Higher independence Vera Fischer University of Vienna February - PowerPoint PPT Presentation

Higher independence Vera Fischer University of Vienna February 4th, 2020 Vera Fischer (University of Vienna) Higher independence February 4th, 2020 1 / 35

Independence on ω Independence Number A family A ⊆ [ ω ] ω is said to be independent for any two non-empty finite disjoint subfamilies A 0 and A 1 the set � � A 0 \ A 1 is infinite. It is a maximal independent family if it is maximal under inclusion and i = min {| A | : A is a m.i.f. } Vera Fischer (University of Vienna) Higher independence February 4th, 2020 2 / 35

Independence on ω Boolean combinations Let A be a independent and let FF( A ) be the set of all finite partial functions from A to 2. For h ∈ FF( A ) define A h = { A h ( A ) : A ∈ dom ( h ) } , � where A h ( A ) = A if h ( A ) = 0 and A h ( A ) = ω \ A if h ( A ) = 1. Remark We refer to { A h : h ∈ FF( A ) } as a set of boolean combinations. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 3 / 35

Independence on ω There is a maximal independent family of cardinality 2 ℵ 0 . ℵ 0 < i ≤ 2 ℵ 0 . If A is a maximal independent family then { A h : h ∈ FF( A ) } is an un-reaped family. Thus r ≤ i . (Shelah) d ≤ i . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 4 / 35

Independence on ω i vs. u In the Miller model u < i , while Shelah devised a special ω ω -bounding poset the countable support iteration of which produces a model of i = ℵ 1 < u = ℵ 2 . a vs. u In the Cohen model a < u , while assuming the existence of a measurable one can show the consistency of u < a . The use of a measurable has been eliminated by Guzman and Kalajdzievski. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 5 / 35

Independence on ω a vs i In the Cohen model a < i = c . Question: Is it consistent that i < a ? Vera Fischer (University of Vienna) Higher independence February 4th, 2020 6 / 35

sp ( i ) Diagonalization A -diagonalization filters (F ., Shelah) Let A be an independent family. A filter U is said to be an A -diagonalization filter if ∀ F ∈ U ∀ B ∈ BC ( A )( | F ∩ B | = ω ) and is maximal with respect to the above property. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 7 / 35

sp ( i ) Diagonalization Lemma (F ., Shelah) If U is a A -diagonalization filter and G is M ( U ) -generic and x G = � { s : ∃ F ( s , F ) ∈ G } , then: A ∪{ x G } is independent 1 If y ∈ ([ ω ] ω \ A ) ∩ V is such that A ∪{ y } is independent, then 2 A ∪{ x G , y } is not independent. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 8 / 35

sp ( i ) Diagonalization Corollary (F ., Shelah) Let κ be of uncountable cofinality. Then it is relatively consistent that ℵ 1 < i = κ < c . Theorem (F ., Shelah) Assume GCH. Let κ 1 < ··· < κ n be regular uncountable cardinals. Then it is consistent that { κ i } n i = 1 ⊆ Sp ( i ) . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 9 / 35

sp ( i ) Theorem (F ., Shelah) Let κ 1 < κ 2 < ··· < κ n be measurable cardinals witnessed by κ i -complete ultrafilters D i ⊆ P ( κ i ) . Then there is a ccc generic extension in which { κ i } n i = 1 = sp ( i ) = {| A | : A m.i.f. } . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 10 / 35

Preservation properties Sp ( i ) can be small Theorem (F ., Shelah) Assume GCH. Let λ be a cardinal of uncountable cofinality. Let G be P -generic filter, where P is the countable support product of Sacks forcing of length λ . Then V [ G ] � Sp ( i ) = { ℵ 1 , λ } . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 11 / 35

Diagonalization and Homogeniety sp ( i ) can be large Lemma Let A be an independent family, U a diagonalization filter for A . For each i ∈ n , let U i = U and let G = ∏ i ∈ n G i be a ∏ i ∈ n M ( U i ) -generic filter, x i a M ( U i ) -generic real. Then in V [ G ] : A ∪{ x i } i ∈ n is independent. For all y ∈ V ∩ [ ω ] ω such that A ∪{ y } is independent and each i ∈ n , the family A ∪{ y , x i } is not independent. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 12 / 35

Diagonalization and Homogeniety sp ( i ) can be large Definition (F ., Shelah) Let θ be an uncountable cardinal and let S ⊆ θ < ω 1 be an θ -splitting tree of height ω 1 . For each α ∈ ω 1 let S α denote the α -th level of S . Recursively, define a finite support iteration P S = � P α , Q α : α ≤ ω 1 , β < ω 1 � as follows: Let P 0 = { / 0 } , Q 0 be a P 0 -name for the trivial poset. Let A 0 = / 0 be the empty independent family and let U 0 be an A 0 -diagonalizing real. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 13 / 35

Diagonalization and Homogeniety sp ( i ) can be large 0 ) let U η = U 0 and let Q 1 = ∏ < ω For η ∈ S 1 = succ S ( / η ∈ S 1 M ( U η ) . In V P 1 ∗ ˙ Q 1 = V P 2 let a η be the M ( U η ) -generic real. Let α ≥ 2 and in V P α for each η ∈ S α let A η = { a ν : ν ∈ succ S ( η ↾ ξ ) , ξ < α } be an ind. family. For each η ∈ S α , let U η be an A η -diagonalisation filter and let Q α = ∏ < ω η ∈ S α M ( U η ) . In V P α ∗ Q α let a η be the M ( U η ) -generic real. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 14 / 35

Diagonalization and Homogeniety sp ( i ) can be large Theorem(F ., Shelah) In V P S for each branch η of S the family A η = { a ν : ν ∈ succ ( η ↾ ξ ) , ξ < σ } is a maximal independent family of cardinality θ . Remark The idea can be extended to adjoin with a sufficiently homogenous forcing witnesses for many distinct values in sp ( i ) simultaneously. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 15 / 35

Diagonalization and Homogeniety sp ( i ) can be large Theorem (V.F ., Shelah, 2019) Assume GCH. Let A ⊆ { ℵ n } n ∈ ω . Then there is a ccc generic extension in which sp ( i ) = A . Remark A more involved argument shows that sp ( i ) can be quite arbitrary. A remaining open question is the consistency of min sp ( i ) = i = ℵ ω . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 16 / 35

Higher Independence Definition Let κ be a regular uncountable cardinal. Let FF < ω , κ ( A ) be the set of all finite partial functions with domain included in A and range the set { 0 , 1 } . For each h ∈ FF < ω , κ ( A ) let A h = � { A h ( A ) : A ∈ dom( h ) } where A h ( A ) = A if h ( A ) = 0 and A h ( A ) = κ \ A if h ( A ) = 1. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 17 / 35

Higher Independence Definition A family A ⊆ [ κ ] κ is said to be κ -independent if for each 1 h ∈ FF < ω , κ ( A ) , A h is unbounded. It is maximal κ -independent family if it is κ -independent, maximal under inclusion. The least size of a maximal κ -independent family is denoted i ( κ ) . 2 Vera Fischer (University of Vienna) Higher independence February 4th, 2020 18 / 35

Higher Independence Lemma (V.F ., D. Montoya) Let κ be a regular infinite cardinal. There is a maximal κ -independent family of cardinality 2 κ . 1 κ + ≤ i ( κ ) ≤ 2 κ 2 r ( κ ) ≤ i ( κ ) 3 d ( κ ) ≤ i ( κ ) . 4 Corollary If κ is regular uncountable, then if i ( κ ) = κ + also a ( κ ) = κ + . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 19 / 35

Higher Independence Definition A κ -independent family A is densely maximal if for every X ∈ [ κ ] κ \ A and every h ∈ FF < ω , κ ( A ) there is h ′ ∈ FF < ω , κ ( A ) extending h such that either A h ′ ∩ X = / 0 or A h ′ ∩ ( κ \ X ) = / 0 . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 20 / 35

κ -Sacks indestructibility Definition (V.F ., D. Montoya) Let κ be a measurable cardinal and U a normal measure on κ . Let P U be the poset of all pairs ( A , A ) where A is a κ -independent family of cardinality κ , A ∈ U is such that ∀ h ∈ FF < ω , κ ( A ) , A h ∩ A is unbounded. The extension relation is defined as follows: ( A 1 , A 1 ) ≤ ( A 0 , A 0 ) iff A 1 ⊇ A 0 and A 1 ⊆ ∗ A 0 . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 21 / 35

κ -Sacks indestructibility Lemma (V.F ., D. Montoya) Assume 2 κ = κ + . Then P U is κ + -closed and κ ++ -cc and if G is a P U -generic filter, then � A G = { A : ∃ A ∈ U with ( A , A ) ∈ G } is a densely maximal κ -independent family. Vera Fischer (University of Vienna) Higher independence February 4th, 2020 22 / 35

κ -Sacks indestructibility Definition Let A be an independent family. The density independence filter F < ω , κ ( A ) is the filter of all X ∈ U , such that ∀ h ∈ FF < ω , κ ( A ) there is h ′ ∈ FF < ω , κ ( A ) such that h ′ ⊇ h and A h ′ ⊆ X . Vera Fischer (University of Vienna) Higher independence February 4th, 2020 23 / 35