The boundary of determinacy within second order arithmetic. Antonio - PowerPoint PPT Presentation

The boundary of determinacy within second order arithmetic. Antonio Montalb´ an. U. of Chicago (with Richard A. Shore) Berkeley, CA, March 2011 Special session in honor of Leo Harrington. Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

The Question How much determinacy can be proved without using uncountable objects? Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Determinacy Fix a set A ⊆ ω ω . Player I a 0 a 2 · · · let ¯ a = ( a 0 , a 1 , a 2 , a 3 , ... ) · · · Player II a 1 a 3 a ∈ ω ω \ A . Player I wins is ¯ a ∈ A , and Player II wins if ¯ A strategy is a function s : ω <ω → ω . It’s a winning strategy for I if ∀ a 1 , a 3 , a 5 , .... ( f ( ∅ ) , a 1 , f ( a 1 ) , a 3 , ... ) ∈ A A ⊆ ω ω is determined if there is a strategy for either player I or II. For a class of sets of reals Γ ⊆ P ( ω ω ), let Γ-DET: Every A ∈ Γ is determined. Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

History Γ Γ-DET remark Open (Σ 0 1 ) [Gale Stwart 53] G δ (Π 0 2 ) [Wolfe 55] F σδ (Π 0 3 ) [Davis 64] G δσδ (Π 0 4 ) [Paris 72] F σδσδ (Π 0 5 ) needs Power-set axiom [Friedman 71] Borel (∆ 1 1 ) [Martin 75] needs ℵ 1 iterations of Power-set axiom [Friedman 71] Analitic (Σ 1 ∀ x ( x ♯ exists) ⊢ .. 1 ) Martin’s bound is sharp [Harrington 1978] [Martin 70] Full ( ω ω ) False in ZFC [Gale Stwart 53] Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Harrington’s result Sharps: We define the statement: “x ♯ exists” as “In L ( x ), there is an ω 1 -list of indiscernibles.” x ♯ is the the ω -type of this list. Thm: [Kunen] [Jensen] (ZFC) The following are equivalent: 1 0 ♯ exists. 2 There is a proper embedding of L into L . 3 There is an uncountable X ⊆ ON such that ∀ Y Y ⊇ X & | Y | = | X | = ⇒ Y �∈ L . Theorem ( [Harrington 78] ) 1 -DET is equivalent to “ ∀ x ( x ♯ exists)”. Σ 1 Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Countable mathematics Second order arithmetic Z 2 (a.k.a. analysis) consist of ordered semi-ring axioms for N induction for all 2 nd -order formulas comprehension for all 2 nd -order formulas Most of classical mathematics can be expressed and proved in Z 2 . Thm: ZFC − is Σ 1 4 -conservative over Z 2 , where ZFC − is ZFC without the Power-set axiom. (Obs: Borel-DET and Π 0 k -DET are Π 1 3 -statements.) Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Determinacy without countable objects Thm: [Friedman 71, Martin] Z 2 � ⊢ Π 0 4 -DET. Theorem (essentially due to Martin) Given n ∈ N , Z 2 (and also ZFC − ) can prove that every Boolean combination of n Π 0 3 sets is determined where F σδ = Π 0 3 = intersection of unions of closed sets But.... The larger the n , the more axioms are needed. Theorem (MS) Z 2 (and also ZFC − ) cannot prove that every Boolean combination of Π 0 3 sets is determined Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Reverse Mathematics in a nutshell The main question of Reverse Mathematics is: What axioms of Z 2 are necessary for classical mathematics? Using a base theory as RCA 0 , one can often prove that theorems are equivalent to axioms. Most theorems are equivalent to one of 5 subsystems. Most theorems of classical mathematics can be proved in Π 1 1 -CA 0 . where in Π 1 1 -CA 0 , induction and comprehension are restricted to Π 1 1 -formulas. No example of a classical theorem of Z 2 needed more than Π 1 3 -CA 0 . We provide a hierarchy of natural statements that need axioms all the way up in Z 2 . Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Strength of Determinacy in Second order arithmetic Γ strength of Γ-DET base ∆ 0 ATR 0 [Steel 78] RCA 0 1 Σ 0 ATR 0 [Steel 78] RCA 0 1 Π 1 Σ 0 1 ∧ Π 0 1 -CA 0 [Tanaka 90] RCA 0 1 ∆ 0 Π 1 1 -TR 0 [Tanaka 91] RCA 0 2 Π 0 Σ 1 1 -ID 0 [Tanaka 91] ATR 0 2 ∆ 0 [Σ 1 1 ] TR -ID 0 Π 1 [MedSalem, Tanaka 08] 1 -TI 0 3 Π 0 Π 1 ∆ 1 3 -CA 0 ⊢ .. 3 -CA 0 � ⊢ .. [Welch 09] 3 Π 0 Z 2 �⊢ .. [Martin] [Friedman 71] 4 Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Difference hierarchy Def: A ⊆ ω ω is m- Π 0 3 if there are Π 0 3 sets A 0 ⊇ A 1 ⊇ ... ⊇ A m = ∅ A = ( ... ((( A 0 \ A 1 ) ∪ A 2 ) \ A 3 ) ∪ ... ) s.t.: i.e. x ∈ A ⇐ ⇒ (least i ( x �∈ A i )) is odd. � Obs: (Boolean combinations of Π 0 m - Π 0 3 ) = 3 . m ∈ ω The difference hierarchy extends through the transfinite. Thm : [Kuratowski 58] ∆ 0 � α - Π 0 4 = 3 . α ∈ ω 1 Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

A closer look at our main theorem Recall: Π 1 n -CA 0 is Z 2 with induction and comprehension restricted to Π 1 n formulas. ∆ 1 n -CA 0 is Z 2 with induction and comprehension restricted to ∆ 1 n sets. Theorem (MS, following Martin’s proof) Π 1 n +2 -CA 0 ⊢ n- Π 0 3 − DET. Theorem (MS) ∆ 1 n +2 -CA 0 �⊢ n- Π 0 3 − DET. [Welch 09] had already proved the cases n = 1. � Π 1 n ∆ 1 Since Z 2 = n -CA 0 = � n -CA 0 : n Corollary: For each n , Z 2 ⊢ n - Π 0 3 − DET , but Z 2 � ⊢ ∀ n ( n - Π 0 3 − DET ). Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Reversals Theorem (MS) Reversals aren’t possible: for each n ∆ 1 ∆ 1 n +2 -CA 0 + n- Π 0 Π 1 n +2 -CA 0 3 -DET n +2 -CA 0 � � Thm: [MedSalem, Tanaka 07] Π 1 1 -CA 0 + Borel-DET �⇒ ∆ 1 2 -CA 0 . Theorem (MS) Let T be a true Σ 1 4 sentence. Then, for n ≥ 2 , ∆ 1 n -CA 0 + T � ⊢ Π 1 n -CA 0 Π 1 n -CA 0 + T � ⊢ ∆ 1 n +1 -CA 0 (even for β -models) This also holds if T is a Σ 1 n +2 theorem of ZFC. Obs: Borel-DET and m - Π 0 3 − DET are Π 1 3 theorems of ZFC. Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

The techniques Def: α is n-admissible if there is no unbounded, Σ n -over- L α -definable function f : δ → α , with δ < α . ⇒ 2 ω ∩ L α | = ∆ 1 α is n -admissible = n +1 -CA 0 (for n ≥ 2) . Let α n be the least n -admissible ordinal. Let Th n =Theory of L α n . Th n �∈ L α n using G¨ odel-Tarski undefinability of truth. Lemma (MS) For n ≥ 2 , there is a ( n- 1) - Π 0 3 game where each player plays a set of sentences, and 1 if I plays Th n , he wins. 2 if I does not play Th n but II does, then II wins. A winning strategy for this game must compute Th n . Hence 2 ω ∩ L α n | = ∆ 1 n +1 -CA 0 & ¬ ( n − 1) - Π 0 3 − DET Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.

Ideas in the proof. Each player has to play a complete, consistent set of formulas including ZF+ V = L α n . We consider the term models of these theories: M and N. M = L α n L α n is the only well-founded model of ZF+ V = L α n . Using differences of Π 0 3 formulas we need to N identify the player playing a well-founded model. L α − A Let L α = N ∩ M . We find a Π 0 3 condition C k and a property P k s.t.: If is α is k -admissible and P k holds , then If C k , we find a descending sequence in N. If ¬ C k , then α is k + 1-admissible and P k +1 Antonio Montalb´ an. U. of Chicago The boundary of determinacy within second order arithmetic.