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

Definable Maximal Families Yurii Khomskii most results joint with J¨ org 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 a 1 , . . . , a n and (different) b 1 , . . . , b m from I , a 1 ∩ · · · ∩ a n ∩ ( ω \ b 1 ) ∩ ( ω \ b m ) 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 a 1 , . . . , a n and (different) b 1 , . . . , b m from I , a 1 ∩ · · · ∩ a n ∩ ( ω \ b 1 ) ∩ ( ω \ b m ) is infinite. = ⇒ maximal independent families (mif). ultrafilters Hausdorff gaps etc. Yurii Khomskii Definable MIFs Arctic 4 2 / 26

Definable maximal families Such maximal families are constructed from a well-order of the reals. V = L → ∃ Σ 1 2 maximal families Usually, Col ( ω, <κ ) � ∄ projective maximal families ( κ inaccessible) L ( R ) Col ( ω,<κ ) (Solovay Model) | = ∄ maximal families The existence of Σ 1 2 / Π 1 1 maximal families is more subtle, and is related to how easy it is to preserve or destroy such a family. Yurii Khomskii Definable MIFs Arctic 4 3 / 26

Preserving vs. Destroying 1 Preserving the definable maximal family: ∃ Σ 1 2 max. family + ¬ CH ∃ Σ 1 2 max. family + ℵ 1 < b , d etc. ∃ Σ 1 2 max. family + all Σ 1 2 sets are measurable/Baire property/etc. 2 Destroying the definable maximal family: ∄ Σ 1 2 max. family ∄ Σ 1 2 max. family but . . . ∄ projective max. family ∄ max. family For the last two points: “can you take Solovay’s inaccessible away?” Yurii Khomskii Definable MIFs Arctic 4 4 / 26

1. Maximal almost disjoint (mad) families Yurii Khomskii Definable MIFs Arctic 4 5 / 26

Preserving Preserving mad families 1 V = L → ∃ Σ 1 2 mad 2 V = L → ∃ Π 1 1 mad (Miller 1989) 3 ∃ Σ 1 2 mad ↔ ∃ Π 1 1 mad (T¨ ornquist 2012) 4 In L , there exists a Cohen-, Sacks- and Miller-indestructible Σ 1 2 -mad family (Raghavan 2009). Therefore, consistently, there exist a Σ 1 2 / Π 1 1 mad together with ℵ 1 < cov ( M ) ≤ d = 2 ℵ 0 , and also with ∆ 1 2 (Baire property). Yurii Khomskii Definable MIFs Arctic 4 6 / 26

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 ℵ 1 < b implies ∄ Σ 1 2 mad. Yurii Khomskii Definable MIFs Arctic 4 7 / 26

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 ℵ 1 < b implies ∄ Σ 1 2 mad. Theorem (Brendle-K, 2011) Con( ℵ 1 < b + ∃ Σ 1 2 / Π 1 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 ℵ 1 -union of perfect almost disjoint sets P α , such that the reinterpreted family remains maximal: A V [ G ] := P V [ G ] � α α< ℵ 1 Yurii Khomskii Definable MIFs Arctic 4 7 / 26

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 a B := least number of Borel a.d. sets whose union is a mad family. Clearly a B ≤ a . Theorem (Brendle-K, 2011) Con( a B < b ) (and as a consequence Con( a B < a ) ). Yurii Khomskii Definable MIFs Arctic 4 8 / 26

Destroying Destroying mad families 1 ∄ Σ 1 1 mad (Mathias 1977) 2 Solovay Model | = ∄ mad (T¨ ornquist 2015) 3 Con( ∄ mad) without inaccessible (Horowitz & Shelah 2017) Moreover: ℵ 1 < a B → ∄ Σ 1 2 mad. Yurii Khomskii Definable MIFs Arctic 4 9 / 26

Destroying Destroying mad families 1 ∄ Σ 1 1 mad (Mathias 1977) 2 Solovay Model | = ∄ mad (T¨ ornquist 2015) 3 Con( ∄ mad) without inaccessible (Horowitz & Shelah 2017) Moreover: ℵ 1 < a B → ∄ Σ 1 2 mad. In general: destroying Σ 1 2 -definable mad is actually not so easy. We know that t ≤ a B (Raghavan) but not much more. Yurii Khomskii Definable MIFs Arctic 4 9 / 26

Questions for destroying mads Question (Raghavan) h ≤ a B ? We can force to increase a B directly, but it’s a bit cumbersome. Question (Brendle-K) Is there some . . . . . . such that TFAE: ∄ Σ 1 2 / Π 1 1 mad ∀ r ∈ ω ω , there exists . . . . . . over L [ r ]? Question (Brendle, Raghavan, T¨ ornquist, Schrittesser) How is ∄ Σ 1 2 mad related to other regularity properties? Compare: Schrittesser’s talk. Does Σ 1 2 ( Ramsey ) imply ∄ Σ 1 2 mad? Yurii Khomskii Definable MIFs Arctic 4 10 / 26

2. Maximal independent families (mif’s) Yurii Khomskii Definable MIFs Arctic 4 11 / 26

Maximal independent families Definition A family I ⊆ [ ω ] ω is an independent if for any a 1 , . . . , a n and different b 1 , . . . , b m from I , the intersection a 1 ∩ · · · ∩ a n ∩ ( ω \ b 1 ) ∩ ( ω \ b m ) 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: V = L → ∃ Σ 1 2 mif 1 V = L → ∃ Π 1 1 mif (Miller 1989) 2 Yurii Khomskii Definable MIFs Arctic 4 13 / 26

Preserving and Destroying mifs Heuristic: mif’s are harder to preserve, easier to destroy. Preserving mif: V = L → ∃ Σ 1 2 mif 1 V = L → ∃ Π 1 1 mif (Miller 1989) 2 Theorem (Brendle-Fischer-K) ∃ Σ 1 2 mif ↔ ∃ Π 1 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 Σ 1 2 / Π 1 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 Σ 1 2 -definable mif in L . This shows the consistency of ∃ Π 1 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. Yurii Khomskii Definable MIFs Arctic 4 15 / 26

Borel independence number But we don’t really have more preservation results. Definition i B := least number of Borel independent sets whose union is a mif family. Clearly i B ≤ i , and ℵ 1 < i B → ∄ Σ 1 2 mif. We don’t really know how to keep i B small. Yurii Khomskii Definable MIFs Arctic 4 15 / 26

Destroying On the other hand, destroying a mif is easier. ∄ Σ 1 1 mif (Miller 1989) Miller’s proof actually shows Σ 1 n (Baire propery) → ∄ Σ 1 n mif. Hence: In the iterated Hechler (or “Amoeba”) model, there is no Σ 1 2 mif = ∄ mif Solovay Model | = ∄ mif Shelah’s model for Baire Property w/out inaccessible | Yurii Khomskii Definable MIFs Arctic 4 16 / 26

Cohen model But in fact, we prove something stronger: Theorem (Brendle-Fischer-K) In the Cohen model V C ω 1 there is no projective mif Yurii Khomskii Definable MIFs Arctic 4 17 / 26

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 = ∅ . A Perfect AC Perfect AD Yurii Khomskii Definable MIFs Arctic 4 18 / 26

Sketch of proof Lemma 1 Σ 1 n (AD/AC-dichotomy) → ∄ Σ 1 n mif. Lemma 2 C adds a perfect AD and a perfect AC set of Cohen reals. Lemma 3 C ω 1 � Proj(AD/AC-dichotomy). Yurii Khomskii Definable MIFs Arctic 4 19 / 26

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 C κ (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 = ∅ ) Yurii Khomskii Definable MIFs Arctic 4 20 / 26