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Class of Coanalytic (= U,')4 Sets Based on the (Known) Determinateness of Open (= Iol) Sets

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Class of Coanalytic (= U,')4 Sets Based on the (Known) Determinateness of Open (= Iol) Sets 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) <f(y).) The effect of the prewellordering theorem below is to force the behavior of the higher levels in the projective hierarchy back to that of the lo' and [lo' sets and thus to make an inner-quantifier rather than an outer-quantifier notation more convenient in presenting our results. Accordingly, we let Ek' (epsilon-i-k) and Tk' (upsilon-1ik) be the classes of sets definable in (k + l)-function-quantifier forms on a recursive matrix with inner quanti- fier existential and universal, respectively. Thus Ea 1 = 2;2,, T2k' = H2k', E2k + 11 = HU + 11 T2U + 11 = 22k + 1', and An' = 2;fln lln = E flnTn'. The classes boldface Ek', TO' are defined similarly. A subset C of w X 9 is Ekl-universal for 9C iff C is Ek' and for every Ek' subset A of 9C there is some n E Xso that x E A *-* (n,x) E C. It is easy to see that if C is Ek'-uni- versal for Qw X $C, then for every Ekl subset A of $X there is some n E co and some a E low so that x C A *-* (n,a,x) E C. For each X and each k choose a fixed Ekl subset Ck(Q ) of X X $ which is EkL-universal for $C. An Ek -code for a subset A of $C is any n C w such that x E A +-* (n,x) e Ck(C). An EJ' code for a subset A of $C is any (n,a) C w X low such that t E A -* (nac,x) E Ck(ww X $). A Akl-code (Akl-code) for A is any (xy) such that x and y are Ekl-codes (Ekl- codes) for A and $C - A, respectively. (2) THE PREWELLORDERING THEOREM.-Let $C be anyfinite product of copies of wand 'w, let 1 be an even number inw, and let kbelor 1+1. If every All subset of Ax is determinate, then there is a subset Wk of w X $C, Ekl-universal for SC, and a pre- wellordering <k of Wk whose initial segments are uniformly Akl, i.e., for which there is a primitive recursive f such that for each w C Wk, f(w) is a Akl-code for {Z: ZCEWk&Z<kW}. Proof: By induction on 1. For 1 = 0, let Wo (= tz: 3mR(zm)} for re- cursive R) be 240-universal for 9C (Wo C woX 9C), and put z <ow ÷- z,w E Wo & the least m such that R(z,m) < the least n such that R(w,n). Induction step. Case I: k (= 1) is even. Let z E Wk +-*3 a (za) C Wk- and let z < k W (- zW C Wk & the < k-l-least (z, a) < k-1 the <k-l-least (w43). Case II: k (= 1 + 1) is odd. Let z E Wk - Va (za) C Wk-1 and let Z <k W "-ZW £ Wk & (3 strategy r for II)(V strategy orfor I) [(z, (a * T)o) <k-1 (WY (a * 7)1) ]- Downloaded by guest on October 3, 2021 710 MATHEMATICS: ADDISON AND MOSCHOVAKIS PROC. N. A. S. 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 <k are uniformly Ak' follows easily by induction using the determinateness hypothesis on All. That <k is a prewellordering is immediate in the basis, easy in Case I of the induction step, and follows by an elementary argument from Al determinateness in Case II of the induction step. (3) Logical Analysis of the Argument.-The argument above is intimately involved with quantifiers on ordered pairs of variables or binary quantifiers, including the standard "decomposable" quantifiers 33, ]V, V3, and VV (where 33ab = 3a 3b, etc.). The conditional 3Va,b 4- V3b,a 4) is a basic principle of logic; on the other hand, many central ideas in mathematics revolve about a special circumstance where the not logically valid converse conditional V3'b,aa 3Va,b 4 holds. For example, let < be a prewellordering of 9C; then {(x,y): 3v'a,/ (x,a) < (yS)} equals (x,y): V 3, a (x,a) < (y,3)} and prewellorders St. (i) This can be viewed as a key combinatorial fact behind the odd-to-even steps of the theorem. The dual principle with 3V13,a and V3a,# (which corresponds to taking suprema instead of infima) fails, suggesting a search for other quantifiers TV and 3V squeezing so closely together between 3V and V3 that they more often coincide, i.e., so that for more S 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.
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