SOME CONSEQUENCES OF THE AXIOM OF DEFINABLE DETERMINA TENESS*
BY J. W. ADDISON AND YIANNIs N. MOSCHOvAKIS
UNIVERSITY OF CALIFORNIA, BERKELEY, AND UNIVERSITY OF CALIFORNIA, LOS ANGELES Communicated by David Blkckwell, January 11, 1968
Introduction.-With each sut set A of the set "w of functions from X to W we associate the following game: Players I and II successively choose natural numbers with I starting, and if the resulting infinite sequence is in A, then I wins; otherwise II wins. The set A is determinate' iff either I or II has a winning strategy. It can be shown in Zermelo-Fraenkel set theory with the axiom of choice (ZFC) that nondeterminate sets exist;l however, no definable set has been proved in ZFC to be nondeterminate.2 Recently, Blackwell3 found an elegant proof of the reduction principle for the class of coanalytic (= U,')4 sets based on the (known) determinateness of open (= Iol) sets. In this note we show that the hypothesis of determinateness for projective (= 2;, for some k) sets yields consequences about the projective hier- archy on 'o and the analytical (= 2,' for some k) hierarchies on cW and 'W: in particular, the separation and reduction problems at all levels of these hier- archies are settled and hierarchies are constructed for the classes Ak' (all k > 0) and Ak' (all odd k). One of the oldest puzzles in the theory of definability has been the strange "flip-flop" behavior of separation principles in the projective hieratchy-since 1935 they have been known to hold for Ho,0111, and H12' sets and to fail for oe,1 I111, and Z2' sets.5 In 1962, one of us (J. W. A.) found a clue to the hidden forces responsible for this behavior in the discovery of a proof of the strong separation theorem for Z11 sets6 which was a "vectorization" of the corresponding theorem for Hol sets; he proposed' that this vectorization be extended to 121 and 1131 sets as a method of "smoking out" a new axiom of set theory.8 Blackwell's proof suggested to him a way of developing the vectorization technique to carry the first separation principle from s-l (where true) to tI+ki1, for all k.9 Upon inspection of this argument, the sought-for axiom was apparent. When one of us (Y. N. M.) saw this work, he recognized that the determinateness of projective sets implies the prewellordering theorem below (for k > 0), which then settles separation, reduction, and several other definability questions at all levels of the hierarchies. D. A. Martin, also inspired by Blackwell's proof, independently found several of the results below, including the separation and reduction theorems.10 Steinhauss" has urged for several years the adoption as an axiom of set theory of "the axiom of determinateness," according to which all subsets of '0w are determinate. But since this axiom contradicts the axiom of choice, it has not found wide support. We would propose instead the axiom of definable deter- minateness, according to which every subset of 'w ordinal definable from elements of Wa is determinate. For several reasons, including its spontaneous emergence as 708 Downloaded by guest on October 3, 2021 VOL. 59,1968 MATHEMATICS: ADDISON AND MOSCHOVAKIS 709
a key to the rich and orderly consequences given below, we foresee an increas- ingly important role for it in the future of set theory. (1) Terminology.-The universes whose subsets we study are all product spaces t= X1 X . . . X Xi, where each Xi is either w or low. We let x, y, z, w be variables over SC, let k, 1, m, n be variables over co, and let a, fl, -y be variables over '%o. If x is an i-tuple and y a j-tuple, by (xy) we mean the (i + j)-tuple whose first i components are the components of x and whose last j components are the components of y. Similarly, if S = X. X ... X Xiand% = Y, X... X Yithen9C X = X, X ... XXiX Y X ... X Yi. Strategies o, r are functions from the set Sq (= I -w:n E w }) of finite sequences of natural numbers to co. By a * r (read: sigma-clash-tau) we mean the play resulting by pitting a against r-formally (a * r)o(n) = a((a * T)iln), (a * r)i(n) = T((T * T)oln+i), (where (a)o(n) = a(2n) and (a),(n) = a(2n + 1)). A prewellordering of a set S is a reflexive, transitive, connected, well-founded relation with field S-from being a wellordering it lacks only antisymmetry. (A prewellordering _ of S is determined by a unique order-preserving function f from S to an ordinal: x < y <- f(x) Note that in defining Wk and < k for $C we use the induction hypothesis for 9 X co. That W1 is Ek-universal for $t is immediate and that the initial segments of 3V13,a S(al3) -* 3V/,a S(a,3) V3a,g> S(a,3) -- V3a,3 S(a,3).12 (#) Now the "game quantifiers" 3V and V3 defined by 3V3,a S(a,13) -> (3 -rEzw)(VEw) S((a * r)o, (a * r)1) and its dual satisfy (#a) and satisfy (# *-) just if S is codeferminate. Corre- spondingly, under a determinateness assumption we have a dual of (i), namely {(x,y): 3Vi3,a (x,a) < (y43)} equals (x,y): V3 a, 3 (x,a) < (y,43) } and prewellorders $. (ii) This can be viewed as a key combinatorial fact behind the odd-to-even steps of the theorem."3 (4) Consequences of the Prewellordering Theorem.-Proofs of the results below from the prewellordering theorem follow the pattern of those given by one of us (Y. N. M.), first for the hierarchy on A21 in Rogers14 and later in more abstract settings. " The notation and hypotheses here are those of the theorem. (I) (Reduction for Ekl.) Let A, B be Ekl subsets of $t. There exist Ekl subsets A', B' of OC such that (i) A' C A) B' C B, (ii) A U B = A' U B', and (iii) A,' n B' = A. (Similarly for Ekl.) (II) (First separation for Tkl.) Let A, B be Tk1 subsets of $X and assume that Downloaded by guest on October 3, 2021 VOL. 59, 1968 MATHEMATICS: ADDISON AND MOSCHOVAKIS 71 A n B = A. There exists a A,' subset C of D such that A C C andB f C = A. (Similarly for Tk1 and Akl.) This fails for Ek', even if we replace Ak' by Ak'. (III) (Boundedness for Ekl subsets of xi, when k > 0.) Let k > 0, assume that 9C is a product of copies of co, let A be an Ekl subset of OC, and let n be such that x EA -(n,x) C Wk. Then A is AkI ifffor some w C Wk and all x C A, (n,x) 3 Blackwell, D., these PROCEEDINGS, 58, 1836-1837 (1967). 4 A subset of co or loco is Lk1 (Hlkl) iff it is explicitly definable by a formula with k + 1 alternat- ing function quantifiers applied to a recursive matrix with outer quantifier existential (universal). (We include here, via their representing sets, functions with domain and range either of these spaces.) A set or function is Ak1 iff it is both Zk1 and ils'. Boldface, A, 1, A indicate that the matrix can be recursive in an arbitrary element of loc. Cf. Addison, J. W., Fund. Math., 46, 123-135 (1958), noting that Zol = 20 easily. 6 A discussion of this puzzle (including an argument that the classes Zol, Hol should be de- fined as indicated in the preceding footnote) is given in "Current problems in descriptive set theory," Axiomatic Set Theory: Proceedings of the 1967 Summer Institute at Los Angeles, (American Mathematical Society, in press). 6 This says that disjoint 1l1 sets can be separated by a Borel set. 7 Presented at the International Symposium on the Theory of Models, Berkeley (1963), but, because of space limitations, not included in the article as published in the proceedings of the symposium. 8 That a new axiom would be needed followed from the known consistency with ZFC of the denial of the separation principle for 1131. See Addison, J. W., loc. cit., and Fund. Math., 46, 337-357 (1959). 9 Also the reduction principle from WIl (where true) to Xs+11. l0 Martin, D. A., "The axiom of determinateness and reduction principles and the analyt- ical hierarchy," Bull. Am. Math. Soc., in press. 11 Mycielski, J., and H. Steinhaus, Bull. Acad. Polon. Sci., S6ries Math. Astr. Phys. 10, 1-3 (1962). 12 A very short, direct proof that determinateness of Ak' sets and the first separation principle for Ilk' sets imply the first separation principle for 24+11 sets can be based on (#). 13 A proof of (a refinement of) the prewellordering theorem can be given by finding an Tk1 relation (b) Z Z `Ok W ()V3#,a(Zit) ` Z