆ V 0 , n , V 1 , n ) B , A holds for all n ∈ ω , then ( ⋆ V 0 ,ω , V 1 ,ω ) holds. B , A Vera Fischer MAD families, splitting families and large continuum

Introduction Adding a mad family Con ( b = a = κ < s = λ ) Increasing s The forcing construction Iterating Suslin Posets Open Questions Lemma Let P , Q be partial orders, such that P is completely embedded into Q . Let ˙ A be a P -name for a forcing notion, ˙ B a Q -name for a forcing notion such that � Q ˙ A ⊆ ˙ B , and every maximal antichain of A in V P is a maximal antichain of ˙ ˙ B in V Q . Then P ∗ ˙ A < ◦ Q ∗ ˙ B . Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions Let f : { η < λ : η ≡ 1 mod 2 } → κ be an onto mapping, such that for all α < κ , f − 1 ( α ) is cofinal in λ . Recursively define a system of finite support iterations �� P α,ζ : α ≤ κ, ζ ≤ λ � , � ˙ Q α,ζ : α ≤ κ, ζ < λ �� as follows. For all α, ζ let V α,ζ = V P α,ζ . Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions (1) If ζ = 0, then for all α ≤ κ , P α, 0 is Hechler’s poset for adding an a.d. family A α = { A β } β<α , (2) If ζ = η + 1, ζ ≡ 1 mod 2, then � P α,η ˙ Q α,η = M ˙ U α,η where ˙ U α,η is a P α,η -name for an ultrafilter and for all α < β ≤ κ , � P β,η ˙ U α,η ⊆ ˙ U β,η , (3) If ζ = η + 1, ζ ≡ 0 mod 2, then if α ≤ f ( η ), ˙ Q α,η is a P α,η -name for the trivial forcing notion; if α > f ( η ) then ˙ Q α,η is a P α,η -name for D V f ( η ) ,η . (4) If ζ is a limit, then for all α ≤ κ , P α,ζ is the finite support iteration of � P α,η , ˙ Q α,η : η < ζ � . Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions Furthermore the construction will satisfy the following two properties: (a) ∀ ζ ≤ λ ∀ α < β ≤ κ , P α,ζ is a complete suborder of P β,ζ , (b) ∀ ζ ≤ λ ∀ α < κ ( ⋆ V α,ζ , V α +1 ,ζ ) holds. A α , A α Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions Proceed by recursion on ζ . For ζ = 0, α ≤ κ let P α, 0 = P α . Then clearly properties ( a ) and ( b ) above hold. Let ζ = η + 1 be a successor ordinal and suppose ∀ α ≤ κ , P α,η has been defined. Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions If ζ ≡ 1 mod 2 define ˙ Q α,η by induction on α ≤ κ as follows. ◮ If α = 0, let ˙ U 0 ,η be a P 0 ,η -name for an ultrafilter, ˙ Q 0 ,η a U 0 ,η and let P 0 ,ζ = P 0 ,η ∗ ˙ P 0 ,η -name for M ˙ Q 0 ,η . ◮ If α = β + 1 and ˙ U β,η has been defined, by the ind. hyp. and Lemma C there is a P α,η -name ˙ U α,η for an ultrafilter such that � P α,η ˙ U β,η ⊆ ˙ U α,η , every maximal antichain of M ˙ U β,η in U α,η and ( ⋆ V β,ζ , V β +1 ,ζ V β,η is a maximal antichain of M ˙ ). A β , A β holds. Let P β,ζ = P β,η ∗ M ˙ U β,η . In particular P β,ζ < ◦ P α,ζ . Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) The forcing construction Open Questions ◮ If α is limit and for all β < α ˙ U β,η has been defined (and so ˙ Q β,η = M ˙ U β,η ) consider the following two cases. ◮ If cf( α ) = ω , find a P α,η -name ˙ U α,η for an ultrafilter such that for all β < α , � P α,η ˙ U β,η ⊆ ˙ U α,η and every maximal antichain of M ˙ U β,η from V β,η is a maximal antichain of M U α,η (in V α,η ) and the relevant ⋆ -property is preserved. ◮ If cf( α ) > ω , then let ˙ U α,η be a P α,η -name for � β<α U β,η . Let ˙ U α,η and let P α,ζ = P α,η ∗ ˙ Q α,η be a P α,η -name for M ˙ Q α,η . Vera Fischer MAD families, splitting families and large continuum