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A BRIEF HISTORY of DETERMINACY §1. Introduction 0001 0002 0003 A BRIEF HISTORY OF DETERMINACY 0004 0005 0006 PAUL B. LARSON 0007 0008 0009 0010 x1. Introduction. Determinacy axioms are statements to the effect that 0011 certain games are determined, in that each player in the game has an optimal 0012 strategy. The commonly accepted axioms for mathematics, the Zermelo{ 0013 Fraenkel axioms with the Axiom of Choice (ZFC; see [Jec03, Kun83]), imply 0014 the determinacy of many games that people actually play. This applies in 0015 particular to many games of perfect information, games in which the 0016 players alternate moves which are known to both players, and the outcome 0017 of the game depends only on this list of moves, and not on chance or other 0018 external factors. Games of perfect information which must end in finitely 0019 many moves are determined. This follows from the work of Ernst Zermelo 0020 [Zer13], D´enesK}onig[K}on27]and L´aszl´oK´almar[Kal1928{29], and also 0021 from the independent work of John von Neumann and Oskar Morgenstern 0022 (in their 1944 book, reprinted as [vNM04]). 0023 As pointed out by Stanis law Ulam [Ula60], determinacy for games of 0024 perfect information of a fixed finite length is essentially a theorem of logic. 0025 If we let x1,y1,x2,y2,::: ,xn,yn be variables standing for the moves made by 0026 players player I (who plays x1,::: ,xn) and player II (who plays y1,::: ,yn), 0027 and A (consisting of sequences of length 2n) is the set of runs of the game 0028 for which player I wins, the statement 0029 0030 (1) 9x18y1 ::: 9xn8ynhx1; y1; : : : ; xn; yni 2 A 0031 essentially asserts that the first player has a winning strategy in the game, 0032 and its negation, 0033 0034 (2) 8x19y1 ::: 8xn9ynhx1; y1; : : : ; xn; yni 62 A 0035 essentially asserts that the second player has a winning strategy.1 0036 0037 The author is supported in part by NSF grant DMS-0801009. This paper is a revised 0038 version of [Lar12]. 0039 1If there exists a way of choosing a member from each nonempty set of moves of 0040 the game, then these statements are actually equivalent to the assertions that the 0041 corresponding strategies exist. Otherwise, in the absence of the Axiom of Choice the statements above can hold without the corresponding strategy existing. 0042 0043 Large Cardinals, Determinacy, and Other Topics: The Cabal Seminar, Volume IV Edited by A. S. Kechris, B. L¨owe, J. R. Steel c 2014, Association for Symbolic Logic 1 2 PAUL B. LARSON We let ! denote the set of natural numbers 0; 1; 2;::: ; for brevity we will 0044 often refer to the members of this set as \integers". Given sets X and Y , 0045 XY denotes the set of functions from X to Y . The Baire space is the 0046 space !!, with the product topology. The Baire space is homeomorphic to 0047 the space of irrational real numbers (see [Mos09, p. 9], for instance), and 0048 we will often refer to its members as \reals" (though in various contexts 0049 the Cantor space !2, the set of subsets of ! (}(!)) and the set of infinite 0050 subsets of ! ([!]!) are all referred to as \the reals"). 0051 ! Given A ⊆ !, we let G!(A) denote the game of perfect information of 0052 length ! in which the two players collaborate to define an element f of !! 0053 (with player I choosing f(0), player II choosing f(1), player I choosing f(2), 0054 and so on), with player I winning a run of the game if and only if f is an 0055 element of A. A game of this type is called an integer game, and the set 0056 A is called the payoff set.A strategy in such a game for player player I 0057 (player II) is a function Σ with domain the set of sequences of integers of 0058 even (odd) length such that for each a 2 dom(Σ), Σ(a) is in !. A run of 0059 the game (partial or complete) is said to be according to a strategy Σ for 0060 player player I (player II) if every initial segment of the run of odd (nonzero 0061 even) length is of the form a_hΣ(a)i for some sequence a. A strategy Σ 0062 for player player I (player II) is a winning strategy if every complete run 0063 of the game according to Σ is in (out of) A. We say that a set A ⊆ !! 0064 is determined (or the corresponding game G!(A) is determined) if there 0065 exists a winning strategy for one of the players. These notions generalize 0066 naturally for games in which players play objects other than integers (for 0067 instance, real games, in which they play elements of !!) or games which 0068 run for more than ! many rounds (in which case player player I typically 0069 plays at limit stages). 0070 The study of determinacy axioms concerns games whose determinacy is 0071 neither proved nor refuted by the Zermelo{Fraenkel axioms ZF (without the 0072 Axiom of Choice). Typically such games are infinite. Axioms stating that 0073 infinite games of various types are determined were studied by Stanis law 0074 Mazur, Stefan Banach and Ulam in the late 1920s and early 1930s; were 0075 reintroduced by David Gale and Frank Stewart [GS53] in the 1950s and 0076 again by Jan Mycielski and Hugo Steinhaus [MS62] in the early 1960s; gained 0077 interest with the work of David Blackwell [Bla67] and Robert Solovay in 0078 the late 1960s; and attained increasing importance in the 1970s and 1980s, 0079 finally coming to a central position in contemporary set theory. 0080 Mycielski and Steinhaus introduced the Axiom of Determinacy (AD), 0081 ! which asserts the determinacy of G!(A) for all A ⊆ !. Work of Banach in 0082 the 1930s shows that AD implies that all sets of reals satisfy the property of 0083 Baire. In the 1960s, Mycielski and Stanislaw Swierczkowski´ proved that AD 0084 implies that all sets of reals are Lebesgue measurable, and Mycielski showed 0085 0086 A BRIEF HISTORY OF DETERMINACY 3 that AD implies countable choice for reals. Together, these results show that 0087 determinacy provides a natural context for certain areas of mathematics, 0088 notably analysis, free of the paradoxes induced by the Axiom of Choice. 0089 Unaware of the work of Banach, Gale and Stewart [GS53] had shown that 0090 AD contradicts ZFC. However, their proof used a wellordering of the reals 0091 given by the Axiom of Choice, and therefore did not give a nondetermined 0092 game of this type with definable payoff set. Starting with Banach's work, 0093 many simply definable payoff sets were shown to induce determined games, 0094 culminating in D. Anthony Martin's celebrated 1974 result [Mar75] that all 0095 games with Borel payoff set are determined. This result came after Martin 0096 had used measurable cardinals to prove the determinacy of games whose 0097 payoff set is an analytic sets of reals. 0098 The study of determinacy gained interest from two theorems in 1967, the 0099 first due to Solovay and the second to Blackwell. Solovay proved that under 0100 AD, the first uncountable cardinal !1 is a measurable cardinal, setting off a 0101 study of strong Ramsey properties on the ordinals implied by determinacy 0102 axioms. Blackwell used open determinacy (proved by Gale and Stewart) to 0103 reprove a classical theorem of Kazimierz Kuratowski. This also led to the 0104 application, by John Addison, Martin, Yiannis Moschovakis and others, of 0105 stronger determinacy axioms to produce structural properties for definable 0106 1 sets of reals. These axioms included the determinacy of ∆n sets of reals, for 0107 n ≥ 2, statements which would not be proved consistente relative to large 0108 cardinals until the 1980s. 0109 The large cardinal hierarchy was developed over the same period, and 0110 came to be seen as a method for calibrating consistency strength. In the 0111 1 1970s, various special cases of ∆2 determinacy were located on this scale, 0112 in terms of the large cardinalse needed to prove them. Determining the 0113 consistency (relative to large cardinals) of forms of determinacy at the 0114 1 level of ∆2 and beyond would take the introduction of new large cardinal 0115 concepts.e Martin (in 1978) and W. Hugh Woodin (in 1984) would prove Π1- 0116 2 L(R) e 0117 determinacy and AD respectively, using hypotheses near the very top 0118 of the large cardinal hierarchy. In a dramatic development, the hypotheses 0119 for these results would be significantly reduced through work of Woodin, 0120 Martin and John Steel. The initial impetus for this development was a 0121 seminal result of Matthew Foreman, Menachem Magidor and Saharon Shelah 0122 which showed, assuming the existence of a supercompact cardinal, that 0123 there exists a generic elementary embedding with well-founded range and 0124 critical point !1. Combined with work of Woodin, this yielded the Lebesgue 0125 measurability of all sets in the inner model L(R) from this hypothesis. 0126 Shelah and Woodin would reduce the hypothesis for this result further, to 0127 the assumption that there exist infinitely many Woodin cardinals below a 0128 measurable cardinal. 0129 4 PAUL B. LARSON Woodin cardinals would turn out to be the central large cardinal concept 0130 for the study of determinacy. Through the study of tree representations for 0131 1 sets of reals, Martin and Steel would show that Πn+1-determinacy follows 0132 from the existence of n Woodin cardinals belowe a measurable cardinal, 0133 and that this hypothesis was not sufficient to prove stronger determinacy 0134 results for the projective hierarchy.
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