Mad families, definability, and ideals (Part 2) David Schrittesser (KGRC) joint with Asger Törnquist and Karen Haga Descriptive Set Theory in Torino August 17 DST Torino ´ Schrittesser Mad families, part 2 17 1 / 15
Overview Definable mad families under Projective Determinacy 1 J -mad families, for Borel ideals J other than Fin. 2 DST Torino ´ Schrittesser Mad families, part 2 17 2 / 15
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: Every projective set is Suslin 1 Reasonably definable forcing cannot change the projective theory 2 DST Torino ´ Schrittesser Mad families, part 2 17 3 / 15
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: Every projective set is Suslin 1 Reasonably definable forcing cannot change the projective theory 2 DST Torino ´ Schrittesser Mad families, part 2 17 3 / 15
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: Every projective set is Suslin 1 Reasonably definable forcing cannot change the projective theory 2 DST Torino ´ Schrittesser Mad families, part 2 17 3 / 15
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 ’. x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ 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. DST Torino ´ Schrittesser Mad families, part 2 17 4 / 15
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 ’. x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ 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. DST Torino ´ Schrittesser Mad families, part 2 17 4 / 15
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 ’. x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ 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. DST Torino ´ Schrittesser Mad families, part 2 17 4 / 15
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 ’. x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ 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. DST Torino ´ Schrittesser Mad families, part 2 17 4 / 15
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 ’. x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ 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. DST Torino ´ Schrittesser Mad families, part 2 17 4 / 15
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 ’. x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ 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. DST Torino ´ Schrittesser Mad families, part 2 17 4 / 15
Lemma x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ Note that it is easier to see that � M A + ˙ x G 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 ] E 0 . Show that x ∈ V , i.e., x ∈ A . Contradiction! DST Torino ´ Schrittesser Mad families, part 2 17 5 / 15
Lemma x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ Note that it is easier to see that � M A + ˙ x G 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 ] E 0 . Show that x ∈ V , i.e., x ∈ A . Contradiction! DST Torino ´ Schrittesser Mad families, part 2 17 5 / 15
Lemma x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ Note that it is easier to see that � M A + ˙ x G 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 ] E 0 . Show that x ∈ V , i.e., x ∈ A . Contradiction! DST Torino ´ Schrittesser Mad families, part 2 17 5 / 15
Lemma x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ Note that it is easier to see that � M A + ˙ x G 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 ] E 0 . Show that x ∈ V , i.e., x ∈ A . Contradiction! DST Torino ´ Schrittesser Mad families, part 2 17 5 / 15
Lemma x G is a.d. from every x ∈ ( p [ T ]) V [ G ] . � M A + ˙ Note that it is easier to see that � M A + ˙ x G 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 ] E 0 . Show that x ∈ V , i.e., x ∈ A . Contradiction! DST Torino ´ Schrittesser Mad families, part 2 17 5 / 15
There are two non-trivial steps in the previous sketch: |{ x ∈ p [ T ] | x ∩ x G / ∈ Fin }| ≤ 1. 1 This uses heavily some property of the ideal Fin. If a real x in V [ G ] is definable from [ x G ] E 0 , x is in V . 2 This uses that ◮ that A + is σ -closed (uses that A is infinite!) ◮ thus, M A + is σ ∗ -closed in second component ◮ that M A + is homogeneous ‘under finite changes’ DST Torino ´ Schrittesser Mad families, part 2 17 6 / 15
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. DST Torino ´ Schrittesser Mad families, part 2 17 7 / 15
Other Borel ideals Martin Goldstern asked: Question: Is there an analytic J -mad family where J is the harmonic ideal: � X ∈ J ⇐ ⇒ 1 / n < ∞ n ∈ X The answer in this case is no; and as before the proof lifts under PD (and lifts further under AD). DST Torino ´ Schrittesser Mad families, part 2 17 8 / 15
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: ∈ J V [ G ] � M ( J , A )+ ˙ x G / � M ( J , A )+ { x ∈ p [ T ] | ˙ x G ∩ x ∈ J V [ G ] } has at most one element. DST Torino ´ Schrittesser Mad families, part 2 17 9 / 15
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