Maximal objects in the projective hierarchy J org Brendle Kobe - PowerPoint PPT Presentation

Maximal objects in the projective hierarchy J¨ org Brendle Kobe University Sant Bernat, November 18, 2018 Reflections on Set Theoretic Reflection In celebration of Joan Bagaria’s 60th birthday J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like ultrafilters on ω J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like ultrafilters on ω maximal almost disjoint families (mad families) J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like ultrafilters on ω maximal almost disjoint families (mad families) maximal independent families (mifs) J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like ultrafilters on ω maximal almost disjoint families (mad families) maximal independent families (mifs) towers J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like ultrafilters on ω maximal almost disjoint families (mad families) maximal independent families (mifs) towers Typically need fragment of AC for construction of such objects, J¨ org Brendle Maximal objects in the projective hierarchy

Objects with maximality properties Sets of reals with maximality properties like ultrafilters on ω maximal almost disjoint families (mad families) maximal independent families (mifs) towers Typically need fragment of AC for construction of such objects, i.e., they cannot be very definable. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: basic results A ⊆ [ ω ] ω is an almost disjoint (a.d.) family if | A ∩ B | < ω for A � = B from A J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: basic results A ⊆ [ ω ] ω is an almost disjoint (a.d.) family if | A ∩ B | < ω for A � = B from A A ⊆ [ ω ] ω is mad if A is a.d. and maximal with this property, i.e., for all X ∈ [ ω ] ω there is A ∈ A such that | X ∩ A | = ω J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: basic results A ⊆ [ ω ] ω is an almost disjoint (a.d.) family if | A ∩ B | < ω for A � = B from A A ⊆ [ ω ] ω is mad if A is a.d. and maximal with this property, i.e., for all X ∈ [ ω ] ω there is A ∈ A such that | X ∩ A | = ω Fact. A Σ 1 n mad is also ∆ 1 n . J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: basic results A ⊆ [ ω ] ω is an almost disjoint (a.d.) family if | A ∩ B | < ω for A � = B from A A ⊆ [ ω ] ω is mad if A is a.d. and maximal with this property, i.e., for all X ∈ [ ω ] ω there is A ∈ A such that | X ∩ A | = ω Fact. A Σ 1 n mad is also ∆ 1 n . Theorem 1 (T¨ ornquist ’12) If there is a Σ 1 2 mad then there is a Π 1 1 mad. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: basic results A ⊆ [ ω ] ω is an almost disjoint (a.d.) family if | A ∩ B | < ω for A � = B from A A ⊆ [ ω ] ω is mad if A is a.d. and maximal with this property, i.e., for all X ∈ [ ω ] ω there is A ∈ A such that | X ∩ A | = ω Fact. A Σ 1 n mad is also ∆ 1 n . Theorem 1 (T¨ ornquist ’12) If there is a Σ 1 2 mad then there is a Π 1 1 mad. Theorem 2 (Mathias ’70’s) There are no Σ 1 1 mad families. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: basic results A ⊆ [ ω ] ω is an almost disjoint (a.d.) family if | A ∩ B | < ω for A � = B from A A ⊆ [ ω ] ω is mad if A is a.d. and maximal with this property, i.e., for all X ∈ [ ω ] ω there is A ∈ A such that | X ∩ A | = ω Fact. A Σ 1 n mad is also ∆ 1 n . Theorem 1 (T¨ ornquist ’12) If there is a Σ 1 2 mad then there is a Π 1 1 mad. Theorem 2 (Mathias ’70’s) There are no Σ 1 1 mad families. Theorem 3 (Miller ∼ ’90) There are Π 1 1 mads in L. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. In particular, the existence of Π 1 1 mads is consistent with c > ω 1 . J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. In particular, the existence of Π 1 1 mads is consistent with c > ω 1 . Kunen: Under CH, there is a mad family which survives arbitrary Cohen extensions. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. In particular, the existence of Π 1 1 mads is consistent with c > ω 1 . Kunen: Under CH, there is a mad family which survives arbitrary Cohen extensions. For many forcing notions P there are P -indestructible mads (B.-Yatabe). J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. In particular, the existence of Π 1 1 mads is consistent with c > ω 1 . Kunen: Under CH, there is a mad family which survives arbitrary Cohen extensions. For many forcing notions P there are P -indestructible mads (B.-Yatabe). Thus: many ¬ CH models with Π 1 1 mads. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. In particular, the existence of Π 1 1 mads is consistent with c > ω 1 . Kunen: Under CH, there is a mad family which survives arbitrary Cohen extensions. For many forcing notions P there are P -indestructible mads (B.-Yatabe). Thus: many ¬ CH models with Π 1 1 mads. Adding a dominating real destroys all ground model mads. J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: what if CH fails? Theorem 4 (Kunen ’70’s + Folklore) In the Cohen model (over V = L) there is a Σ 1 2 and thus Π 1 1 mad. In particular, the existence of Π 1 1 mads is consistent with c > ω 1 . Kunen: Under CH, there is a mad family which survives arbitrary Cohen extensions. For many forcing notions P there are P -indestructible mads (B.-Yatabe). Thus: many ¬ CH models with Π 1 1 mads. Adding a dominating real destroys all ground model mads. Can we have b > ω 1 together with Π 1 1 mads? J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: some cardinals b := min {| F | : F ⊆ ω ω and ∀ g ∈ ω ω ∃ f ∈ F ∃ ∞ n ( g ( n ) < f ( n )) } the unbounding number J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: some cardinals b := min {| F | : F ⊆ ω ω and ∀ g ∈ ω ω ∃ f ∈ F ∃ ∞ n ( g ( n ) < f ( n )) } the unbounding number a := min {|A| : A ⊆ [ ω ] ω is an infinite mad family } the almost disjointness number J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: some cardinals b := min {| F | : F ⊆ ω ω and ∀ g ∈ ω ω ∃ f ∈ F ∃ ∞ n ( g ( n ) < f ( n )) } the unbounding number a := min {|A| : A ⊆ [ ω ] ω is an infinite mad family } the almost disjointness number a closed := min {|F| : F infinite family of closed sets, � F mad } J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: some cardinals b := min {| F | : F ⊆ ω ω and ∀ g ∈ ω ω ∃ f ∈ F ∃ ∞ n ( g ( n ) < f ( n )) } the unbounding number a := min {|A| : A ⊆ [ ω ] ω is an infinite mad family } the almost disjointness number a closed := min {|F| : F infinite family of closed sets, � F mad } a Borel := min {|F| : F infinite family of Borel sets, � F mad } J¨ org Brendle Maximal objects in the projective hierarchy

Mad families: some cardinals b := min {| F | : F ⊆ ω ω and ∀ g ∈ ω ω ∃ f ∈ F ∃ ∞ n ( g ( n ) < f ( n )) } the unbounding number a := min {|A| : A ⊆ [ ω ] ω is an infinite mad family } the almost disjointness number a closed := min {|F| : F infinite family of closed sets, � F mad } a Borel := min {|F| : F infinite family of Borel sets, � F mad } Fact. ω 1 ≤ b ≤ a ≤ c and ω 1 ≤ a Borel ≤ a closed ≤ a J¨ org Brendle Maximal objects in the projective hierarchy