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Descriptive Set Theory and the Wadge order

Reducibility notions for the Scott domain PhDs in Logic XI, April 25th, 2019, Bern, Switzerland Louis Vuilleumier

University of Lausanne and University Paris Diderot

April 25th, 2019, Bern

Descriptive Set Theory

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Measure Problem Question (Lebesgue 1902)

Does there exist a function m which associates to each bounded set of real numbers X a non-negative real number m(X) such that the following conditions hold? i) m is not always 0; ii) m is invariant under translation; iii) m is σ-additive.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Measure Problem Question (Lebesgue 1902)

Does there exist a function m which associates to each bounded set of real numbers X a non-negative real number m(X) such that the following conditions hold? i) m is not always 0; ii) m is invariant under translation; iii) m is σ-additive.

Answer (Vitali 1905)

No! Under ZFC, there exists a non-measurable set.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive set theory is the study of “definable sets” in Polish (i.e. separable completely metrizable) spaces. In this theory, sets are classified in hierarchies, accord- ing to the complexity of their definitions [...].

• A. Kechris

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive set theory is the study of “definable sets” in Polish (i.e. separable completely metrizable) spaces. In this theory, sets are classified in hierarchies, accord- ing to the complexity of their definitions [...].

• A. Kechris

Examples

R, C, IN and separable Banach spaces are Polish.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive set theory is the study of “definable sets” in Polish (i.e. separable completely metrizable) spaces. In this theory, sets are classified in hierarchies, accord- ing to the complexity of their definitions [...].

• A. Kechris

Examples

R, C, IN and separable Banach spaces are Polish. The Baire space ωω and the Cantor space 2ω are also Polish.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Cantor Space

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive Set Theory The Borel hierarchy for metrizable spaces

Let X be a metrizable space, we set for 2 ≤ α < ω1, Σ0

1(X) =

• U ⊆ X : U is open
• ,

Σ0

α(X) = n∈ω

A∁

n : An ∈ Σ0 βn(X), βn < α

• ,

Π0

α(X) =

• A : A∁ ∈ Σ0

α(X)

• ,

∆0

α(X) = Σ0 α(X) ∩ Π0 α(X).

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Π0

2

∆0

2

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Π0

2

∆0

2

. . .

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Π0

2

∆0

2

. . .

Σ0

α

α < ω1

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Π0

2

∆0

2

. . .

Σ0

α

α < ω1

Π0

α

∆0

α

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Π0

2

∆0

2

. . .

Σ0

α

α < ω1

Π0

α

∆0

α

. . .

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy

Σ0

1

Π0

1

∆0

1

Σ0

2

Π0

2

∆0

2

. . .

Σ0

α

α < ω1

Π0

α

∆0

α

. . . B

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition

Let X be a topological space, and A, B ⊆ X. The set A is Wadge reducible to B, denoted by A ≤w B, if there exists a continuous function f : X → X such that f −1(B) = A.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition

Let X be a topological space, and A, B ⊆ X. The set A is Wadge reducible to B, denoted by A ≤w B, if there exists a continuous function f : X → X such that f −1(B) = A.

Remarks

◮ Wadge reducibility is topologically natural.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition

Let X be a topological space, and A, B ⊆ X. The set A is Wadge reducible to B, denoted by A ≤w B, if there exists a continuous function f : X → X such that f −1(B) = A.

Remarks

◮ Wadge reducibility is topologically natural. ◮ It induces a quasi-order on P(X).

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition

Let X be a topological space, and A, B ⊆ X. The set A is Wadge reducible to B, denoted by A ≤w B, if there exists a continuous function f : X → X such that f −1(B) = A.

Remarks

◮ Wadge reducibility is topologically natural. ◮ It induces a quasi-order on P(X). ◮ It induces a partial order on the Wadge degrees

called the Wadge order (P(X)/≡w, ≤w).

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks

◮ The Wadge order refines the Borel hierarchy.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks

◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com-

plexity.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks

◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com-

plexity.

◮ The Wadge order is topologically meaningful:

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks

◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com-

plexity.

◮ The Wadge order is topologically meaningful:

The Wadge hierarchy is the ultimate analysis of P(ωω) in terms of topological complexity [...]. Andretta and Louveau.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks

◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com-

plexity.

◮ The Wadge order is topologically meaningful:

The Wadge hierarchy is the ultimate analysis of P(ωω) in terms of topological complexity [...]. Andretta and Louveau.

Problem

Understanding the shape of the Wadge order on differ- ent topological spaces.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · ·

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn xn+1

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn xn+1 s

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn xn+1 s · · · · · ·

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn xn+1 s · · · · · · x = (xn)n∈ω y = (yn)n∈ω

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn xn+1 s · · · · · · x = (xn)n∈ω y = (yn)n∈ω The Wadge game Gw(A, B) where xn, yn ∈ S.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news

If X = ωω or X = 2ω, we can use games!

Definition

Let S ∈ {2, ω} and A, B ⊆ Sω. I II x0 y0 x1 s x2 y1 · · · · · · xn yn xn+1 s · · · · · · x = (xn)n∈ω y = (yn)n∈ω The Wadge game Gw(A, B) where xn, yn ∈ S. We say that II wins the game iff x ∈ A ↔ y ∈ B.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Strategies Definition

A strategy for II in the Wadge game Gw(A, B) is a func- tion: σ : S<w → S ∪ {s} (xn)n≤k → σ

• (xn)n≤k
• .

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Strategies Definition

A strategy for II in the Wadge game Gw(A, B) is a func- tion: σ : S<w → S ∪ {s} (xn)n≤k → σ

• (xn)n≤k
• .

A strategy σ is winning if it makes II win whatever I plays.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Strategies Definition

A strategy for II in the Wadge game Gw(A, B) is a func- tion: σ : S<w → S ∪ {s} (xn)n≤k → σ

• (xn)n≤k
• .

A strategy σ is winning if it makes II win whatever I plays.

Theorem (Wadge)

If X = ωω or X = 2ω, then II has a w.s. ⇔ A ≤w B.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Theorem (Borel determinacy, Martin)

If A, B are Borel subsets of Sω, then Gw(A, B) is deter- mined, i.e., one player has a winning strategy.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Theorem (Borel determinacy, Martin)

If A, B are Borel subsets of Sω, then Gw(A, B) is deter- mined, i.e., one player has a winning strategy.

Theorem (Wadge’s Lemma, Wadge)

If A, B are Borel sets and A w B, then B ≤w Sω \ A. In particular, A w B and B w A implies A ≡w Sω \ B.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Theorem (Borel determinacy, Martin)

If A, B are Borel subsets of Sω, then Gw(A, B) is deter- mined, i.e., one player has a winning strategy.

Theorem (Wadge’s Lemma, Wadge)

If A, B are Borel sets and A w B, then B ≤w Sω \ A. In particular, A w B and B w A implies A ≡w Sω \ B.

Theorem (Martin)

The quasi-order ≤w is well-founded on the Borel sub- sets of Sω.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order on the Cantor space and the Baire space

[∅]W [2ω]W · · · · · · · · ·

limit levels

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order on the Cantor space and the Baire space

[∅]W [2ω]W · · · · · · · · ·

limit levels

[∅]W [ωω]W · · · cof (λ) = ω · · · cof (λ) > ω · · ·

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order on Polish spaces Definition

Let X be a Polish space. Then X is zero-dimensional if it admits a basis consisting of clopen sets.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order on Polish spaces Definition

Let X be a Polish space. Then X is zero-dimensional if it admits a basis consisting of clopen sets.

Theorem (Schlicht)

Let X be a Polish space. Then (B(X)/≡w, ≤w) is a well- quasi-order if and only if X is zero-dimensional.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain Recall

The Cantor space 2ω is the set Πn∈N2 equipped with the product of the discrete topology on 2 = {0, 1}.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain Recall

The Cantor space 2ω is the set Πn∈N2 equipped with the product of the discrete topology on 2 = {0, 1}.

Definition

The Scott domain Pω is the set Πn∈N2 equipped with the product of the Sierpi ´ nski topology TS on 2 = {0, 1}, i.e., TS =

• ∅, {1}, {0, 1}
• .

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain Recall

The Cantor space 2ω is the set Πn∈N2 equipped with the product of the discrete topology on 2 = {0, 1}.

Definition

The Scott domain Pω is the set Πn∈N2 equipped with the product of the Sierpi ´ nski topology TS on 2 = {0, 1}, i.e., TS =

• ∅, {1}, {0, 1}
• .

Remark

The topology of the Cantor space 2ω is the topology of positive and negative information, whereas the topol-

• gy of the Scott domain Pω is the topology of positive

information only.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain and quasi-Polish spaces Definition

A quasi-Polish space is second countable completely quasi-metrizable space, where a quasi-metric is a met- ric whose symmetry condition has been dropped.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain and quasi-Polish spaces Definition

A quasi-Polish space is second countable completely quasi-metrizable space, where a quasi-metric is a met- ric whose symmetry condition has been dropped. The majority of this paper will be dedicated to showing the naturalness of extending the descriptive set theory

• f Polish spaces to the class of quasi-Polish spaces.
• M. de Brecht

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain and quasi-Polish spaces Definition

A quasi-Polish space is second countable completely quasi-metrizable space, where a quasi-metric is a met- ric whose symmetry condition has been dropped. The majority of this paper will be dedicated to showing the naturalness of extending the descriptive set theory

• f Polish spaces to the class of quasi-Polish spaces.
• M. de Brecht

Remark

The Scott domain Pω was the first denotational seman- tic of the λ-calculus. The Scott domain Pω is universal among quasi-Polish spaces.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain Problem

Understand the shape of the Wadge order on the Scott domain Pω.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Scott domain Problem

Understand the shape of the Wadge order on the Scott domain Pω.

Theorem (V.)

The Wadge order on the ∆0

2-subsets of Pω has an infi-

nite strictly decreasing sequence and an infinite set of pairwise incomparable elements.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Other reducibility notions Definition

If X is a topological space, then F ⊆ {f : X → X} is a reducibility notion if it contains the identity and is closed under composition.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Other reducibility notions Definition

If X is a topological space, then F ⊆ {f : X → X} is a reducibility notion if it contains the identity and is closed under composition. In this case, if A, B ⊆ X, A is F-reducible to B, written A ≤F B, whenever there exists a function f ∈ F such that f −1(B) = A.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Other reducibilities

F2 =

• f : P(N) → P(N) : f −1(U) ∈ Σ0

2 for all U ∈ Σ0 1

• .

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Other reducibilities

F2 =

• f : P(N) → P(N) : f −1(U) ∈ Σ0

2 for all U ∈ Σ0 1

• .

Theorem (V.)

There exists a reducibility notion F ⊆ F2 such that the quasi-order ≤F on the Borel subsets of the Scott do- main is well-founded and admits maximal antichains

• f size 2.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Other reducibilities

F2 =

• f : P(N) → P(N) : f −1(U) ∈ Σ0

2 for all U ∈ Σ0 1

• .

Theorem (V.)

There exists a reducibility notion F ⊆ F2 such that the quasi-order ≤F on the Borel subsets of the Scott do- main is well-founded and admits maximal antichains

• f size 2.

In particular, if F′ is a reducibility notion such that F2 ⊆ F′, then the quasi-order ≤F′ on the Borel subsets

• f the Scott domain is well-founded and admits maxi-

mal antichains of size 2.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Further Directions

• 1. Give a complete description of (∆0

2(Pω)/≡w, ≤w); Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Further Directions

• 1. Give a complete description of (∆0

2(Pω)/≡w, ≤w);

• 2. Investigate other reducibilities;

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Further Directions

• 1. Give a complete description of (∆0

2(Pω)/≡w, ≤w);

• 2. Investigate other reducibilities;
• 3. Generalize to other quasi-Polish spaces.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Further Directions

• 1. Give a complete description of (∆0

2(Pω)/≡w, ≤w);

• 2. Investigate other reducibilities;
• 3. Generalize to other quasi-Polish spaces.

Thank you.

Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern