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( b = ℵ 1 < s = ℵ 2 ) ◮ In 1984 S. Shelah obtained the above consistency using an almost ω ω - bounding version of Mathias forcing, in which the pure Mathias condition is supplied with additional structure in the form of a finite logarithmic measure. ◮ The countable support iteration of proper almost ω ω -bounding posets is weakly bounding, which implies that in such extensions the ground model reals remain an unbounded family. 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 ◮ A modification of the preceding argument produces the consistency of b = ℵ 1 < a = s = ℵ 2 . 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( b = κ < s = κ + ) ◮ Obtain a ccc suborder of Shalah’s poset, which behaves sufficiently similarly to the larger forcing notion. 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 (V. F., J. Stepr¯ ans) Let κ be a regular, uncountable cardinal, ∀ µ < κ (2 µ ≤ κ ) , cov ( M ) = κ and let H be an unbounded directed family of size κ . Then there is an ultrafilter U H on ω such that the relativized Mathias poset M ( U H ) , preserves the unboundedness of H . 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 ◮ If H is an unbounded family, such that every countable subfamily of H is dominated by a element of the family, then in order to preserve the unboundedness of H in finite support iterated forcing construction, it is sufficient to preserve the family unbounded at each successor stage of the iteration. ◮ If H is unbounded and P is a poset of size smaller than the cardinality of H , then H remains unbounded in V P . 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 ◮ Add κ many Hechler reals to a model of GCH to obtain a directed unbounded family H of size κ . ◮ Proceed with a finite support iteration of length κ + alternating C κ , M ( U H ) and restricted Hechler forcing. ◮ An appropriate bookkeeping function will guarantee that in the final generic extension there are no unbounded families of size < κ and so H will remain a witness to b = κ . ◮ Since cofinally often we add reals not split by the ground model reals, s = κ + in the final generic extension. 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 (V. F., J. Stepr¯ ans) Let κ be a regular uncountable cardinal. Then there is a ccc generic extension in which b = κ < s = c = κ + . 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 (J. Brendle) Let κ be a regular uncountable cardinal. Then there is a ccc generic extension in which b = κ < a = c = κ + . The iteration techniques of the last two models can be combined to produce the consistency of b = κ < s = a = κ + . 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 (V.F., J. Stepr¯ ans) Assume CH. There is a countably closed, ℵ 2 -c.c. poset P which adds a C ω 2 -name for an ultrafilter U such that in V P × C ω 2 the relativized Mathias poset M ( U ) preserves the unboundedness of all families of Cohen reals of size ω 1 . 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 ◮ How to iterate ( P × C ( ω 2 )) × M ( U )? ◮ How to force an entire forcing construction with the desired properties? 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 The method of matrix iteration was introduced by S. Shelah and A. Blass in their work on the ultrafilter and dominating number. Using this technique they establish the consistency of u = κ < d = λ for κ < λ arbitrary regular uncountable cardinals. 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 These are systems of finite support iterations �� P α,ζ : α ≤ κ, ζ ≤ λ � , � ˙ Q α,ζ : α ≤ κ, ζ < λ �� such that: ◮ For all α ≤ κ , �� P α,ζ : ζ ≤ λ � , � ˙ Q α,ζ : ζ < λ �� is a finite support iteration of ccc posets. ◮ For all α 1 ≤ α 2 and ζ ≤ λ , P α 1 ,ζ is a complete suborder of P α 2 ,ζ . Thus for all α 1 ≤ α 2 , ζ 1 ≤ ζ 2 we have P α 1 ,ζ 1 < ◦ P α 2 ,ζ 2 . 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 . 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 (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 Con ( b = a = κ < s = λ ) Adding a mad family The forcing construction Increasing s 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 . Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) Adding a mad family The forcing construction Increasing s Open Questions 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 Con ( b = a = κ < s = λ ) Adding a mad family The forcing construction Increasing s 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 Con ( b = a = κ < s = λ ) Adding a mad family The forcing construction Increasing s 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 . Vera Fischer MAD families, splitting families and large continuum

Introduction Con ( b = a = κ < s = λ ) Adding a mad family The forcing construction Increasing s Open Questions 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 Con ( b = a = κ < s = λ ) Adding a mad family The forcing construction Increasing s 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