Mad families, definability, and ideals (Part 2) David Schrittesser - - PowerPoint PPT Presentation

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

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

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

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

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

The analytic case

Theorem

There are no infinite Σ1

1 mad families.

Proof ideas Suppose A = p[T] is an infinite a.d.-family. Let MA+ be Mathias forcing ‘relative to A’. MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. The formula Σ1

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 Σ1

n using PD.

Schrittesser Mad families, part 2 DST Torino ´ 17 4 / 15

The analytic case

Theorem

There are no infinite Σ1

1 mad families.

Proof ideas Suppose A = p[T] is an infinite a.d.-family. Let MA+ be Mathias forcing ‘relative to A’. MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. The formula Σ1

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 Σ1

n using PD.

Schrittesser Mad families, part 2 DST Torino ´ 17 4 / 15

The analytic case

Theorem

There are no infinite Σ1

1 mad families.

Proof ideas Suppose A = p[T] is an infinite a.d.-family. Let MA+ be Mathias forcing ‘relative to A’. MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. The formula Σ1

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 Σ1

n using PD.

Schrittesser Mad families, part 2 DST Torino ´ 17 4 / 15

The analytic case

Theorem

There are no infinite Σ1

1 mad families.

Proof ideas Suppose A = p[T] is an infinite a.d.-family. Let MA+ be Mathias forcing ‘relative to A’. MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. The formula Σ1

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 Σ1

n using PD.

Schrittesser Mad families, part 2 DST Torino ´ 17 4 / 15

The analytic case

Theorem

There are no infinite Σ1

1 mad families.

Proof ideas Suppose A = p[T] is an infinite a.d.-family. Let MA+ be Mathias forcing ‘relative to A’. MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. The formula Σ1

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 Σ1

n using PD.

Schrittesser Mad families, part 2 DST Torino ´ 17 4 / 15

The analytic case

Theorem

There are no infinite Σ1

1 mad families.

Proof ideas Suppose A = p[T] is an infinite a.d.-family. Let MA+ be Mathias forcing ‘relative to A’. MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. The formula Σ1

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 Σ1

n using PD.

Schrittesser Mad families, part 2 DST Torino ´ 17 4 / 15

Lemma

MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. Note that it is easier to see that MA+ ˙ xG is a.d. from every x ∈ (p[T])V, i.e., from A. Proof ideas Let Z = {x ∈ p[T] | x ∩ xG / ∈ Fin}. Show |Z| ≤ 1. Supposing Z = ∅, let x be its unique element. x is definable from [xG]E0. Show that x ∈ V, i.e., x ∈ A. Contradiction!

Schrittesser Mad families, part 2 DST Torino ´ 17 5 / 15

Lemma

MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. Note that it is easier to see that MA+ ˙ xG is a.d. from every x ∈ (p[T])V, i.e., from A. Proof ideas Let Z = {x ∈ p[T] | x ∩ xG / ∈ Fin}. Show |Z| ≤ 1. Supposing Z = ∅, let x be its unique element. x is definable from [xG]E0. Show that x ∈ V, i.e., x ∈ A. Contradiction!

Schrittesser Mad families, part 2 DST Torino ´ 17 5 / 15

Lemma

MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. Note that it is easier to see that MA+ ˙ xG is a.d. from every x ∈ (p[T])V, i.e., from A. Proof ideas Let Z = {x ∈ p[T] | x ∩ xG / ∈ Fin}. Show |Z| ≤ 1. Supposing Z = ∅, let x be its unique element. x is definable from [xG]E0. Show that x ∈ V, i.e., x ∈ A. Contradiction!

Schrittesser Mad families, part 2 DST Torino ´ 17 5 / 15

Lemma

MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. Note that it is easier to see that MA+ ˙ xG is a.d. from every x ∈ (p[T])V, i.e., from A. Proof ideas Let Z = {x ∈ p[T] | x ∩ xG / ∈ Fin}. Show |Z| ≤ 1. Supposing Z = ∅, let x be its unique element. x is definable from [xG]E0. Show that x ∈ V, i.e., x ∈ A. Contradiction!

Schrittesser Mad families, part 2 DST Torino ´ 17 5 / 15

Lemma

MA+ ˙ xG is a.d. from every x ∈ (p[T])V[G]. Note that it is easier to see that MA+ ˙ xG is a.d. from every x ∈ (p[T])V, i.e., from A. Proof ideas Let Z = {x ∈ p[T] | x ∩ xG / ∈ Fin}. Show |Z| ≤ 1. Supposing Z = ∅, let x be its unique element. x is definable from [xG]E0. Show that x ∈ V, i.e., x ∈ A. Contradiction!

Schrittesser Mad families, part 2 DST Torino ´ 17 5 / 15

There are two non-trivial steps in the previous sketch:

1

|{x ∈ p[T] | x ∩ xG / ∈ Fin}| ≤ 1. This uses heavily some property of the ideal Fin.

2

If a real x in V[G] is definable from [xG]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

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

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

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)+ ˙ xG / ∈ J V[G] M(J ,A)+ {x ∈ p[T] | ˙ xG ∩ x ∈ J V[G]} has at most one element.

Schrittesser Mad families, part 2 DST Torino ´ 17 9 / 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: M(J ,A)+ ˙ xG / ∈ J V[G] M(J ,A)+ {x ∈ p[T] | ˙ xG ∩ x ∈ J V[G]} has at most one element.

Schrittesser Mad families, part 2 DST Torino ´ 17 9 / 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: M(J ,A)+ ˙ xG / ∈ J V[G] M(J ,A)+ {x ∈ p[T] | ˙ xG ∩ x ∈ J V[G]} has at most one element.

Schrittesser Mad families, part 2 DST Torino ´ 17 9 / 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: M(J ,A)+ ˙ xG / ∈ J V[G] M(J ,A)+ {x ∈ p[T] | ˙ xG ∩ x ∈ J V[G]} has at most one element.

Schrittesser Mad families, part 2 DST Torino ´ 17 9 / 15

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

n Φ(X ∩ 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 ´ 17 10 / 15

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, MJ ,A+ is ‘homogeneous under changes

• n sets in J ’ and σ∗-closed on the second part.

MJ ,A+ φ( ˙ xG) = ∞, i.e., ˙ xG / ∈ J V[G] MJ ,A+ {x ∈ p[T] | ˙ xG ∩ x ∈ J V[G]} has at most one element.

Schrittesser Mad families, part 2 DST Torino ´ 17 11 / 15

What about ideals which are not of this form?

Let J be any ideal. Define an ideal Jω on ω × ω by: X ∈ Jω ⇐ ⇒ (∀n ∈ ω) X(n) ∈ J That is, Jω the ideal generated by sets of the form

• n∈ω

{n} × Jn, where (Jn)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 ´ 17 12 / 15

What about ideals which are not of this form?

Let J be any ideal. Define an ideal Jω on ω × ω by: X ∈ Jω ⇐ ⇒ (∀n ∈ ω) X(n) ∈ J That is, Jω the ideal generated by sets of the form

• n∈ω

{n} × Jn, where (Jn)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 ´ 17 12 / 15

What about ideals which are not of this form?

Let J be any ideal. Define an ideal Jω on ω × ω by: X ∈ Jω ⇐ ⇒ (∀n ∈ ω) X(n) ∈ J That is, Jω the ideal generated by sets of the form

• n∈ω

{n} × Jn, where (Jn)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 ´ 17 12 / 15

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 Finα, for every α < ω1. Each Finα is a definable ideal (in fact, Borel). The Finα appear cofinally in the Borel hierarchy.

Theorem

For each α < ω1, There is no analytic Finα-mad family Under the Axiom of Projective Determinacy, there is no infinite projective Finα-mad family Under the Axiom of Determinacy, there is no infinite Finα-mad family in L(R).

Schrittesser Mad families, part 2 DST Torino ´ 17 13 / 15

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 ´ 17 14 / 15

Grazie mille!

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