Annals of the Japan Association for Philosophy of Science Vol.25 (2017) 35～44 35 A Survey of Determinacy of Infinite Games in Second Order Arithmetic Dedicated to Professor Tanaka’s 60th birthday Keisuke Yoshii∗ Abstract In the conference of Computability Theory and Foundations of Mathematics 2015, we had special sessions on Professor Kazuyuki Tanaka’s work in honor of his 60th birthday. It was a great honor for me to give a talk about determinacy of infinite games in that session. In this paper, accordance with works by Professor Tanaka on determinacy, we introduce a collection of related researches. Key words: weak determinacy, inductive definitions 1. A Historical Introduction Tanaka started the works on determinacy of infinite games as the graduate study under the supervision of Leo Harrington at Berkeley in 1980’s. The infinite games, so called Gale-Stewart games, are very simple: For a formula ϕ with a distinct variable f N ranging over N , we associate a two-person game Gϕ (or simply denote ϕ)asfollows: player I and player II alternately choose a natural number (starting with player I) to form an infinite sequence f ∈ NN and player I (resp. II) wins iff ϕ(f) (resp. ¬ϕ(f)). We say that ϕ is determinate if one of the players has a winning strategy σ : N<N → N in the game ϕ.ForaclassC of formulas, C-Det is the axiom which states that any game in C is determinate. In 1953, D. Gale and F. M. Stewart [3] introduced the infinite game and proved the open determinacy. In such a game, it may be natural to think that computing a winning strategy becomes harder if a class C becomes more complicated. Indeed, H. Friedman [2] showed that Borel determinacy is not provable from set theory ZF in 1971, and D. Martin [9] proved, in 1975, that it is derivable from ZFC. 0 Moreover, H. Friedman [2] showed that Σ5 determinacy is not provable from ∗ National Institute of Technology, Okinawa College [email protected] —35— 36 Keisuke Yoshii Vol. 25 0 second order arithemetic, Z2, and Martin [8] improved it as Z2 does not prove Σ4 0 determinacy. Finally, J. Steel proved the earliest result of reverse mathematics, Σ1 determinacy is equivalent to ATR with guides of S. G. Simpson. Thus, researches on determinacy have strong connection with the birth of reverse mathematics. Shore [13] gives a beautiful introduction of reverse mathematics and, it also contains details about determinacy. Tanaka started his researches on determinacy at almost the same time as the dawn of reverse mathematics. In this paper, accordance with works by Tanaka on determinacy, we see an overview of some results of determinacy for lower levels of Borel hierarchy. 2. Preliminaries In this section, we recall some basic definitions and facts about second order arithmetic. The language L2 of second order arithmetic is a two-sorted language with number variables x,y,z,... and unary function variables f,g,h,..., consisting of constant symbols 0, 1, +, ·, =,<.We also use set variables X,Y,Z,...,intending to range over the {0, 1}-valued functions, that is, the characteristic functions of sets. The formulas can be classified as follows: 0 • ϕ is bounded (Π0) if it is built up from atomic formulas by using propositional connectives and bounded number quantifiers (∀x<t), (∃x<t), where t does not contain x. 1 1 • ϕ is Π0 if it does not contain any function quantifier. Π0-formulas are called arithmetical formulas. i i •¬ϕ is Σn if ϕ is a Πn-formula (i ∈{0, 1},n∈ ω). 0 0 •∀x1 ···∀xkϕ is Πn+1 if ϕ is a Σn-formula (n ∈ ω), 1 1 •∀f1 ···∀fkϕ is Πn+1 if ϕ is a Σn-formula (n ∈ ω). i i We loosely say that a formula is Σn (resp. Πn)ifitisequivalentoverabase i i theory (such as ACA0)toaψ ∈ Σn (resp. Πn). We now define some popular axiom schemata of second order arithmetic. Definition 2.1. Let C be a set of L2-formulas. (1) C-IND: (ϕ(0) ∧∀x(ϕ(x) → ϕ(x + 1))) →∀xϕ(x), where ϕ(x) belongs to C. (2) C-TI: for any well-ordering <X , (∀x(∀y<X xϕ(y) → ϕ(x))) →∀xϕ(x), where ϕ(x) belongs to C. (3) C-CA : ∃X∀x(x ∈ X ↔ ϕ(x)), where ϕ(x) belongs to C and X does not occur freely in ϕ(x). (4) C∩C−-CA : ∀x(ϕ(x) ↔ ψ(x)) →∃X∀x(x ∈ X ↔ ϕ(x)), —36— A Survey of Determinacy of Infinite Games in Second Order Arithmetic 37 where ϕ(x) and ¬ψ(x) belong to C and X does not occur freely in ϕ(x). (5) C-AC : ∀x∃Xϕ(x, X) →∃X∀xϕ(x, Xx), where ϕ(x, X) belongs to C and Xx = {y :(x, y) ∈ X}. The system ACA0 consists of the ordered semiring axioms for (ω,+, ·, 0, 1,<), 0 0 Σ1-CA and Σ1-IND. For a set Λ of sentences, Λ0 denotes the system consisting of ACA0 plus Λ. i i i − By Δn-CA,wedenoteΣn ∩ (Σn) -CA. We can easily show that for any k ≥ 0, 1 1 Δk-CA0 ⊂ Σk-AC0. Moreover, if k = 2, the above two axioms are known to be equivalent to each other. 0 0 0 0 3. Characterization of Determinacy for Σ1, Σ1 ∧ Π1 and Δ2 sets One of the important things about Tanaka’s works on determinacy could be that he placed more emphasis on the differences which occur among the boldface and lightface statements. Before his research, J. Steel worked on the determinacy and subsystems of analysis with guides of S. Simpson. Steel [15] proved that open 0 0 determinacy, more specifically Δ1 and Σ1 determinacy, is equivalent to arithmetical transfinite recursion over ACA. Definition 3.1. The axiom scheme arithmetical transfinite recursion asserts that for any well-ordering and any arithmetical formula ψ(x, X), there exists H ⊆ N such that the following are satisfied: • If b is the minimum element with respect to , (H)b = ∅. • If b is the immediate successor of a with respect to , ∀n ∈ N(n ∈ (H)b ↔ ψ(n, (H)a)). • If b is a limit ordinal, ∀a∀x ∈ N((x, a) ∈ (H)b ↔ a ≺ b ∧ x ∈ (H)a)). The formal system ATR0 consists of the following: 1. ACA0, 2. Arithmetical transfinite recursion. 0 0 Theorem 3.2 (J. Steel, [15], [17]). ACA0 ATR ↔ Δ1-Det ↔ Σ1-Det Steel proved the boldface version of the above with full induction, and Tanaka invented a proof with restricted induction. In 1990, Tanaka [17] pointed out that, in 1 0 lightface, Π1-CA can be obtained from determinacy for Boolean combinations of Σ1 sets, but the converse is not possible over ACA0. 0 0 1 Theorem 3.3 (K. Tanaka, [17]). 1. ACA0 Π1 ∧ Σ1-Det → Π1-CA 1 0 0 2. ATR0 Π1-CA → Π1 ∧ Σ1-Det —37— 38 Keisuke Yoshii Vol. 25 We note that Steel had announced that determinacy for Boolean combinations 0 1 of Σ1 sets are characterized by Π1-CA, but the proof was not published. For the second statement, Tanaka pointed out that the base theory can not be weaker than 0 0 ATR0, because, from Theorem 3.2, in order to prove Π1 ∧ Σ1-Det,wehavetoprove 1 the existence of hyperarithmetical hierarchies with a Π1 oracle. 1 In the same paper [17], he introduced a new axiom Π1-TR andprovedthefol- lowing. 1 0 Theorem 3.4 (K. Tanaka, [17]). ACA0 Π1-TR ↔ Δ2-Det 1 1 The axiom Π1-TR is transfinite recursion of Π1-CA. The above theorem is im- portant in terms of the relation between determinacy and transfinite recursion. By 0 0 0 showing Δ2 formulas coincide with transfinite Boolean combinations of Π1 and Σ1 0 formulas (Hausdorff’s difference classes), Tanaka showed that determinacy for Δ2 1 formula can be proved by Π1 transfinite recursion. This plays important roles to characterize determinacy for Δ(C) formula by using transfinite recursion of inductive definitions. 1 4. Introducing a new axiom system Σ1-MI0 It may be natural to consider the characterization of more stronger determinacy, 0 0 such as Σ2-Det. However, it seems not possible to characterize Σ2-determinacy with 1 1 comprehension axioms because of the complicated structure of Δ2-CA,e.g.Δ2-CA 1 and Π1-TR are not comparable (cf. Tanaka [16]). Thus, in order to characterize 0 1 determinacy for Σ2 or higher, Tanaka introduced somewhat unusual axiom, Σ1-MI. 1 The following is the formal definition of Σ1-MI. An operator Γ : P (N) → P (N) belongs to a class C of formulas iff its graph {(x, X):x ∈ Γ(X)} is defined by a formula in C. Γ is said to be monotone iff Γ(X) ⊂ Γ(Y ) whenever X ⊂ Y .Bymon-C, we will denote the class of monotone operators in C. A relation W is a pre-ordering iff it is reflexive, connected and transitive. W is a pre-well-ordering iff it is a well-founded pre-ordering. The field of W is the set F = {x : ∃y(x, y) ∈ W ∨ (y,x) ∈ W }. An axiom of inductive definition asserts the existence of a pre-well-ordering constructed by iterative applications of a given operator. Definition 4.1 (K. Tanaka, [18]). Let C be a set of L2 formulas. C-ID asserts that for any operator Γ ∈C, there exists a set W ⊂ N × N such that 1. W is a pre-well-ordering on its field F , 2. ∀x ∈ FWx =Γ(W<x) ∪ W<x, 3.