Mad families, definability, and ideals (Part 2)

Transcription

1 Mad families, definability, and ideals (Part 2) David Schrittesser (KGRC) joint with Asger Törnquist and Karen Haga Descriptive Set Theory in Torino August 17 Schrittesser Mad families, part 2 DST Torino 17 1 / 15

2 Overview 1 Definable mad families under Projective Determinacy 2 J -mad families, for Borel ideals J other than Fin. Schrittesser Mad families, part 2 DST Torino 17 2 / 15

3 Definable mad families under Projective Determinacy Theorem Assuming the Axiom of Projective Determinacy (PD), there are no infinite projective mad families. Our methods also give results under AD, but not all of them are new: Theorem (Neeman-Norwood) Assuming the Axiom of Determinacy holds in L(R), there are no infinite mad families in L(R). We concentrate on PD. Our proofs use: 1 Every projective set is Suslin 2 Reasonably definable forcing cannot change the projective theory Schrittesser Mad families, part 2 DST Torino 17 3 / 15

4 Definable mad families under Projective Determinacy Theorem Assuming the Axiom of Projective Determinacy (PD), there are no infinite projective mad families. Our methods also give results under AD, but not all of them are new: Theorem (Neeman-Norwood) Assuming the Axiom of Determinacy holds in L(R), there are no infinite mad families in L(R). We concentrate on PD. Our proofs use: 1 Every projective set is Suslin 2 Reasonably definable forcing cannot change the projective theory Schrittesser Mad families, part 2 DST Torino 17 3 / 15

5 Definable mad families under Projective Determinacy Theorem Assuming the Axiom of Projective Determinacy (PD), there are no infinite projective mad families. Our methods also give results under AD, but not all of them are new: Theorem (Neeman-Norwood) Assuming the Axiom of Determinacy holds in L(R), there are no infinite mad families in L(R). We concentrate on PD. Our proofs use: 1 Every projective set is Suslin 2 Reasonably definable forcing cannot change the projective theory Schrittesser Mad families, part 2 DST Torino 17 3 / 15

6 The analytic case Theorem There are no infinite Σ 1 1 mad families. Proof ideas Suppose A = p[t ] is an infinite a.d.-family. Let M A + be Mathias forcing relative to A. MA + ẋg is a.d. from every x (p[t ]) V [G]. The formula Σ 1 2 formula V [G] ( x)( y A) x y x y Fin is true in V [G] and so by Shoenfied absoluteness, also in V. This proof straightforwardly lifts to Σ 1 n using PD. Schrittesser Mad families, part 2 DST Torino 17 4 / 15

7 The analytic case Theorem There are no infinite Σ 1 1 mad families. Proof ideas Suppose A = p[t ] is an infinite a.d.-family. Let M A + be Mathias forcing relative to A. MA + ẋg is a.d. from every x (p[t ]) V [G]. The formula Σ 1 2 formula V [G] ( x)( y A) x y x y Fin is true in V [G] and so by Shoenfied absoluteness, also in V. This proof straightforwardly lifts to Σ 1 n using PD. Schrittesser Mad families, part 2 DST Torino 17 4 / 15

8 The analytic case Theorem There are no infinite Σ 1 1 mad families. Proof ideas Suppose A = p[t ] is an infinite a.d.-family. Let M A + be Mathias forcing relative to A. MA + ẋg is a.d. from every x (p[t ]) V [G]. The formula Σ 1 2 formula V [G] ( x)( y A) x y x y Fin is true in V [G] and so by Shoenfied absoluteness, also in V. This proof straightforwardly lifts to Σ 1 n using PD. Schrittesser Mad families, part 2 DST Torino 17 4 / 15

9 The analytic case Theorem There are no infinite Σ 1 1 mad families. Proof ideas Suppose A = p[t ] is an infinite a.d.-family. Let M A + be Mathias forcing relative to A. MA + ẋg is a.d. from every x (p[t ]) V [G]. The formula Σ 1 2 formula V [G] ( x)( y A) x y x y Fin is true in V [G] and so by Shoenfied absoluteness, also in V. This proof straightforwardly lifts to Σ 1 n using PD. Schrittesser Mad families, part 2 DST Torino 17 4 / 15

10 The analytic case Theorem There are no infinite Σ 1 1 mad families. Proof ideas Suppose A = p[t ] is an infinite a.d.-family. Let M A + be Mathias forcing relative to A. MA + ẋg is a.d. from every x (p[t ]) V [G]. The formula Σ 1 2 formula V [G] ( x)( y A) x y x y Fin is true in V [G] and so by Shoenfied absoluteness, also in V. This proof straightforwardly lifts to Σ 1 n using PD. Schrittesser Mad families, part 2 DST Torino 17 4 / 15

11 The analytic case Theorem There are no infinite Σ 1 1 mad families. Proof ideas Suppose A = p[t ] is an infinite a.d.-family. Let M A + be Mathias forcing relative to A. MA + ẋg is a.d. from every x (p[t ]) V [G]. The formula Σ 1 2 formula V [G] ( x)( y A) x y x y Fin is true in V [G] and so by Shoenfied absoluteness, also in V. This proof straightforwardly lifts to Σ 1 n using PD. Schrittesser Mad families, part 2 DST Torino 17 4 / 15

12 Lemma MA + ẋg is a.d. from every x (p[t ]) V [G]. Note that it is easier to see that MA + ẋg is a.d. from every x (p[t ]) V, i.e., from A. Proof ideas Let Z = {x p[t ] x x G / Fin}. Show Z 1. Supposing Z, let x be its unique element. x is definable from [x G ] E0. Show that x V, i.e., x A. Contradiction! Schrittesser Mad families, part 2 DST Torino 17 5 / 15

13 Lemma MA + ẋg is a.d. from every x (p[t ]) V [G]. Note that it is easier to see that MA + ẋg is a.d. from every x (p[t ]) V, i.e., from A. Proof ideas Let Z = {x p[t ] x x G / Fin}. Show Z 1. Supposing Z, let x be its unique element. x is definable from [x G ] E0. Show that x V, i.e., x A. Contradiction! Schrittesser Mad families, part 2 DST Torino 17 5 / 15

14 Lemma MA + ẋg is a.d. from every x (p[t ]) V [G]. Note that it is easier to see that MA + ẋg is a.d. from every x (p[t ]) V, i.e., from A. Proof ideas Let Z = {x p[t ] x x G / Fin}. Show Z 1. Supposing Z, let x be its unique element. x is definable from [x G ] E0. Show that x V, i.e., x A. Contradiction! Schrittesser Mad families, part 2 DST Torino 17 5 / 15

15 Lemma MA + ẋg is a.d. from every x (p[t ]) V [G]. Note that it is easier to see that MA + ẋg is a.d. from every x (p[t ]) V, i.e., from A. Proof ideas Let Z = {x p[t ] x x G / Fin}. Show Z 1. Supposing Z, let x be its unique element. x is definable from [x G ] E0. Show that x V, i.e., x A. Contradiction! Schrittesser Mad families, part 2 DST Torino 17 5 / 15

16 Lemma MA + ẋg is a.d. from every x (p[t ]) V [G]. Note that it is easier to see that MA + ẋg is a.d. from every x (p[t ]) V, i.e., from A. Proof ideas Let Z = {x p[t ] x x G / Fin}. Show Z 1. Supposing Z, let x be its unique element. x is definable from [x G ] E0. Show that x V, i.e., x A. Contradiction! Schrittesser Mad families, part 2 DST Torino 17 5 / 15

17 There are two non-trivial steps in the previous sketch: 1 {x p[t ] x x G / Fin} 1. This uses heavily some property of the ideal Fin. 2 If a real x in V [G] is definable from [x G ] E0, x is in V. This uses that that A + is σ-closed (uses that A is infinite!) thus, MA + is σ -closed in second component that MA + is homogeneous under finite changes Schrittesser Mad families, part 2 DST Torino 17 6 / 15

18 Other ideals Let J be an ideal on ω. Two sets A, A ω are called J -almost disjoint iff A A J. Let J + = P(ω) \ J A set A P(ω) is called a J -almost disjoint family iff A J + and any two distinct sets in A are J -almost disjoint. J -mad families are defined analogously. Schrittesser Mad families, part 2 DST Torino 17 7 / 15

19 Other Borel ideals Martin Goldstern asked: Question: Is there an analytic J -mad family where J is the harmonic ideal: X J n X 1/n < The answer in this case is no; and as before the proof lifts under PD (and lifts further under AD). Schrittesser Mad families, part 2 DST Torino 17 8 / 15

20 Fix a σ -closed Borel ideal J and a Suslin infinite J -a.d. family A P(ω), A = p[t ]. Denote (J, A) + by the co-ideal of the ideal generated by A J. Lemma (J, A) + is σ -closed. The obvious forcing M (J,A) + is homogeneous under changes in J and σ -closed in the second part We need more to show two more crucial properties: M(J,A) + ẋg / J V [G] M(J,A) + {x p[t ] ẋ G x J V [G] } has at most one element. Schrittesser Mad families, part 2 DST Torino 17 9 / 15

21 Fix a σ -closed Borel ideal J and a Suslin infinite J -a.d. family A P(ω), A = p[t ]. Denote (J, A) + by the co-ideal of the ideal generated by A J. Lemma (J, A) + is σ -closed. The obvious forcing M (J,A) + is homogeneous under changes in J and σ -closed in the second part We need more to show two more crucial properties: M(J,A) + ẋg / J V [G] M(J,A) + {x p[t ] ẋ G x J V [G] } has at most one element. Schrittesser Mad families, part 2 DST Torino 17 9 / 15

22 Fix a σ -closed Borel ideal J and a Suslin infinite J -a.d. family A P(ω), A = p[t ]. Denote (J, A) + by the co-ideal of the ideal generated by A J. Lemma (J, A) + is σ -closed. The obvious forcing M (J,A) + is homogeneous under changes in J and σ -closed in the second part We need more to show two more crucial properties: M(J,A) + ẋg / J V [G] M(J,A) + {x p[t ] ẋ G x J V [G] } has at most one element. Schrittesser Mad families, part 2 DST Torino 17 9 / 15

23 Fix a σ -closed Borel ideal J and a Suslin infinite J -a.d. family A P(ω), A = p[t ]. Denote (J, A) + by the co-ideal of the ideal generated by A J. Lemma (J, A) + is σ -closed. The obvious forcing M (J,A) + is homogeneous under changes in J and σ -closed in the second part We need more to show two more crucial properties: M(J,A) + ẋg / J V [G] M(J,A) + {x p[t ] ẋ G x J V [G] } has at most one element. Schrittesser Mad families, part 2 DST Torino 17 9 / 15

24 Submeasures and ideals A submeasure is a function Φ: P(ω) [0, ] such that X Y Φ(X) Φ(Y ) for any X, Y P(ω) Φ(X Y ) Φ(X) + Φ(Y ) for any X, Y P(ω) Φ({n}) < for any n ω Φ is lower semi-continuous. I.e., Φ(X) = lim Φ(X n) n Any submeasure gives rise to an ideal Fin(Φ) by letting Fin(Φ) = {X P(ω) Φ(X) < } Schrittesser Mad families, part 2 DST Torino / 15

25 Generalization to finite part of submeasure The previous arguments go through for ideals of the form Fin(Φ) where Φ is a submeasure. In particular this answers Goldstern s question. If J = Fin(Φ) + J is σ -closed. For any J -a.d. family A, M J,A + is homogeneous under changes on sets in J and σ -closed on the second part. MJ,A + φ(ẋ G ) =, i.e., ẋ G / J V [G] MJ,A + {x p[t ] ẋ G x J V [G] } has at most one element. Schrittesser Mad families, part 2 DST Torino / 15

26 What about ideals which are not of this form? Let J be any ideal. Define an ideal J ω on ω ω by: X J ω ( n ω) X(n) J That is, J ω the ideal generated by sets of the form {n} J n, n ω where (J n ) n ω is any sequence from J. Observation Let J be any non-trivial ideal. There is a countable J ω -mad family, namely {{ n} ω n ω}. In particular such ideals appear cofinally in the Borel hierarchy. Schrittesser Mad families, part 2 DST Torino / 15

27 What about ideals which are not of this form? Let J be any ideal. Define an ideal J ω on ω ω by: X J ω ( n ω) X(n) J That is, J ω the ideal generated by sets of the form {n} J n, n ω where (J n ) n ω is any sequence from J. Observation Let J be any non-trivial ideal. There is a countable J ω -mad family, namely {{ n} ω n ω}. In particular such ideals appear cofinally in the Borel hierarchy. Schrittesser Mad families, part 2 DST Torino / 15

28 What about ideals which are not of this form? Let J be any ideal. Define an ideal J ω on ω ω by: X J ω ( n ω) X(n) J That is, J ω the ideal generated by sets of the form {n} J n, n ω where (J n ) n ω is any sequence from J. Observation Let J be any non-trivial ideal. There is a countable J ω -mad family, namely {{ n} ω n ω}. In particular such ideals appear cofinally in the Borel hierarchy. Schrittesser Mad families, part 2 DST Torino / 15

29 Fubini product ideals Consider the ideal Fin Fin on ω ω, defined as follows: For X ω ω (letting X(n) = {m (n, m) X}), define X Fin Fin {n ω X(n) / Fin} Fin Using Fubini products, we can define Fin α, for every α < ω 1. Each Fin α is a definable ideal (in fact, Borel). The Fin α appear cofinally in the Borel hierarchy. Theorem For each α < ω 1, There is no analytic Fin α -mad family Under the Axiom of Projective Determinacy, there is no infinite projective Fin α -mad family Under the Axiom of Determinacy, there is no infinite Fin α -mad family in L(R). Schrittesser Mad families, part 2 DST Torino / 15

30 Question: Can we characterize the analytic, or at least the Borel ideals J P(ω) such that there is no analytic infinite J -mad family? Schrittesser Mad families, part 2 DST Torino / 15

31 Grazie mille! Schrittesser Mad families, part 2 DST Torino / 15