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) ::: ! I (Gale-Stewart games) A ⊆ X . I wins a play of G(A) if hf (0); g(0); f (1); g(1);::: i 2 A Otherwise II wins. ! I (Lipschitz games) A; B ⊆ X . II wins a play of GL(A; B) if hf (0); f (1);::: i 2 A iff hg(0); g(1);::: i 2 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) ::: ! I (Wadge games) A; B ⊆ X . II wins a play of GW (A; B) if hf (0); f (1);::: i 2 A iff hg(0); g(1);::: i 2 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 = 8A ⊆ !!, the Gale-Stewart game G(A) is determined, i.e. either I or II has a winning strategy. I (Wadge, 1972) ! ADL=W = 8A; B ⊆ ! , the game GL=W (A; B) is determined. I Similarly for pointclasses: open determinacy, Borel determinacy, projective determinacy, etc. ! I AD =) ADL=W ; BP; LM; PSP; :AC; AC!(! ) I (Solovay's conjecture) ZF + V = L(R) proves AD () ADW Introduction. Lipschitz and Wadge reducibility ! I (L/W reducibility) Given A; B ⊆ X , ! ! A ≤L=W B iff there is a Lipschitz/continuous F : X ! X such that A = F −1(B). ! I (L/W degrees) Given A ⊆ X , we define ! [A]L=W = fB ⊆ X : B ≡L=W Ag ! ! I (Wadge's lemma) A; B ⊆ 2 or ! . 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) ! c SLOL=W = For all A; B ⊆ ! ; 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 : ! SLOW =) PSP; :AC; AC!(! ). 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 p p ∆1 0 p p Π1 0 0 p p Σ1 [ Π1 ^ p p Df2 \ Df2 p p Df2 ^ p p 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. 9σI8σII [Inf !:(A(f ) $ B(g))] _ 9σII8σI [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. 9σII8σI [Inf ^ (A(f ) $ B(g))] _ 9σII8σI [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 $ '(f ) 2 Π1. Open set $ '(f ) 2 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 fx : '(x) 2 Σ1g exists. Using '(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 '(x). Finally, fx : '(x)g 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. 8α; 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.