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

Maximal objects in the projective hierarchy

J¨

• rg Brendle

Kobe University

Sant Bernat, November 18, 2018 Reflections on Set Theoretic Reflection In celebration of Joan Bagaria’s 60th birthday

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• rg Brendle

Maximal objects in the projective hierarchy

Objects with maximality properties

Sets of reals with maximality properties like

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Objects with maximality properties

Sets of reals with maximality properties like ultrafilters on ω

J¨

• rg 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¨

• rg 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¨

• rg 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

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• rg 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¨

• rg 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.

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• rg 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

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• rg 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

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• rg 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 Σ1

n mad is also ∆1 n.

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• rg 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 Σ1

n mad is also ∆1 n.

Theorem 1 (T¨

• rnquist ’12)

If there is a Σ1

2 mad then there is a Π1 1 mad.

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• rg 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 Σ1

n mad is also ∆1 n.

Theorem 1 (T¨

• rnquist ’12)

If there is a Σ1

2 mad then there is a Π1 1 mad.

Theorem 2 (Mathias ’70’s) There are no Σ1

1 mad families.

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• rg 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 Σ1

n mad is also ∆1 n.

Theorem 1 (T¨

• rnquist ’12)

If there is a Σ1

2 mad then there is a Π1 1 mad.

Theorem 2 (Mathias ’70’s) There are no Σ1

1 mad families.

Theorem 3 (Miller ∼’90) There are Π1

1 mads in L.

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• rg 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 Σ1

2 and thus Π1 1 mad.

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• rg 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 Σ1

2 and thus Π1 1 mad.

In particular, the existence of Π1

1 mads is consistent with c > ω1.

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• rg 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 Σ1

2 and thus Π1 1 mad.

In particular, the existence of Π1

1 mads is consistent with c > ω1.

Kunen: Under CH, there is a mad family which survives arbitrary Cohen extensions.

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• rg 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 Σ1

2 and thus Π1 1 mad.

In particular, the existence of Π1

1 mads is consistent with c > ω1.

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).

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• rg 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 Σ1

2 and thus Π1 1 mad.

In particular, the existence of Π1

1 mads is consistent with c > ω1.

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 Π1

1 mads.

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• rg 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 Σ1

2 and thus Π1 1 mad.

In particular, the existence of Π1

1 mads is consistent with c > ω1.

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 Π1

1 mads.

Adding a dominating real destroys all ground model mads.

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• rg 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 Σ1

2 and thus Π1 1 mad.

In particular, the existence of Π1

1 mads is consistent with c > ω1.

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 Π1

1 mads.

Adding a dominating real destroys all ground model mads. Can we have b > ω1 together with Π1

1 mads?

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∃∞n (g(n) < f (n))} the unbounding number

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃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

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃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 aclosed := min{|F| : F infinite family of closed sets, F mad}

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃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 aclosed := min{|F| : F infinite family of closed sets, F mad} aBorel := min{|F| : F infinite family of Borel sets, F mad}

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃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 aclosed := min{|F| : F infinite family of closed sets, F mad} aBorel := min{|F| : F infinite family of Borel sets, F mad}

• Fact. ω1 ≤ b ≤ a ≤ c and ω1 ≤ aBorel ≤ aclosed ≤ a

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃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 aclosed := min{|F| : F infinite family of closed sets, F mad} aBorel := min{|F| : F infinite family of Borel sets, F mad}

• Fact. ω1 ≤ b ≤ a ≤ c and ω1 ≤ aBorel ≤ aclosed ≤ a
• Fact. If aBorel > ω1, then there are no Σ1

2 mads.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mad families: some cardinals

b := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃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 aclosed := min{|F| : F infinite family of closed sets, F mad} aBorel := min{|F| : F infinite family of Borel sets, F mad}

• Fact. ω1 ≤ b ≤ a ≤ c and ω1 ≤ aBorel ≤ aclosed ≤ a
• Fact. If aBorel > ω1, then there are no Σ1

2 mads.

Can we have aBorel < b?

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

• Fact. b ≤ d ≤ c and ω1 ≤ s ≤ d

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

• Fact. b ≤ d ≤ c and ω1 ≤ s ≤ d

Say ¯ A = (Aα,n : α < ω1, n < ω) is a tail-splitting sequence of partitions (tssp) if the Aα,n, n ∈ ω, are pairwise disjoint and for all B ∈ [ω]ω there is α < ω1 such that all Aβ,n, β ≥ α, n ∈ ω, split B.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

• Fact. b ≤ d ≤ c and ω1 ≤ s ≤ d

Say ¯ A = (Aα,n : α < ω1, n < ω) is a tail-splitting sequence of partitions (tssp) if the Aα,n, n ∈ ω, are pairwise disjoint and for all B ∈ [ω]ω there is α < ω1 such that all Aβ,n, β ≥ α, n ∈ ω, split B.

• Facts. (1) the existence of a tssp implies s = ω1.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

• Fact. b ≤ d ≤ c and ω1 ≤ s ≤ d

Say ¯ A = (Aα,n : α < ω1, n < ω) is a tail-splitting sequence of partitions (tssp) if the Aα,n, n ∈ ω, are pairwise disjoint and for all B ∈ [ω]ω there is α < ω1 such that all Aβ,n, β ≥ α, n ∈ ω, split B.

• Facts. (1) the existence of a tssp implies s = ω1.

(2) d = ω1 implies the existence of a tssp.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

• Fact. b ≤ d ≤ c and ω1 ≤ s ≤ d

Say ¯ A = (Aα,n : α < ω1, n < ω) is a tail-splitting sequence of partitions (tssp) if the Aα,n, n ∈ ω, are pairwise disjoint and for all B ∈ [ω]ω there is α < ω1 such that all Aβ,n, β ≥ α, n ∈ ω, split B.

• Facts. (1) the existence of a tssp implies s = ω1.

(2) d = ω1 implies the existence of a tssp. (3) There is a tssp in the Hechler model.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mad families: a combinatorial principle

d := min{|F| : F ⊆ ωω and ∀g ∈ ωω ∃f ∈ F ∀∞n (g(n) < f (n))} the dominating number s := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (|X ∩ Y | = |Y \ X| = ω)} the splitting number

• Fact. b ≤ d ≤ c and ω1 ≤ s ≤ d

Say ¯ A = (Aα,n : α < ω1, n < ω) is a tail-splitting sequence of partitions (tssp) if the Aα,n, n ∈ ω, are pairwise disjoint and for all B ∈ [ω]ω there is α < ω1 such that all Aβ,n, β ≥ α, n ∈ ω, split B.

• Facts. (1) the existence of a tssp implies s = ω1.

(2) d = ω1 implies the existence of a tssp. (3) There is a tssp in the Hechler model. Theorem 5 The existence of a tssp implies aclosed = ω1.

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Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1. Corollary 6 (B. and Khomskii ’11) In the Hechler model aclosed = ω1.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1. Corollary 6 (B. and Khomskii ’11) In the Hechler model aclosed = ω1. In particular aclosed < b is consistent.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1. Corollary 6 (B. and Khomskii ’11) In the Hechler model aclosed = ω1. In particular aclosed < b is consistent. In the Hechler model over V = L there is a Π1

1 mad.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1. Corollary 6 (B. and Khomskii ’11) In the Hechler model aclosed = ω1. In particular aclosed < b is consistent. In the Hechler model over V = L there is a Π1

1 mad.

In particular, the existence of Π1

1 mads is consistent with b > ω1.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1. Corollary 6 (B. and Khomskii ’11) In the Hechler model aclosed = ω1. In particular aclosed < b is consistent. In the Hechler model over V = L there is a Π1

1 mad.

In particular, the existence of Π1

1 mads is consistent with b > ω1.

Corollary 7 (Raghavan and Shelah) d = ω1 implies aclosed = ω1.

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• rg Brendle

Maximal objects in the projective hierarchy

Mad families: what if b > ω1?

Theorem 5 The existence of a tssp implies aclosed = ω1. Corollary 6 (B. and Khomskii ’11) In the Hechler model aclosed = ω1. In particular aclosed < b is consistent. In the Hechler model over V = L there is a Π1

1 mad.

In particular, the existence of Π1

1 mads is consistent with b > ω1.

Corollary 7 (Raghavan and Shelah) d = ω1 implies aclosed = ω1. Question 1 Does s = ω1 imply aclosed = ω1?

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Maximal objects in the projective hierarchy

Mad families: non-existence

Theorem 8 (Horowitz and Shelah ’16) It is consistent on the basis of ZFC that there are no (projective) mads.

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Maximal objects in the projective hierarchy

Mad families: non-existence

Theorem 8 (Horowitz and Shelah ’16) It is consistent on the basis of ZFC that there are no (projective) mads.

• Note. This was originally proved by Mathias under the assumption
• f the consistency of a Mahlo cardinal, and by T¨
• rnquist, under

the assumption of the consistency of an inaccessible cardinal.

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• rg Brendle

Maximal objects in the projective hierarchy

Maximal independent families: definitions

A ⊆ [ω]ω is an independent (ind.) family if whenever F, G ⊆ A are finite and disjoint then σ(F, G) := F ∩

A∈G(ω \ A) is infinite

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• rg Brendle

Maximal objects in the projective hierarchy

Maximal independent families: definitions

A ⊆ [ω]ω is an independent (ind.) family if whenever F, G ⊆ A are finite and disjoint then σ(F, G) := F ∩

A∈G(ω \ A) is infinite

A ⊆ [ω]ω is a mif if A is ind. and maximal with this property, i.e., for all X ∈ [ω]ω there are F, G ⊆ A finite and disjoint such that either σ(F, G) ⊆∗ X or σ(F, G) ∩ X =∗ ∅

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• rg Brendle

Maximal objects in the projective hierarchy

Maximal independent families: definitions

A ⊆ [ω]ω is an independent (ind.) family if whenever F, G ⊆ A are finite and disjoint then σ(F, G) := F ∩

A∈G(ω \ A) is infinite

A ⊆ [ω]ω is a mif if A is ind. and maximal with this property, i.e., for all X ∈ [ω]ω there are F, G ⊆ A finite and disjoint such that either σ(F, G) ⊆∗ X or σ(F, G) ∩ X =∗ ∅

• Fact. A Σ1

n mif is also ∆1 n.

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Maximal objects in the projective hierarchy

Mifs: basic results

Theorem 9 (B. and Khomskii ’17) If there is a Σ1

2 mif then there is a Π1 1 mif.

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: basic results

Theorem 9 (B. and Khomskii ’17) If there is a Σ1

2 mif then there is a Π1 1 mif.

Theorem 10 (Miller ∼’90) There are no Σ1

1 mifs.

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: basic results

Theorem 9 (B. and Khomskii ’17) If there is a Σ1

2 mif then there is a Π1 1 mif.

Theorem 10 (Miller ∼’90) There are no Σ1

1 mifs.

Theorem 11 (Miller ∼’90) There are Π1

1 mifs in L.

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: an implication

Theorem 12 (B. and Khomskii ’15) If all Σ1

n sets have the property of Baire, then there is no Σ1 n mif.

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Maximal objects in the projective hierarchy

Mifs: an implication

Theorem 12 (B. and Khomskii ’15) If all Σ1

n sets have the property of Baire, then there is no Σ1 n mif.

Proof idea. Analyze proof of Miller’s result.

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Maximal objects in the projective hierarchy

Mifs: an implication

Theorem 12 (B. and Khomskii ’15) If all Σ1

n sets have the property of Baire, then there is no Σ1 n mif.

Proof idea. Analyze proof of Miller’s result.

• Question. Does Σ1

n measurability imply no Σ1 n mifs?

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Maximal objects in the projective hierarchy

Mifs: an implication

Theorem 12 (B. and Khomskii ’15) If all Σ1

n sets have the property of Baire, then there is no Σ1 n mif.

Proof idea. Analyze proof of Miller’s result.

• Question. Does Σ1

n measurability imply no Σ1 n mifs?

Question 2 Does the Σ1

n Ramsey property imply no Σ1 n mads?

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Maximal objects in the projective hierarchy

Mifs: non-existence

Theorem 13 (B. and Khomskii ’15) In the Cohen model W there are no projective mifs. In L(R)W there are no mifs.

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Maximal objects in the projective hierarchy

Mifs: non-existence

Theorem 13 (B. and Khomskii ’15) In the Cohen model W there are no projective mifs. In L(R)W there are no mifs. This is based on the same combinatorics as the previous result + homogeneity of Cohen forcing.

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: what if CH fails?

Theorem 14 (Fischer ’18) In the Sacks model (over V = L) there is a Σ1

2 and thus Π1 1 mif.

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: what if CH fails?

Theorem 14 (Fischer ’18) In the Sacks model (over V = L) there is a Σ1

2 and thus Π1 1 mif.

In particular, the existence of Π1

1 mifs is consistent with c > ω1.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: what if CH fails?

Theorem 14 (Fischer ’18) In the Sacks model (over V = L) there is a Σ1

2 and thus Π1 1 mif.

In particular, the existence of Π1

1 mifs is consistent with c > ω1.

Under CH, there is a mif which survives arbitrary Sacks extensions.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: what if CH fails?

Theorem 14 (Fischer ’18) In the Sacks model (over V = L) there is a Σ1

2 and thus Π1 1 mif.

In particular, the existence of Π1

1 mifs is consistent with c > ω1.

Under CH, there is a mif which survives arbitrary Sacks extensions. Adding an unbounded real ... destroys all ground model mifs.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: what if CH fails?

Theorem 14 (Fischer ’18) In the Sacks model (over V = L) there is a Σ1

2 and thus Π1 1 mif.

In particular, the existence of Π1

1 mifs is consistent with c > ω1.

Under CH, there is a mif which survives arbitrary Sacks extensions. Adding an unbounded real ... destroys all ground model mifs. Can we have d > ω1 together with Π1

1 mifs?

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: what if CH fails?

Theorem 14 (Fischer ’18) In the Sacks model (over V = L) there is a Σ1

2 and thus Π1 1 mif.

In particular, the existence of Π1

1 mifs is consistent with c > ω1.

Under CH, there is a mif which survives arbitrary Sacks extensions. Adding an unbounded real ... destroys all ground model mifs. Can we have d > ω1 together with Π1

1 mifs?

In fact, many forcings destroy all ground model mifs.

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif }

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif }

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif } r := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (X ⊆∗ Y or |X ∩ Y | < ω)} the reaping number

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• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif } r := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (X ⊆∗ Y or |X ∩ Y | < ω)} the reaping number non(meager) is the least size of a non-meager set of reals

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif } r := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (X ⊆∗ Y or |X ∩ Y | < ω)} the reaping number non(meager) is the least size of a non-meager set of reals cov(meager) is the least size of a family of meager sets covering the reals

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif } r := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (X ⊆∗ Y or |X ∩ Y | < ω)} the reaping number non(meager) is the least size of a non-meager set of reals cov(meager) is the least size of a family of meager sets covering the reals

• Fact. cov(meager) ≤ d, r ≤ i ≤ c and non(meager) ≤ i.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif } r := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (X ⊆∗ Y or |X ∩ Y | < ω)} the reaping number non(meager) is the least size of a non-meager set of reals cov(meager) is the least size of a family of meager sets covering the reals

• Fact. cov(meager) ≤ d, r ≤ i ≤ c and non(meager) ≤ i.

Theorem 15 (B. and Khomskii ’15) cov(meager) ≤ iBorel.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some cardinals

i := min{|A| : A ⊆ [ω]ω is a mif} the independence number iclosed := min{|F| : F collection of closed ind. families, F mif } iBorel := min{|F| : F collection of Borel ind. families, F mif } r := min{|F| : F ⊆ [ω]ω and ∀Y ∈ [ω]ω ∃X ∈ F (X ⊆∗ Y or |X ∩ Y | < ω)} the reaping number non(meager) is the least size of a non-meager set of reals cov(meager) is the least size of a family of meager sets covering the reals

• Fact. cov(meager) ≤ d, r ≤ i ≤ c and non(meager) ≤ i.

Theorem 15 (B. and Khomskii ’15) cov(meager) ≤ iBorel. In particular, cov(meager) > ω1 implies there are no Σ1

2 mifs.

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some questions

Question 3

1 Is d ≤ iBorel? Or is iclosed < d consistent? J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some questions

Question 3

1 Is d ≤ iBorel? Or is iclosed < d consistent?

Is d > ω1 consistent with the existence of a Π1

1 mif?

J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some questions

Question 3

1 Is d ≤ iBorel? Or is iclosed < d consistent?

Is d > ω1 consistent with the existence of a Π1

1 mif?

2 Is r ≤ iBorel? J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some questions

Question 3

1 Is d ≤ iBorel? Or is iclosed < d consistent?

Is d > ω1 consistent with the existence of a Π1

1 mif?

2 Is r ≤ iBorel? 3 Is non(meager) ≤ iBorel? J¨

• rg Brendle

Maximal objects in the projective hierarchy

Mifs: some questions

Question 3

1 Is d ≤ iBorel? Or is iclosed < d consistent?

Is d > ω1 consistent with the existence of a Π1

1 mif?

2 Is r ≤ iBorel? 3 Is non(meager) ≤ iBorel? 4 Is iBorel = i? J¨

• rg Brendle

Maximal objects in the projective hierarchy

Happy birthday, Joan!

J¨

• rg Brendle

Maximal objects in the projective hierarchy