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SETS of ORDINALS CONSTRUCTIBLE from TREES and the THIRD VICTORIA DELFINO PROBLEM Howards. Becker(!) and Alexander S. Kechris(Z)

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SETS of ORDINALS CONSTRUCTIBLE from TREES and the THIRD VICTORIA DELFINO PROBLEM Howards. Becker(!) and Alexander S. Kechris(Z) Contemporary Mathematics Volume 31, 1984 SETS OF ORDINALS CONSTRUCTIBLE FROM TREES AND THE THIRD VICTORIA DELFINO PROBLEM HowardS. Becker(!) and Alexander S. Kechris(z) A very important part of the structure theory of z~ sets of reals is based on their close interrelationship with the GBdel constructible universe L. The fundamental fact underlying this connection is the theorem of Shoenfield which asserts that every z1 set of reals is Souslin over L. This means that given any z1 subset of the reals (= ww in this paper), there is a tree T on w x A (A some ordinal, which can be taken to be ~l here) such that TeL and A= p[T] = {a € ww: :iff € Aw vn(arn, trn) € T}. In 1971 Moschovakis [14] introduced the notion of scale and showed that under the hypothesis of Projective Determinacy (PD) the pointclasses TI~n+l' l z2n+2 for all n .:._ 0 have the Sea 1e Property. To take for instance the case of TI~n+l this asserts that given any TI~n+l set As ww there is a sequence of norms ~m= A+ Ordinals (which we are always assuming here to be regular, i.e., having as range an initial segment of the ordinals) such that the follow­ ing properties hold: (i) If a0,a1, ... eA, ai+a and ~m(ai)+Am (inthesensethat '.Pm(ai) = Am for all large enough i), then a e A and ~m(a) 2_ Am. (ii) The relations (on m, a, S): a 2_~* 8 ~a € A 1\ (S ~ Avq:>m(o:) 2.~m{S)), * m 1 a:< S ~a € A 1\ (S ~A v ~(a)<~ (S)) are TI n+l' Cl'm m m 2 Such a sequence {cpm} is called a (regular) TI~n+ 1 -saale on A. Now any scale ~ = ~m} on a set A (i.e. a sequence of norms satisfying (i) above) gives rise to a tree T= T{A,~ on w x A, where A= sup{range(~ )}, given by m m 1980 Mathematics Subject Classification. 03D70,03El5,03E45,03Eb0,04Al5. 1 { )Research partially supported by NSF Grant MCS 82-11328 2 ( )F.esearch partially supported by NSF Grant MCS 81-17804 © 1984 American Mathematical Society 0271-4132/84 $1.00 + $.25 P"r page 13 14 BECKER AND KECHRIS It is then easy to check the key fact that A= p[T], so that T verifies that A is Souslin. Moreover {~m} can be canonically recovered from T, since for a e A, (~0 (a),~ 1 (a), ... ) = the leftmost branch of T(a), where T(a) = {(n0, ... ,nn): (ann+l),(n0, ... ,nn)) e T 1\ new}. Thus T is a canonical embodiment of the scale {~m} as (essentially) a set of ordinals. w 1 Now, always assuming PO, let Pc:=w be a complete I12n+ 1 set of reals 1 p. and {~m} =~ a regular TI2n+1-scale on LPt T2n+ 1 (p,q)) be the associated 1 1 tree. This is a tree on w x ~2n+l' where ~2n+l = sup{t_; : t_; is the length of a 1 ~2n+l prewellordering of ww}. If we consider the model L[T 2n+ 1 (p,q;-} J. 1 of sets constructible from T2n+l(p,q)), then by [9], 9A- 2 every ~ 2 n+ 2 set 1 is Souslin over this model. Thus the ~ 2 n+ 2 sets have the same relationship to this model as the ~~ sets have with respPct to L, and it is natural to consider L[T2n+l(p,q))] as a higher level analog of the counstructible uni- verse. Indeed Moschovakis (see [9], p. 42) showed (in ZF + DC) that L[T1(p,q))] = L. In particular L[T1(p,q))] is independent of the choice of the IT~-complete set 1 P and the TI 1-scale ~ on p. The question whether the same invariance holds for 2n+l > 1 arose almost immediately after the introduction of these models and Moschovakis conjectured in 1971 that this was indeed the case. This problem of the invariance of L[T2n+l(p,q))] was eventually formulated as the 3rd Victoria Delfino Problem (see [10], p. 280) for the first open case 2n + 1 = 3. (It has been already proved in [5] that L[T2n+l(p,q))] n ww is invariant for all n.) We provide in this paper an affirmative solution of this problem for all 2n + 1. THIRD VICTORIA DELFINO PROBLEM 15 Theorem 1. Assume Projective Determinacy when n ~ 1. Let p be a complete 1 1 rr 2n+l set of reals and 'i> = {cpm} a regular rr2n+ 1-scale on p. Then the model L[T 2 n+l(p,~] is independent of the choice of p, ~· Thus one can legitimately refer for each n to the model without any embellishments. Since the tree T2 n+l(p,~) is a natural set theoretic manifestation of the scale ~ on p this result shows also a strong uniqueness property inherent in 1 the concept of rr2n+1-scale. Actually the way the theorem is proved establishes quite a bit more. To explain the stronger statement let us go back to one of the immediate conse­ quences of the Souslin representation of ~~ sets by trees in L, namely the fact that every ~~ subset of w is constructible. Following work of Solovay (see [16]) it was shown in Kechris-Moschovakis [8] that every ~~ subset of - 1 ~, - ~l is also constructible. (Apparently H. Friedman has also independently proved that theorem without publishing it--see also [4]). As usual, when we say that a set X<;;. ~l is ~~ we mean that given a I1~-norm q>: p ontq ~~ on a complete rr~ set p' the set X* = {w e p : cp(w) e X} 1 is ~ 2 . It is not hard to ch~ck that this notion is intrinsic, i.e., independent of the choice of p, 1:1>· Following the work of Harrington-Kechris [5] it has become possible to generalize this notion to higher levels, assuming PD. We call a subset X of 1 ~~n+ 1 ~~n+2 if given a IT~n+ ,-norm q> : P ontq ~~n+ 1 on a complete rr 2n+l set p the set X* = {w e p : cp(w) e X} is Z~n+ 2 . (It is worth recalling here that the length ot any such norm is 1 exactly ~ 2 n+l' see [12], p. 216.) From [5], p. 125 it follows that this notion 16 BECKER AND KECHRIS is again independent of P, ~· We can now generalize the result about Z~ subsets of ~ 1 and L as fo 11 ows. Theorem 2. Assume Projective Determinacy when n > 1. Let p be a complete 1 - 1 - n2n+l set of reals and ~ = {q,m} a regular n2n+1-scale on p. Then every 1 1 z 2n+2 subset of ~ 2 n+l is in L[T 2 n+l(p,~)]. Now it has been verified (see [5], p. 131 or [12], p. 164) that any T 2 n+i(P,~ (viewed by some simple coning of tuples as a subset of ~~n+l) is indeed Z~n+2 , thus Theorem 2 immediately implies Theorem l. Moschovakis [12] has introduced (under PD) the models H2n+l' which are defined as L[P2n+2], where p2n+2 <:::. w X ~~n+l is universal z~n+2" It follows immediately from Theorem 2 and the remarks following it that L[T2n+l] = H2n+l" Martin showed in 1976 (see an exposition in [6], p. 67) that if the tree T 2 n+l(p,~ has a special property, namely it is homogeneous with fully (i.e., 1 ~ 2 n+l-) additive measures, then indeed L[T 2 n+l(p,~] satisfies Theorem 2 and thus also H2n+l = L[T 2 n+l(?,~] for such a T 2 n+l(p,~. (For the definition of homogeneous tree see [6]). He also succeeded in 1982 (unpublished) in proving the existence of such trees. Thus Theorems 1 and 2 have been known to be equivalent. Martin's proof of the special case of Theorem 2 for the homogeneous trees used an idea for a game which has found several applications in the study of the models H2n+l' and is also being used in our proof as well. The models L[T2 n+l(p,~] and H2n+l are defined and studied in Section 8G of [12]. Other work on these models can be found in the following references: [1], [2], [3], [5], [6], [9], and [15]. These models have a very interesting and useful structure theory, and they are related to many topics in descriptive set theory. Our demonstration of Theorem 2 is actually quite general and applies to many other pointclasses "resembling IT~." In fact our main result, from which everything else is an immediate corollary (when combined with already known theorems) is a very simple and general constructibility theorem for the tree associated with a scale, which is just a theorem of ZF + DC. We formulate and prove this result in§ 1. In §2 we derive as corollaries Theorem 2 and its generalizations to move general pointclasses, and in §3 we discuss some analogs of Theorem 2 for the even levels of the projective hierarchy. Finally in §4 we prove the n1 analog of Kleene's theorem that IT~ equals inductive on the structure (w,<), solving an open problem raised in Kechris-Martin [7]. THIRD VICTORIA DELFINO PROBLEM 17 §1. The Main Theorem. We will follow below basically the notation and termi­ nology of Moschovakis [12], except for calling ww the set of reaZs and denoting it by IR.
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