Definable Maximal Families Yurii Khomskii most results joint with J - - PowerPoint PPT Presentation

Definable Maximal Families

Yurii Khomskii

most results joint with J¨

• rg Brendle and Vera Fischer

Arctic Set Theory 4

Yurii Khomskii Definable MIFs Arctic 4 1 / 26

Maximal families

Families of reals with maximality properties have many applications in set theory and mathematics.

A is almost disjoint if a ∩ b is finite for all a, b ∈ A = ⇒ maximal almost disjoint (mad) families.

Yurii Khomskii Definable MIFs Arctic 4 2 / 26

Maximal families

Families of reals with maximality properties have many applications in set theory and mathematics.

A is almost disjoint if a ∩ b is finite for all a, b ∈ A = ⇒ maximal almost disjoint (mad) families. I is independent if for a1, . . . , an and (different) b1, . . . , bm from I, a1 ∩ · · · ∩ an ∩ (ω \ b1) ∩ (ω \ bm) is infinite. = ⇒ maximal independent families (mif).

Yurii Khomskii Definable MIFs Arctic 4 2 / 26

Maximal families

Families of reals with maximality properties have many applications in set theory and mathematics.

A is almost disjoint if a ∩ b is finite for all a, b ∈ A = ⇒ maximal almost disjoint (mad) families. I is independent if for a1, . . . , an and (different) b1, . . . , bm from I, a1 ∩ · · · ∩ an ∩ (ω \ b1) ∩ (ω \ bm) is infinite. = ⇒ maximal independent families (mif).

ultrafilters Hausdorff gaps etc.

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Definable maximal families

Such maximal families are constructed from a well-order of the reals. V = L → ∃Σ1

2 maximal families

Usually, Col(ω, <κ) ∄ projective maximal families (κ inaccessible) L(R)Col(ω,<κ) (Solovay Model) | = ∄ maximal families The existence of Σ1

2/Π1 1 maximal families is more subtle, and is

related to how easy it is to preserve or destroy such a family.

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Preserving vs. Destroying

1 Preserving the definable maximal family:

∃Σ1

2 max. family + ¬CH

∃Σ1

2 max. family + ℵ1 < b, d etc.

∃Σ1

2 max. family + all Σ1 2 sets are measurable/Baire property/etc.

2 Destroying the definable maximal family:

∄Σ1

2 max. family

∄Σ1

2 max. family but . . .

∄ projective max. family ∄ max. family

For the last two points: “can you take Solovay’s inaccessible away?”

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• 1. Maximal almost disjoint (mad) families

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Preserving

Preserving mad families

1 V = L → ∃Σ1

2 mad

2 V = L → ∃Π1

1 mad (Miller 1989)

3 ∃Σ1

2 mad ↔ ∃Π1 1 mad (T¨

• rnquist 2012)

4 In L, there exists a Cohen-, Sacks- and Miller-indestructible Σ1

2-mad

family (Raghavan 2009). Therefore, consistently, there exist a Σ1

2/Π1 1

mad together with ℵ1 < cov(M) ≤ d = 2ℵ0, and also with ∆1

2(Baire

property).

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Dominating reals

However, if P adds a dominating real and A is mad, then P A is not mad (this is similar to b ≤ a). Therefore, one might expect that ℵ1 < b implies ∄Σ1

2 mad.

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Dominating reals

However, if P adds a dominating real and A is mad, then P A is not mad (this is similar to b ≤ a). Therefore, one might expect that ℵ1 < b implies ∄Σ1

2 mad.

Theorem (Brendle-K, 2011) Con(ℵ1 < b + ∃Σ1

2/Π1 1 mad).

We used the Hechler partial order D (canonical ccc forcing for adding dominating reals). Although D destroys the maximality of a ground model mad family, we can construct an ℵ1-union of perfect almost disjoint sets Pα, such that the reinterpreted family remains maximal: AV [G] :=

• α<ℵ1

PV [G]

α

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Borel a.d. number

While a = least size of a mad family, a more important concept for preservation and destruction turns out to be the following: Definition aB := least number of Borel a.d. sets whose union is a mad family. Clearly aB ≤ a. Theorem (Brendle-K, 2011) Con(aB < b) (and as a consequence Con(aB < a)).

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Destroying

Destroying mad families

1 ∄Σ1

1 mad (Mathias 1977)

2 Solovay Model |

= ∄ mad (T¨

• rnquist 2015)

3 Con(∄ mad) without inaccessible (Horowitz & Shelah 2017)

Moreover: ℵ1 < aB → ∄Σ1

2 mad.

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Destroying

Destroying mad families

1 ∄Σ1

1 mad (Mathias 1977)

2 Solovay Model |

= ∄ mad (T¨

• rnquist 2015)

3 Con(∄ mad) without inaccessible (Horowitz & Shelah 2017)

Moreover: ℵ1 < aB → ∄Σ1

2 mad.

In general: destroying Σ1

2-definable mad is actually not so easy. We know

that t ≤ aB (Raghavan) but not much more.

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Questions for destroying mads

Question (Raghavan) h ≤ aB? We can force to increase aB directly, but it’s a bit cumbersome. Question (Brendle-K) Is there some . . . . . . such that TFAE: ∄Σ1

2/Π1 1 mad

∀r ∈ ωω, there exists . . . . . . over L[r]? Question (Brendle, Raghavan, T¨

• rnquist, Schrittesser)

How is ∄Σ1

2 mad related to other regularity properties? Compare: Schrittesser’s talk.

Does Σ1

2(Ramsey) imply ∄Σ1 2 mad? Yurii Khomskii Definable MIFs Arctic 4 10 / 26

• 2. Maximal independent families (mif’s)

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Maximal independent families

Definition A family I ⊆ [ω]ω is an independent if for any a1, . . . , an and different b1, . . . , bm from I, the intersection a1 ∩ · · · ∩ an ∩ (ω \ b1) ∩ (ω \ bm) is infinite

        Yurii Khomskii Definable MIFs Arctic 4 12 / 26

Preserving and Destroying mifs

Heuristic: mif’s are harder to preserve, easier to destroy. Preserving mif:

1

V = L → ∃Σ1

2 mif 2

V = L → ∃Π1

1 mif (Miller 1989) Yurii Khomskii Definable MIFs Arctic 4 13 / 26

Preserving and Destroying mifs

Heuristic: mif’s are harder to preserve, easier to destroy. Preserving mif:

1

V = L → ∃Σ1

2 mif 2

V = L → ∃Π1

1 mif (Miller 1989)

Theorem (Brendle-Fischer-K) ∃Σ1

2 mif ↔ ∃Π1 1 mif Yurii Khomskii Definable MIFs Arctic 4 13 / 26

Sacks model

Unlike the situation with mad families, we have very few preservation results. Theorem (Brendle-Fischer-K) In the iterated Sacks-model, there is a Σ1

2/Π1 1 mif

Using a method implicit in Shelah’s proof of Con(i < u), and studied more explicitly by Fischer & Montoya, we construct a Sacks-indestructible Σ1

2-definable mif in L. This

shows the consistency of ∃Π1

1 mif + ¬CH (in fact we can also get i < u in this model). Yurii Khomskii Definable MIFs Arctic 4 14 / 26

Borel independence number

But we don’t really have more preservation results.

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Borel independence number

But we don’t really have more preservation results. Definition iB := least number of Borel independent sets whose union is a mif family. Clearly iB ≤ i, and ℵ1 < iB → ∄Σ1

2 mif.

We don’t really know how to keep iB small.

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Destroying

On the other hand, destroying a mif is easier. ∄Σ1

1 mif (Miller 1989)

Miller’s proof actually shows Σ1

n(Baire propery) → ∄Σ1 n mif.

Hence: In the iterated Hechler (or “Amoeba”) model, there is no Σ1

2 mif

Solovay Model | = ∄ mif Shelah’s model for Baire Property w/out inaccessible | = ∄ mif

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Cohen model

But in fact, we prove something stronger: Theorem (Brendle-Fischer-K) In the Cohen model V Cω1 there is no projective mif

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Sketch of the proof

Definition We say that C ⊆ [ω]ω is perfect almost disjoint if it is perfect and ∀a, b ∈ C (|a ∩ b| < ω) and perfect almost covering if it is perfect and ∀a, b ∈ C (a ∪ b =∗ ω) (i.e., the collection of complements is perfect almost disjoint). A set A ⊆ [ω]ω satisfies the AD/AC-dichotomy if there exists a perfect almost disjoint C ⊆ A or a perfect almost covering D ∩ A = ∅.

Perfect AC

A

Perfect AD

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Sketch of proof

Lemma 1 Σ1

n(AD/AC-dichotomy) → ∄Σ1 n mif.

Lemma 2 C adds a perfect AD and a perfect AC set of Cohen reals. Lemma 3 Cω1 Proj(AD/AC-dichotomy).

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Sketch of proof

I will only show the proof of Lemma 3. In fact, we prove a generally useful and little-known fact: sometimes you can use the Cohen model instead of the Solovay model. Lemma Suppose X is some absolutely-definable collection of Borel sets of reals, such that C ∃H ∈ X ∀c ∈ X (c is Cohen). Then in the forcing extension by Cκ (for any regular, uncountable κ) every projective (or in L(R)) set of reals A satisfies the following homogeneity property: ∃H ∈ X (H ⊆ A ∨ H ∩ A = ∅)

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Sketch of proof

Proof. Let Gκ be Cκ generic over V and W := V [Gκ]. Suppose A = {x : φ(x)}, wlog. with parameters in V . Let c be Cohen over V and assume w.l.o.g. that W | = φ(c). By the forcing theorem V [c] | = “p Q φ(ˇ c)” where Q is the remainder forcing from V [c] to W and p is some Q-condition. However, since Cκ is the product forcing, Q is isomorphic to Cκ. Moreover, since Cκ is homogeneous we can assume that p is the trivial condition, hence: V [c] | = “1 Cκ φ(ˇ c)” Let Φ(c) abbreviate the above statement. By the forcing theorem, there is a Cohen-condition in V forcing Φ( ˙ xgen) (over V ) and wlog. this is the trivial condition.

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Sketch of proof

Proof. By assumption, in V [c] there exists H ∈ X in V [c], such that V [c] | = ∀x ∈ H (x is Cohen over V ). By our absoluteness assumptions, the above statement is upwards absolute, hence it holds in W as well. Now take any other x ∈ H from W . Then x is Cohen over V , therefore V [x] satisfies what is forced, namely Φ(x). Therefore: V [x] | = “1 Cκ φ(ˇ x)” But the remainder forcing leading from V [x] to W is isomorphic to Cκ, so W | = φ(x). Therefore W | = H ⊆ {x : φ(x)}.

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Consequence

Corollary It is consistent that there are no projective mif’s but other things survive, e.g., ∃Π1

1 mad family (also, ¬Σ1 2(Baire Property)).

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Questions about mif’s

Question Are there any other forcings that keep iB small? In particular, is iB < i consistent? Question Is the existence of a Π1

1 mif consistent with ℵ1 < i?

Question Is there some . . . . . . such that TFAE: ∄Σ1

2/Π1 1 mif

∀r ∈ ωω, there exists . . . . . . over L[r]? Question How is ∄Σ1

2 mif related to other regularity properties? Yurii Khomskii Definable MIFs Arctic 4 24 / 26

Other maximal families

There is a lot to investigate concerning other maximal families (or, in general, sets of reals constructed from a well-order of the reals). For example: there are Hausdorff gaps which are preserved by all forcing notions. In fact: Theorem (K, 2011) TFAE:

1

∄Σ1

2 Hausdorff gap 2

∄Π1

1 Hausdorff gap 3

∀r (ωL[r]

1

< ω1) Fischer & Schilhan have interesting results on definable towers. What about “∃∆1

2 ultrafilter”? This is related to the Borel-version of the ultrafilter

number, uB. There are some partial results of Schilhan about this.

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Thank you!

Yurii Khomskii yurii@deds.nl

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