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Proceedings of the International Congress of Mathematicians Helsinki, 1978 Infinite Games Donald A. Martin* 1. Introduction. If G and T are finite or infinite sequences, <T-<T means that T extends o. A tree T is a collection of finite sequences such that, if cr£ T and T-<G9 then T£JT. We will be especially concerned with the tree Seq of all finite sequences of natural numbers. If T is a tree, [T] is the set of all infinite sequences x such that, for all n£co9 x\n^T9 where x\n is the a of length n such that a<x. [Seq] we identify with coœ9 the set of all functions from the natural numbers to the natural numbers. Suppose T is a tree such that every element of T has a proper extension belonging to T. Let A^[T]. We define a game G, the game with payoff A9 as follows. Two players, I and II, take turns moving as follows : ZQ %1 Z2 I III Z3 ... II... Each sequence (z 0 , ..., z„) must belong to T. I wins & play of G just in case the sequence {zt: i^co)£A. The notions of strategy for I (or II) for G and winning strategy for I (or II) for G are defined in the obvious way. G is determined if either I or II has a winning strategy for G. Gale and Stewart [3] introduced the games G and proved, using the axiom of choice, that there is an undetermined game with T = Seq. To consider more restricted games, they put a topology on [T] by letting the basic open sets be those sets of the * This paper was supported in part by Grant Number MCS 76-05525 from the National Science Foundation of the United States. 270 Donald Ai. Martin . » , • , i forms {x: <r<x} for G£T. Let us say that G is open, Borei etc., just in case the payoff A is open, Borei, etc. 1.1 THEOREM (GALE-STEWART). All open games are determined. After further results by Wolfe [10], Davis [1], and Paris [8], Martin [6] proved the following result. 1.2 THEOREM. All Borei games are determined. The method of proof is to associate, with an ^ c [T] of Borei rank a, an A * s [T *\ with A* open and to prove that the game G with payoff A and the game G* with payoffs* are equivalent: whoever has a winning strategy for one has a winning strategy for the other. T* is much bigger than T: if T has size 3fy, then T7* has size roughly mß+a. Individual moves in G* represent complex commitments as to how the players will move in an associated play of G. Results of Friedman [2] showed that, even for r=Seq, some kind of appeal to uncountable cardinals would be necessary to prove all Borei games are determined. If T is a class of subsets of coœ9 let Det (T) be the assertion that all games with r = S e q and payoff in T are determined. Recent work (see [7]) has shown that Det (Projective) is a very powerful hypothesis in descriptive set theory. For example, Det (il^)=Det (CPCA) implies that all 2?J sets of real numbers are Lebesgue measurable and yields a complete structural theory for levels three and four of the projective hierarchy. J. Mycielski observed that a result of [1] implies that Det (n\) is not provable in the usual set theory ZFC. If one assumes large cardinal axioms, one gets more determinacy: 1.3 THEOREM (MARTIN [5]). If a measurable cardinal exists, Det(17j). 1.4 THEOREM. If there are 2 (actually 1 \) supercompact cardinals, then Det (A (n\)) where A is operation A. 1.5 THEOREM. If there is a non-trivial iterable elementary embedding of a rank Rk into itself, then Det (U^. L. Harrington has proved the converse of a slightly sharper version of Theorem 1.3. It is known from work of J. Green, Martin, W. Mitchell, J. Simms, and R. Solovay that much stronger hypotheses than those of Theorem 1.3 are needed to prove the conclusions of Theorems 1.4 and 1.5. The rest of this paper is devoted to sketching the proof of Theorem 1.5. Iterability will be explained in §2 below. The hypothesis of Theorem 1.5 is strictly weaker (barring inconsistency) than the existence of an elementary j : V^M with M transitive and j\Rx^identity and j(Rx) = RÀ. Kunen [4] shows that J(^A+I) = ^ + I impossible. Infinite Games 271 2. Iterable emfyeddings. For the rest of this paper let j\Rk-+Rk be an elementary embedding first moving K. Let xQ=x and K I+1 =7(ty)- It follows by [4] that A=sup,ty or A = s u p / « / + l . We assume the former. If Y^RX9 let j(Y) = \Ji j(YnRx). Clearly j(j) is an elementary embedding of RÀ into i?A, first moving x1. Let jQ=j and ,/JI+i =./,,(/,,)• L e t 7Will = identity and Jntm^i—Jm°Jntm f ° r / 2 ^ ' " - Here o denotes composition. Let Mj = RÀ9 / = 0 , 1,.... As long as direct limits are well-founded, we can iterate the system (Mi9jnm) to get a system (Ma9jpy) for ordinals a, /?, y with /f<y, where .fy : Mß-+My is elementary and each Ma is transitive. When this can be done, we say that j is iterable. Set 7or=7a>a+1. 2.1 LEMMA. If hi^n, jmu=f•«—m PROOF. ltzeR^jqojq (z) = (jq(jq))ojq(z) by elementarily, and this is just Applying this fact repeatedly yields the lemma. 2.2 LEMMA. Suppose j is iterable. Suppose a^ß Ja,P°Ja + n PROOF. = jq+1ojq(z) are ordinals and nZw. Jß + n°J«,ß- For z£Ma (=M a + „), jai^oyll+n(z)=C/ai^a+»))%^(z)- Bu ^«,/>0« +w ) =./>+,,• 3. /7-embeddings. If /? is an ordinal, a ß-embedding is an elementary embedding fc: .#«+0 -> Ra>+ß. first moving a>ß. Set v(k) = a and v'(Z:) = a'. If A: is a ^-embedding and y-fl</? define a 0-1 measure /i* as follows: rt(X) = l~k\RHk)+7£k(X). It is easily checked that ptky is v(fc)-complete and concentrates in y-embeddings k' with v(k')^v(k) and v'(k') = v(k). The following lemmas are easily verified. 3.1 LEMMA. If k is a ß-embedding and y + l^ß!^ß2, then ^=/x^ i?v(k)+ "i. 3.2 LEMMA. Let k be a ß-embedding and let y i ^ ^ and y2+l<ß. li\(X) = \. Then pkn{z: z\Rv(z) + n£X} = l. Suppose 4. A normal form for III s e t s - F ° r ^ e rest of tliis paper, let A ^ of* be a fixed Il\ set. Let Seq* be the collection of nonempty elements of Seq. For o-ÇSeq, let lh(<7) be the length of G. Let Seq*2={<<7,T>: G9 i6Seq*&lh((7)=lh(T)}. The following lemma is just a restatement of Shoenfield's analysis of Ft] sets [9]. 4.1 LEMMA. There is a function Q: Seq*2-*Cü such that lh (G) = \ -+Q(G9 T ) = 0 and lh(o-)>J -+Q(G9 T)H-1 <lh (G) and, for any uncountable cardinal v\ and any 272 Donald A. Martin x^o)°>, x€A if and only if there are Ft: [itf^-Hf for z^Seq* such that, if <3ç0<...<:aIh(T)<^ and %' is a one-term extension of T, then ^'{«Os •••> a lh(r)} < ^ t { a 0 ? •••> ae(xHh(r'),t')> •••' a lh(t)}n Here [rç] is the collection of all size n subsets of r\. 4.2 LEMMA. If j is iterable and xÇœ^.xdA such that if and only if there is an #:Seq*-*Jl (1) lh(T) = l - ^ ( T ) < x 1 ; (2) if T' ö ß one-term extension of T, fAe« PROOF. H(x')^jQij!cnh{:zf)^H{z). Assume the FT exist with n = n. If lh(T)=«, let H(T) = (j0t„(Fxj){x0,...9xn-1}. (1) is immediate, and applications of Lemma 2.1 yield (2). Now assume that H exists. If « = lh(r) and a 0 <...<a J i _ 1 , set F t {a 0 , ...,<*„_!} =j a „_ 2+ i ia „_ 1 o...o7 ao+ljai oj 0>ao (/f(T)). Apphcations of Lemma 2.2 show that, with suitable rj9 the Fv are as required. 5. Proof of Theorem 1.5. Let G be the game with payoff A. Let T 0 , T15 ..., be an enumeration of Seq* suchthat T / -<T / -»-I'« I /. Let G* be played as follows : («o, «o) I w i ("2 » «i) II I "a («4, «a) • • II I ... Let x(i)=n( and ^ ( r / ) = a / . I www G* just in case H obeys the constraints of Lemma 4.2. 5.1 LEMMA. G* is determined. PROOF. G* is closed. 5.2 LEMMA. If I has a winning strategy for G*, then I has a winning strategy for G. 5.3 LEMMA. If II has a winning strategy for G*, then II has a winning strategy for G. Let s* be a winning strategy for II for G*. We define a strategy s for II for G. Let o- b e a position in G with II to move. Let ß09 ...,ßm be even ordinals such that, if G is extended to a position G* in G* ,by setting a,—/?,, then I is not already lost at <r*. We define an iterated product measure on the set of all sequences (kQ9...9km)9 where each k( is a ßt -embedding with v'(fc,)<A. To do this we assign to each i a measure space, which may depend upon PROOF. Infinite Games 273 (/r0, . . . J / C ^ J ) . If Hi (T7) = 1, then the measure for / is ^ 1 , / ? *i + ^ + 2. If rr is a one-term extension of T{9 then the measure for /', is pJpf'\ where a = Q(tr\lh{i:v)9 ir). Let s (er) be the constant value of s* (o* (k0, ..., km)) for measure one of (k0,..., k,„)9 where <7*(£0, ..., /<rm) is the result of extending G by setting ai=v(ki). Lemmas 3.1 and 3.2 imply that S(G) is independent of the choice of the /?,.. Suppose that x is a play of G according to s and that x£A. Let H witness that x£A. Let /?, = 27/(1;,). Using the ßi to compute s, we can find a sequence /r 0 ,/r l5 ..., such that, if we set oci=v(ki)9 then we extend x to a play of G* according to s* which is won by I. This contradiction completes the proof. References 1. Morton Davis, Infinite games of perfect information, Advances in Game Theory, Ann. of Math. Studies No. 52, Princeton Univ. Press, Princeton, N . J., 1964, pp. 85—101. 2. H. Friedman, Higher set theory and mathematical practice, Ann. Math. Logic 2(1971), 326—357. 3. D. Gale and F. M. Stewart, Infinite games with perfect information, Contributions to the Theory of Games, Ann. of Math. Studies No. 28, Princeton Univ. Press, Princeton, N . J., 1953, pp. 245—266. 4. IC Kunen, Elementary embedding and infinitary combinatorics, J. Symbolic Logic 36 (1971), 407—413. 5. D. A. Martin, Measurable cardinals and analytic games, Fund. Math. 66 (1970), 287—291. 6. Borei determinacy, Ann. of Math. 102 (1975), 363—371. 7. Y. Moschovakis, New methods and results in descriptive set theory, Proceedings of the International Congress of Mathematicians (Vancouver 1974), Volume I, pp. 251—257. 8. J. Paris, Z F \-E\ determinateness, J. Symbolic Logic 37 (1972), 661—667. 9. J. R. Shoenfield, The problem of predicativity, Essays on the Foundations of Mathematics, Magnes Press, Jerusalem, 1961, pp. 132—139. 10. P. Wolfe, The strict determinateness of certain infinite games, Pacific J. Math. 5 (1955), 841 — 847. UNIVERSITY OF CALIFORNIA Los ANGELES, CALIFORNIA 90024, U.S.A.