A Survey of Determinacy of Infinite Games in Second Order Arithmetic
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
∗ 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 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 —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 —38— A Survey of Determinacy of Infinite Games in Second Order Arithmetic 39 where Wx = {y ∈ F :(y,x) ∈ W } and W 0 1 Theorem 4.2 (K. Tanaka, [18]). 1. ACA0 Σ2-Det → Σ1-MI 1 0 2. ATR0 Σ1-MI → Σ2-Det Then, he conjectured that, for the second statement, the base theory can not be 1 0 weaker than ATR0, i.e. ACA0 Σ1-MI → Σ2-Det. As long as I know, it still remains open. 1 Also, it is worth noticing that Tanaka introduced Σ1-MI as a new system of Z2 in 0 order to pin down Σ2-Det. A theme of reverse mathematics is to find out necessary and sufficient axiom systems to prove theorems of ordinary mathematics. Yet, what Tanaka did was finding the axiom system from the theorem. This theorem made us possible to do researches where we try to find out new axiom systems which are equivalent to weak determinacy on Borel hierarchy. 0 5. Δ3-Det and other systems 0 At the level of Δ3 determinacy, MedSalem and Tanaka first in [6] compared it with many popular systems. These observations lead them to the characterization of 0 Δ3-Det. MedSalem and Tanaka showed the following theorems. 1 1 0 Theorem 5.1 (M.O. MedSalem, K. Tanaka, [6]). 1. Δ3-CA0 +Σ3-IND Δ3-Det 1 1 0 2. Π2-CA0 +Π3-TI Δ3-Det 1 1 Moreover, they showed that neither Σ3-IND nor Π3-TI can be dropped by proving the next theorem. 1 1 Theorem 5.2 (M.O. MedSalem, K. Tanaka, [6]). 1. Π2-CA0 Δ2-MI 1 1 0 2. Δ2-MI0 +Π3-TI Δ3-Det 1 0 3. Δ3-CA0 Δ3-Det The following are other main theorems of [6]. Theorem 5.3 can be seen as a 1 0 refinement of Welch [21], which is Δ3-CA0 Σ3-Det. 1 1 0 Theorem 5.3 (M.O. MedSalem, K. Tanaka, [6]). 1. Δ3-CA0 +Σ∞-IND Σ3-Det 1 1 0 2. Π2-CA0 +Π∞-TI Σ3-Det 1 Theorem 5.4 (M.O. MedSalem, K. Tanaka, [6]). Δ1-Det0 does not prove any of the following axioms. 1 1 1 1. Σ3-IND 2. Π2-TI 3. Δ2-CA0 1 1 Here, Δ1-Det0 means ACA0 +Δ1-Det. From these observations, we could see that —39— 40 Keisuke Yoshii Vol. 25 1 1 0 Σ3-IND or Π3-TI seem to be relevant to characterize Δ3-Det. The following diagram 0 Det 0 shows some relations among Δ3- and other major systems. Note that (Σ2)<ω 0 0 denotes k∈ω(Σ2)k,and(Σ2)<ω-Det states that for any standard natural number k, 0 we have (Σ2)k determinacy. See Definition 6.3. 6. Multiple Inductive definitions 0 Finally, MedSalem and Tanaka [7] gave the characterization to Δ3-Det by in- 1 TR troducing a new axiom [Σ1] -ID, which asserts the existence of inductively defined 1 sets with combination of α-many Σ1-operators. We here just give the definition of 1 k [Σ1] -ID with k = 2, and see [7] for the formal definition. Definition 6.1. Let C0 and C1 are collections of operators. The axiom scheme [C0, C1]-ID0 asserts the following. For any Γ0 ∈C0, Γ1 ∈C1, there exist W ⊆ F1 × F1, x and V : x ∈ F1 ∪{∞} such that the following are all satisfied. 1. W is a pre-well-ordering on F1. 2. ∀x ∈ F1 ∪{∞} x x • V is a pre-well-ordering on its field F0 . x W 3. W∞ = W<∞ = F1. Here, Wx = {y :(x, y) ∈ W }, W —40— A Survey of Determinacy of Infinite Games in Second Order Arithmetic 41 Tanaka and MedSalem [7] introduced multiple inductive definition to character- 0 ize the finite Boolean combinations of Σ2 determinacy. Then, to observe more finer classes of determinacy, we [20] introduced transfinite recursion of multiple inductive definitions, which is Theorem 6.10. Definition 6.2. Let C and C be classes of L2 formulas. We denote the classes of formulas in the form ϕ ∧ ψ (ϕ ∈C,ψ ∈C) as C∧C,and¬ψ (ψ ∈C) as ¬C. 0 0 Definition 6.3. For all n, k ≥ 1, we define the classes of the formulas (Σn)k, (Πn)k as follows. 0 0 0 0 0 • (Σn)1 =Σn, (Σn)k =Σn ∧ (Πn)k−1 if k>1, 0 0 • (Πn)k = ¬(Σn)k. 0 0 0 0 Remark: (Σn)2k =(Σn)2 ∨ (Σn)2 ∨···∨(Σn)2 (k times). Theorem 6.4 (M.O. MedSalem, K. Tanaka, [7]). Assume 0 0 1 k 1. (Σ2)k-Det 2. [Σ1] -ID 1 1 We note that Σ1-ID and Σ1-MI are equivalent over ATR0. 1 0 Lemma 6.5 (M.O. MedSalem, K. Tanaka, [7]). Δ2-MI0 (Σ2)<ω-Det 1 0 Theorem 6.6 (M.O. MedSalem, K. Tanaka, [7]). 1. Π2-CA0 (Σ2)<ω-Det 1 1 <ω 1 2. Δ2-CA0 +[Σ1] -ID Δ2-MI0 1 <ω 1 k Note that [Σ1] -ID denotes k∈ω[Σ1] -ID. The first statement just comes from Theorem 5.2.1 and Lemma 6.5. The second proof is essentially due to the result of 1 0 C. Heinatsch and M. M¨ollerfeld [1], saying Π2-CA0 and (Σ2)<ω-Det are proof theo- retically equivalent. For the transfinite level, they finally showed the next theorem. Here, it is not 1 1 known whether Π3-TI can be dropped or weaken to Σ3-IND. 1 1 TR Theorem 6.7 (M.O. MedSalem, K. Tanaka, [7]). ACA0 +Π3-TI [Σ1] -ID ↔ 0 Δ3-Det 1 k 1 k We close this section with a result with [Σ1] -IDTR. An axiom [Σ1] -IDTR is 1 transfinite recursion of Σ1-ID with k-operators, which is introduced to characterize 0 the determinacy for pointclasses refining the Σ2 difference classes. We omit the formal 1 k definitions of [Σ1] -IDTR0, and see K. Yoshii and K. Tanaka [20] for the detail. Definition 6.8. Let C be a class of formulas. A C-formula ϕ is called a Δ(C)-formula if there exists a ¬C-formula ϕ such that ϕ and ϕ are equivalent over an appropriate —41— 42 Keisuke Yoshii Vol. 25 0 0 system, e.g., ACA0. In particular, we write Δn for Δ(Σn). Definition 6.9. Let C, C be classes of formulas. A formula ϕ(f) is called a Sep(C, C)-formula, if it is written as (ψ(f) ∧ η(f)) ∨ (¬ψ(f) ∧ η(f)) for some ψ ∈C,η ∈¬C and η ∈C. Theorem 6.10 (K. Tanaka, K. Yoshii [20]). For any k>0, we have the following. 0 0 0 1 k RCA0 Δ((Σ2)k+1)-Det ↔ Sep(Δ2, (Σ2)k)-Det ↔ [Σ1] -IDTR. We put some notes on Theorem 6.10. For Theorem 6.10, the equivalence 0 0 0 Sep(Δ2, (Σ2)k)-Det ↔ Δ((Σ2)k+1)-Det is obtained from the following theorem. 0 Theorem 6.11 (K. Tanaka, K. Yoshii [20]). A Δ((Σn)k+1)-formula is equivalent to 0 0 a Sep(Δn, (Σn)k)-formula for n, k ≥ 1, and vice versa. 0 0 0 We remember that Sep(Δn, (Σn)k) is treated as a separated union of (Σn)k and 0 0 0 Δn,andΔn is obtained from transfinite difference of Πn−1 (cf. Tanaka [17]). Thus, 1 k 0 0 axiom of transfinite recursion, such as [Σ1] -IDTR, are used to prove Sep(Δn, (Σn)k)- 0 Det or Δ((Σ2)k+1)-Det. 7. Related Works There are many related results with determinacy in second order arithmetic. Here, we introduce only few results from them. First, we would like to introduce the works by T. Nemoto, M. O. MedSalem, and K. Tanaka [4]. They investigated the relationships between the determinacy in Cantor space and Baire space. 0 ∗ Theorem 7.1 (T. Nemoto, M.O. MedSalem, K.Tanaka [4]). 1. RCA0 Δ2-Det ↔ 0 ∗ 0 Σ2-Det ↔ Σ1-Det ↔ ATR0 0 ∗ 0 2. RCA0 (Σ2)k-Det ↔ (Σ2)k−1-Det 0 ∗ 0 3. Δ3-Det ↔ Δ3-Det ∗ Note that Det represents determinacy in Cantor space. Montalb´an and Shore in [11] showed that the determinacy for infinite Boolean 0 0 combinations of Σ3 sets, denoted as (Σ3)ω-Det, is not provable in Z2 andclarifiedthe “limit” of second order arithmetic. Moreover, they showed that determinacy of kth 0 1 1 level of Σ3 sets is provable from Πk+2-CA0, but not possible from Δk+2-CA0. Theorem 7.2 (A. Montalb´an, R. A. Shore, [11]). For each k ≥ 1, 1 0 1 0 1. Πk+2-CA0 (Σ3)k-Det 2. Δk+2-CA0 (Σ3)k-Det For k = 1, it is due to Welch [21]. Then, in the succeeding paper [12], they mentioned about the discussion with Steel, at Berkeley, “Is the consistency of Z2 —42— A Survey of Determinacy of Infinite Games in Second Order Arithmetic 43 0 provable from (Σ3)ω-Det?” Quite surprisingly, they showed as indeed it is. That 0 0 means, although Z2 and (Σ3)ω-Det are, reverse mathematically, incomparable, (Σ3)ω- 0 Det is proof-theoretically stronger than Z2. Moreover, logical strength of (Σ3)k-Det 1 1 is strictly between Con(Πk+2-CA0) and Con(Δk+2-CA0), and therefore, proof the- 1 1 oretically, it is strictly between Πk+2-CA0 and Δk+2-CA0. Wealsonotethatthey 0 − mentioned that Z2,(Σ3)ω-Det,andZFC are all equiconsistent. Theorem 7.3 (A. Montalb´an, R. A. Shore, [12]). For every k ≥ 1, the following hold: 1 0 1. Πk+2-CA0 ∃β-model of (Σ3)k-Det 0 0 2. There is a β-model of (Σ3)k-Det (Σ3)k-Det 0 1 3. (Σ3)k-Det ∃β-model of Δk+2-CA0 0 1 From Theorem 7.3, we now know that Σ3-Det lies strictly between Con(Δ3-CA0) 1 and Con(Π3-CA0). Thus, it is natural to ask what systems of second order arithmetic 0 can characterize Σ3-Det (28th Question of Montalb´an [10]). To answer this question, by improving results of Welch ([21], [22]), and Montalb´an-Shore ([11], [12]), S. J. 0 1 Hachtman claimed that Σ3-Det is equivalent to the existence of β-model of Π2-MI. 1 0 Theorem 7.4 (J. S. Hachtman, [5]). Π1-CA0 Σ3-Det ↔∃countable coded β-model 1 of Π2-MI. For the future works, next levels of characterizations of determinacy for finite 0 Boolean combinations of Σ3 sets are expected. 8. Acknowledgment I would like to express my sincerely appreciation to editors for a special section of the journal Annals of the Japanese Association for Philosophy of Science, Professor Chong Chi Tat, Professor Stephen G. Simpson, and Professor Makoto Kikuchi, and all committee members of CTFM 2015 for this opportunity to write about works on Professor Tanaka. By writing about Professor Tanaka’s work, I could feel renewed appreciation for his guiding me to this field. I would like to warmly thank Dr. Takako Nemoto and Ms. Wenjuan Li for their valuable comments on this article. I also appreciate the comments which are given by the referee. Again, I would like to express my greatest gratitude to Professor Kazuyuki Tanaka for all his contribution to this research community. References 1 [1] C. Heinatsch, M. M¨ollerfeld, The determinacy strength of Π2-comprehension, Ann. —43— 44 Keisuke Yoshii Vol. 25 Pure Appl. Logic 161, 1462–1470 (2010). [2] H. Friedman, Higher set theory and mathematical practice, Annals of Mathematical Logic 2, 325–357 (1971). [3] D. Gale, F. M. Stewart, Infinite game with perfect information. Contributions to the theory of games, vol. 2, 245–266. Annals of Mathematical Studies, no. 28, Princeton University Press, Princeton, N.J. (1953). [4] T. Nemoto, M. O. MedSalem, K. 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Tanaka, Infinite games and transfinite recursion of multiple inductive definitions, How the world computes, Lecture Notes in Comp. Sci., vol. 7318, Springer, Heidelberg, 374–383 (2012). [21] P. D. Welch, Weak systems of determinacy and arithmetical quasi-inductive definitions, Journal of Symbolic Logic, 76, no.2, 418–436 (2011). [22] P. D. Welch, Gδσ games, Issac Newton Institute Preprint Series No. NI12050-SAS, 1–10 (2012) (Received 2016.8.1; Accepted 2016.11.15) —44—