On the Reverse Mathematics of Lipschitz and Wadge Determinacy

On the reverse mathematics of Lipschitz and Wadge determinacy

Manuel Loureiro Engineering Faculty-ULHT, Lisbon

joint work with A. Cord´on-Franco (Seville) and F.F. Lara-Mart´ın(Seville)

Journ´eessur les Arithm´etiquesFaibles 35 Universidade de Lisboa, 6-7 June 2016 Goal and Plan

I Goal: to calibrate the strength of Lipschitz and Wadge determinacy, as well as the Semi-Linear Ordering principle (SLO), in terms of subsystems of second order arithmetic (Z2)

I Plan

I Introduction

I L/W determinacy. Topological results

I Formalization of L/W determinacy and SLO in Z2

I Results in Z2

I L/W determinacy and SLO in Cantor space

I L/W determinacy and SLO in Baire space

I Concluding remarks Introduction. Gale-Stewart and Lipschitz Games

Player I f (0) f (1) ... Player II g(0) g(1) ...

ω I (Gale-Stewart games) A ⊆ X . I wins a play of G(A) if hf (0), g(0), f (1), g(1),... i ∈ A Otherwise II wins.

ω I (Lipschitz games) A, B ⊆ X . II wins a play of GL(A, B) if hf (0), f (1),... i ∈ A iﬀ hg(0), g(1),... i ∈ B Otherwise I wins. Introduction. Wadge Games

I Variant of Lipschitz games where II is allowed to pass, but she must play inﬁnitely often otherwise she loses

Player I f (0) ... f (k) f (k+1) f (k+2) ... Player II p ··· p g(0) p g(1) ...

ω I (Wadge games) A, B ⊆ X . II wins a play of GW (A, B) if hf (0), f (1),... i ∈ A iﬀ hg(0), g(1),... i ∈ B Otherwise I wins.

I Reduction to Gale-Stewart games:

I I wins GL(A, B) iﬀ I wins G(¬(A ↔ B)) I Similar for GW (A, B) Introduction. Determinacy axioms in set theory

I (Mycielski/Steinhaus, 1962)

AD = ∀A ⊆ ωω, the Gale-Stewart game G(A) is determined, i.e. either I or II has a winning strategy.

I (Wadge, 1972)

ω ADL/W = ∀A, B ⊆ ω , the game GL/W (A, B) is determined.

I Similarly for pointclasses: open determinacy, Borel determinacy, projective determinacy, etc. ω I AD =⇒ ADL/W , BP, LM, PSP, ¬AC, ACω(ω )

I (Solovay’s conjecture) ZF + V = L(R) proves AD ⇐⇒ ADW Introduction. Lipschitz and Wadge reducibility

ω I (L/W reducibility) Given A, B ⊆ X ,

ω ω A ≤L/W B iﬀ there is a Lipschitz/continuous F : X → X such that A = F −1(B).

ω I (L/W degrees) Given A ⊆ X , we deﬁne

ω [A]L/W = {B ⊆ X : B ≡L/W A}

ω ω I (Wadge’s lemma) A, B ⊆ 2 or ω . Then:

1. II wins the game GL(A, B) iﬀ A ≤L B.

2. II wins the game GW (A, B) iﬀ A ≤W B. c 3. If I wins GL/W (A, B) then B ≤L/W A. Introduction. Semi-Linear Ordering Principle

I (Wadge, 1972)

ω c SLOL/W = For all A, B ⊆ ω , A ≤L/W B ∨ B ≤L/W A.

I Wadge’s lemma implies

ADL =⇒ SLOL ⇓ ADW =⇒ SLOW

I Classical consequences of AD have been proved under SLOW : ω SLOW =⇒ PSP, ¬AC, ACω(ω ).

I (Andretta, 2003/4) ZF + BP + DC proves

SLOW ⇐⇒ ADW ⇐⇒ ADL ⇐⇒ SLOL. Introduction. Classical results

I (Martin, 1975) ZF + DC proves determinacy for all Borel sets.

I (Friedman, 1971) Z2 cannot prove that all Borel Gale-Stewart games are determined.

I (Montalb´an/Shore, 2012) For each k, Z2 proves determinacy 0 for Boolean combinations of k many Π3 sets, but not for all 0 ﬁnite Boolean combinations of Π3 sets.

I (Steel, ’77; Tanaka, ’90; Nemoto/MedSalem/Tanaka, ’07) A detailed picture of the reverse mathematics of Gale-Stewart determinacy is known.

I (Louveau/Saint-Raymond, 1987) Z2 does prove that all Borel Lipschitz games are determined. Introduction. Reverse mathematics of L/W games

I In contrast, the situation for L/W games is completely diﬀerent:

I There is no detailed analysis of the strength of L/W determinacy in terms of subsystems of Z2.

I Reverse Mathematics of SLO hasn’t been investigated either.

I Known results on Gale-Stewart determinacy give us upper bounds on the strength of L/W determinacy. But these bounds needn’t be optimal.

I This is certainly a notable gap in our understanding of the reverse mathematics of inﬁnite games. L/W determinacy. Topological results

I By reinterpreting Wadge’s topological analysis of games we obtain direct proofs of L/W determinacy in ZF + DC:

(A, B) Cantor Baire

0 √ √ ∆1 0 √ √ Π1 0 0 √ √ Σ1 ∪ Π1 ^ √ √ Df2 ∩ Df2 √ √ Df2 ^ √ √ Df2 ∪ Df2

I We formalize these proofs in Z2 and obtain a number of results on the strength of L/W determinacy and SLO. Formalizing L/W determinacy and SLO in Z2

I Γ-DetL/W : L/W determinacy for Γ formulas in the Baire space.

∃σI∀σII [Inf → ¬(A(f ) ↔ B(g))] ∨ ∃σII∀σI [Inf ∧ (A(f ) ↔ B(g))]

σI , σII range over strategies for I and II, resp.

I Γ-SLOL/W : L/W semilinear ordering principle for Γ formulas in the Baire space.

∃σII∀σI [Inf ∧ (A(f ) ↔ B(g))] ∨ ∃σII∀σI [Inf ∧ (¬B(f ) ↔ A(g))]

σI , σII range over strategies for I and II, resp.

∗ ∗ I Γ-DetL/W , Γ-SLOL/W : similar for the Cantor space. Results in Z2

Subsystem Cantor Baire 0 ∗ RCA0 ∆1-DetW 0 ∗ ∆1-DetL, 0 0 ∗ WKL0 (∆1, Σ1)-DetL/W , 0 0 ∗ (Σ1, ∆1)-DetL (Σ0 ∪ Π0)-Det∗ , 0 1 1 L/W ∆1-DetW , ACA0 0 ∗ 0 0 Σ1 2 -DetL/W (Σ1 ∪ Π1)-DetW 0 ∆1-DetL, ATR0 0 0 (Σ1 ∪ Π1)-DetL 1 0 0 Π1-CA0 Σ1 2 ∪ ¬ Σ1 2-DetL/W

0 0 0 Σ1 2 stands for diﬀerences of closed sets (= Σ1 ∧ Π1) Results in Z2. Reverse mathematics results

Theorem Over RCA0 the following are equivalent:

0 ∗ 0 ∗ 1. ACA0 2. Σ1 2-DetL 3. Σ1 2-SLOL

Theorem Over ACA0 the following are equivalent:

0 0 0 0 1. ATR0 2. ∆1-DetL 3. ∆1-SLOL 4. Σ1 ∪ Π1 -DetL

Theorem Over RCA0 the following are equivalent:

0 0 1. ATR0 2. Σ1 ∪ Π1 -DetL. L/W determinacy and SLO in 2N. Methodology

0 0 I Closed set ↔ ϕ(f ) ∈ Π1. Open set ↔ ϕ(f ) ∈ Σ1.

I Basic fact: subset F of X N is closed iﬀ it is the body of a tree T on X , i.e. F = [T ].

I The topological structure of sets A, B gives us information ∗ for constructing a winning strategy in GL/W (A, B).

I clopen sets ←→ comparison of associated well-founded trees.

I A = [T ] closed and not open ←→ existence of a path of T with eventual extensions outside T for any initial sequence.

I A = [T0] − [T1] diﬀerence of closed sets ←→ existence of a path of T1 with eventual extensions in T0 which, in turn, have eventual extensions outside T0. L/W determinacy and SLO in 2N. Main theorem

Theorem The following assertions are pairwise equivalent over RCA0:

0 ∗ 0 ∗ 1. ACA0 2. Σ1 2-DetL 3. Σ1 2-SLOL.

Proof. (Sketch) (1 → 2) We construct winning strategies by analysing several (a lot of!) cases of diﬀerences of pruned trees. ∗ ∗ (2 → 3) RCA0 proves that Γ-DetL/W implies Γ-SLOL/W . ∗ ∗ (3 → 1) Show that if II wins GL (A, B) ∨ II wins GL (¬B, A) then 0 0 {x : ϕ(x) ∈ Σ1} exists. Using ϕ(x), we construct (Σ1)2 sets A(f ) ∗ and B(g) s.t. II cannot have a winning strategy in GL (¬B, A). ∗ Then II has a winning strategy in GL (A, B), say σII. We use σII to 0 construct a ∆1 formula equivalent to ϕ(x). Finally, {x : ϕ(x)} 0 exists by ∆1-CA0. L/W determinacy and SLO in NN. New tools

I Methodology: similar to the Cantor space:

I ﬁnite trees ←→ well-founded (and so ranked) trees

I comparison of maximal lenghts ←→ comparison of tree ranks

I (Hirst, 2001) Let α and β be countable well orderings. The following are equivalent over RCA0: 1. ATR0.

2. ∀α, β (α ≤w β ∨ β ≤w α).

I (Hirst, 2000) ATR0 proves: well-founded tree ↔ ranked tree.

I (Increasing functions on trees) S T iﬀ there is a function f : S → T such that s1 ⊂ s2 → f (s1) ⊂ f (s2).

I ACA0 proves: if S T then rk(S) ≤w rk(T ). N L/W determinacy and SLO in N . Reversal for ATR0

Theorem The following are equivalent over ACA0:

0 0 1. ATR0 2. ∆1-DetL 3. ∆1-SLOL

Proof. (Sketch 3 → 1 )

II wins GL(A, B) ∨ II wins GL(¬B, A) → α ≤w β ∨ β ≤w α

α, β −→ ranked trees S(α), T (β) & S(α) T (β) % ↓ σII −→ function f : S(α) → T (β) α ≤w β L/W determinacy and SLO in NN. Main result

Theorem The following are equivalent over RCA0:

0 0 0 0 0 1. ATR0 2. (Σ1 ∪ Π1)-DetL 3. Π1-DetL 4. (∆1, Π1)-DetL

Proof. (Sketch 4 → 1 )

Since we have

0 ∆1-SLOL + ACA0 −→ ATR0, it suﬃces to show

0 0 (∆1, Π1)-DetL −→ ACA0

over the base subsystem RCA0. Concluding remarks

A ﬁrst step towards a better understanding of the reverse mathematics of L/W determinacy ...

I Reinterpreting Wadge’s classical work in descriptive set theory, we developed a topological analysis of the L/W games for the ﬁrst levels of the diﬀerence hierarchy.

I We showed that this analysis can be carried out within natural subsystems of Z2.

I Main results: we obtained new reversals for ACA0 and ATR0 in terms of Lipschitz determinacy and SLO.

I In particular, we investigated the reverse mathematics of the salient Semi-Linear Ordering Principle for the ﬁrst time. Pending questions (I)

I A reversal for WKL0 in terms of L/W determinacy is left pending.

0 ∗ 0 ∗ I Natural candidates: ∆1-DetL or Σ1-DetL.

I Main obstacle: to determine the exact strength of the Dichotomy Principle: BinaryTree(T ) → (TrueClosed(T ) ∨ T deﬁnes a clopen set)

I We developed our analysis of L/W games up to the difference of closed sets. But it seems plausible to extend this analysis to all finite levels of the difference hierarchy. Pending questions (II)

I (Conjecture) The following are equivalent over RCA0:

1. ACA0. 0 ∗ 2. For each natural number k, (Σ1)k -DetL. 0 ∗ 3. For each natural number k, (Σ1)k -SLOL.

I (Conjecture) The following are equivalent over RCA0:

1 1.Π 1-CA0. 0 2. For each natural number k, (Σ1)k -DetL. 0 3. For each natural number k, (Σ1)k -SLOL. Lines of future work

I (Andretta, 2003/4) ZF + BP + DC proves

SLOW ⇐⇒ ADW ⇐⇒ ADL ⇐⇒ SLOL.

(L1) To formalize Andretta’s proof within Z2 in order to obtain equivalences between subsystems of second order arithmetic and Wadge determinacy principles.

I (Louveau/St.-Raymond, ’87) Z2 proves full Borel L/W determinacy. (L2) To study in detail Louveau/St.-Raymond’s proof in order to isolate a natural subsystem of Z2 which can prove full Borel L/W determinacy, and to investigate whether that subsystem would turn out to be actually equivalent to Borel L/W determinacy.