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 A0 and A1 the set

• A0\
• A1

is infinite. It is a maximal independent family if it is maximal under inclusion and i = min{|A | : A is a m.i.f.}

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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 =

• {Ah(A) : A ∈ dom(h)},

where Ah(A) = A if h(A) = 0 and Ah(A) = ω\A if h(A) = 1. Remark We refer to {A h : h ∈ FF(A )} as a set of boolean combinations.

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Independence on ω

There is a maximal independent family of cardinality 2ℵ0. ℵ0 < i ≤ 2ℵ0. If A is a maximal independent family then {A h : h ∈ FF(A )} is an un-reaped family. Thus r ≤ i. (Shelah) d ≤ i.

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Independence on ω

i vs. u In the Miller model u < i, while Shelah devised a special ωω-bounding poset the countable support iteration of which produces a model of i = ℵ1 < u = ℵ2. 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.

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Independence on ω

a vs i In the Cohen model a < i = c. Question: Is it consistent that i < a?

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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.

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sp(i)

Diagonalization

Lemma (F ., Shelah) If U is a A -diagonalization filter and G is M(U )-generic and xG = {s : ∃F(s,F) ∈ G}, then:

1

A ∪{xG} is independent

2

If y ∈ ([ω]ω\A )∩V is such that A ∪{y} is independent, then A ∪{xG,y} is not independent.

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sp(i)

Diagonalization

Corollary (F ., Shelah) Let κ be of uncountable cofinality. Then it is relatively consistent that ℵ1 < i = κ < c. Theorem (F ., Shelah) Assume GCH. Let κ1 < ··· < κn be regular uncountable cardinals. Then it is consistent that {κi}n

i=1 ⊆ Sp(i).

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sp(i)

Theorem (F ., Shelah) Let κ1 < κ2 < ··· < κn be measurable cardinals witnessed by κi-complete ultrafilters Di ⊆ P(κi). Then there is a ccc generic extension in which {κi}n

i=1 = sp(i) = {|A | : A m.i.f.}.

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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) = {ℵ1,λ}.

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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 Ui = U and let G = ∏i∈n Gi be a ∏i∈n M(Ui)-generic filter, xi a M(Ui)-generic real. Then in V[G]: A ∪{xi}i∈n is independent. For all y ∈ V ∩[ω]ω such that A ∪{y} is independent and each i ∈ n, the family A ∪{y,xi} is not independent.

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Diagonalization and Homogeniety

sp(i) can be large

Definition (F ., Shelah) Let θ be an uncountable cardinal and let S ⊆ θ <ω1 be an θ-splitting tree of height ω1. For each α ∈ ω1 let Sα denote the α-th level of S. Recursively, define a finite support iteration PS = Pα,Qα : α ≤ ω1,β < ω1 as follows: Let P0 = {/ 0}, Q0 be a P0-name for the trivial poset. Let A0 = / 0 be the empty independent family and let U0 be an A0-diagonalizing real.

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Diagonalization and Homogeniety

sp(i) can be large

For η ∈ S1 = succS(/ 0) let Uη = U0 and let Q1 = ∏<ω

η∈S1 M(Uη).

In V P1∗ ˙

Q1 = V P2 let aη be the M(Uη)-generic real.

Let α ≥ 2 and in V Pα for each η ∈ Sα let Aη = {aν : ν ∈ succS(η↾ξ),ξ < α} be an ind. family. For each η ∈ Sα, let Uη be an Aη-diagonalisation filter and let Qα = ∏<ω

η∈Sα M(Uη).

In V Pα∗Qα let aη be the M(Uη)-generic real.

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Diagonalization and Homogeniety

sp(i) can be large

Theorem(F ., Shelah) In V PS for each branch η of S the family Aη = {aν : 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.

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Diagonalization and Homogeniety

sp(i) can be large

Theorem (V.F ., Shelah, 2019) Assume GCH. Let A ⊆ {ℵn}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 minsp(i) = i = ℵω.

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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 = {Ah(A) : A ∈ dom(h)} where Ah(A) = A if h(A) = 0 and Ah(A) = κ\A if h(A) = 1.

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Higher Independence

Definition

1

A family A ⊆ [κ]κ is said to be κ-independent if for each h ∈ FF<ω,κ(A ), A h is unbounded. It is maximal κ-independent family if it is κ-independent, maximal under inclusion.

2

The least size of a maximal κ-independent family is denoted i(κ).

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Higher Independence

Lemma (V.F ., D. Montoya) Let κ be a regular infinite cardinal.

1

There is a maximal κ-independent family of cardinality 2κ.

2

κ+ ≤ i(κ) ≤ 2κ

3

r(κ) ≤ i(κ)

4

d(κ) ≤ i(κ). Corollary If κ is regular uncountable, then if i(κ) = κ+ also a(κ) = κ+.

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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.

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κ-Sacks indestructibility

Definition (V.F ., D. Montoya) Let κ be a measurable cardinal and U a normal measure on κ. Let PU 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: (A1,A1) ≤ (A0,A0) iff A1 ⊇ A0 and A1 ⊆∗ A0.

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κ-Sacks indestructibility

Lemma (V.F ., D. Montoya) Assume 2κ = κ+. Then PU is κ+-closed and κ++-cc and if G is a PU -generic filter, then AG =

• {A : ∃A ∈ U with (A ,A) ∈ G}

is a densely maximal κ-independent family.

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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.

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κ-Sacks indestructibility

Definition We refer to a partition E of κ into bounded sets as a bounded partition.

1

If E is a bounded partition of κ, A ∈ [κ]κ is such that for all E ∈ E (|E ∩A| ≤ 1), we say that A is a semi-selector for E .

2

If E is a bounded partition of κ and A ∈ [κ]κ is such that for all E ∈ E , |E ∩A| ≤ 2, then A is called a 2-semi-selector of E . Remark Since U is a normal measure on κ, for each bounded partition E of κ there is a semi-selector of E in U .

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κ-Sacks indestructibility

Definition Let F ⊆ [κ]κ. We say that:

1

F is a κ-P-set if every H ⊆ F of cardinality ≤ κ has a pseudo-intersection in F;

2

F is a κ-Q-set if every bounded partition of κ has a 2-semi-selector in F.

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κ-Sacks indestructibility

Selective Independence at κ

Theorem (V.F ., D. Montoya) The density independence filter of F<ω,κ(AG) is both a κ-Q-set and a κ-P-set, which is generated by {A : ∃A (A ,A) ∈ G}. Theorem (V.F ., D. Montoya) The generic maximal independent family AG adjoined by PU over a model of GCH remains maximal after the κ-support product Sλ

κ.

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κ-Sacks indestructibility

Corollary Let κ be a measurable cardinal. There is a cardinal preserving generic extension in which a(κ) = d(κ) = r(κ) = i(κ) = κ+ < 2κ.

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κ-Sacks indestructibility

Question Let κ be a regular uncountable cardinal. Is it consistent that κ+ < i(κ) < 2κ?

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κ-Sacks indestructibility

Definition Let A be a κ-independent family. A κ-complete filter F is said to be an κ-diagonalization filter for A if ∀F ∈ F∀h ∈ FF<ω,κ(A )|F ∩A h| = κ and F is maximal with respect to the above property. Question Given a κ-independent family A is there a κ-diagonalizazion filter for A ? Is there a large cardinal property which guarantees the existence

• f such maximal filter?

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Strong-κ-independence

Definition Let κ be a regular uncountable cardinal, A ⊆ [κ]κ of size at least κ.

1

Let FF<κ,κ(A ) = {h : A → {0,1} : such that |dom(h)| < κ}.

2

For each h ∈ FF<κ,κ(A ) let A h = {Ah(A) : A ∈ dom(h)} where Ah(A) = A if h(A) = 0 and Ah(A) = κ\A if h(A) = 1.

3

A is said to be strongly-κ-independent if for each h ∈ FF<κ,κ(A ), A h is unbounded.

4

A is maximal strongly-κ-independent family if it is κ-independent, maximal under inclusion.

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Strong-κ-independence

Lemma (V.F ., D. Montoya) Let κ be a regular infinite cardinal.

1

For κ strongly inaccessible, there is a strongly-κ-independent family of cardinality 2κ.

2

If A is strongly-κ-independent and |A | < r(κ) then A is not maximal.

3

Suppose d(κ) is such that for every γ < d(κ), γ<κ < d(κ). If A is strongly-κ-independent and |A | < d(κ) then A is not maximal.

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Strong-κ-independence

Corollary Thus if is(κ) = min{|A | : A maximal strongly-κ-independent family} is defined, then κ+ ≤ is(κ) ≤ 2κ; r(κ) ≤ is(κ); if for every γ < d(κ), γ<κ < d(κ), then d(κ) ≤ is(κ).

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Strong-κ-independence

Theorem (Kunen, 1983)

1

The existence of a maximal strongly-ω1-independent family implies CH and the existence of a weakly inaccessible cardinal between ω1 and 2ω1;

2

The existence of a measurable cardinal is equiconsistent with the existence of a maximal strongly-ω1-independent family.

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Strong-κ-independence

Thank you!

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Strong-κ-independence

• V. Fischer Selective Independence preprint.
• V. Fischer, D. C. Montoya Higher Independence preprint.
• V. Fischer, D. C. Montoya Ideals of Independence 2019.
• V. Fischer, S. Shelah The spectrum of independence 2019.
• V. Fischer, S. Shelah The spectrum of independence II preprint.
• S. Shelah Are a and d your cup of tea? 2000.
• S. Shelah Con(u > i) 1992.

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