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Wadge Hierarchy and Veblen Hierarchy Part I: Borel Sets of Finite Rank Author(s): J. Duparc Source: The Journal of Symbolic Logic, Vol. 66, No. 1 (Mar., 2001), pp. 56-86 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2694911 Accessed: 04/01/2010 12:38 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org THE JOURNAL OF SYMBOLICLOGIC Voluime 66. Number 1, March 2001 WADGE HIERARCHY AND VEBLEN HIERARCHY PART I: BOREL SETS OF FINITE RANK J. DUPARC Abstract. We consider Borel sets of finite rank A C A'" where cardinality of A is less than some uncountable regular cardinal K5. We obtain a "normal form" of A, by finding a Borel set Q, such that A and Q continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum. to multiplication by a countable ordinal, and to ordinal exponentiation of base K;*under the map which sends every Borel set A of finite rank to its Wadge degree. ?1. Introduction. We recall a tree T on A is a subset of ACIDthat is closed under subsequences i.e., satisfies Vu E T Vv (v C u - v E T).When T is some tree, we write [T] to denote the set of its infinite branches: {a E AW: Vn E co (a [ n) E T}. By a Borel set we mean a Borel subset A of A'A,where AA (which contains at least two elements to avoid trivialities) is some well-ordered alphabet of cardinality less than some uncountable regular cardinal K, and the topology on A' is the product topology of the discrete topology on AA: A is open if and only if there exists some U C A"<-"such that A= {ua oAC, : u U}. As soon as it was introduced, the class of Borel sets (the closure by comple- mentation and countable union of the class of open sets) was set up in a hi- erarchy by Baire: V is the class of open sets, and by induction for 4 > 1, Lo; = {UnEOA,1 -An E U(<;V} where -A,] stands for the complement of the set An. The Baire hierarchy of height co, was later subdivided by Hausdorff and Kuratowski into K wil levels (they worked with K = wl) by introducing the sepa- rated difference operation[. But an ultimate refinementwas to come with Wadge's work based on the notion of inverse images of continuous functions. A continuousreduction of A to B is a continuous function f: AC-0 ACBsuch that A - f' B. It is denoted A < w B; intuitively it means that A is less complicated than B -strictly if there is no reduction of B to A; which is denoted <w. If A < w B < w A, in other words, A and B continuously reduce to each other, then A is said Wadge-equivalentto B, denoted _ w. Given two Borel sets A C A(", B C AWB,it is a consequence of Borel determinacy [5] that either A <? B or -B <? A. In case K = wal, Martin proved that <? is wellfounded on Borel sets [1]. This result still holds in case K > col and allows to define the Wadge degree of a Borel set by induction: dw(0) = dw(-0) = 0 and dt?A = sup{dw (B) + 1: B <w A}. Wadge's lemma, as a consequence of Borel Received January 30, 1997; revised May 20, 1999. ? 2001, Association for Symbolic Logic 0022-4812/01/6601 -0002/$4. I0 56 WADGEHIERARCHY AND VEBLENHIERARCHY 57 determinacy, states that two Borel sets have same Wadge degree if and only if they are Wadge equivalent or one is Wadge equivalent to the complement of the other [1]. We recall the definition of a self-dual set: a set A that satisfies A =jw -A. Willing to concentrate on non self dual sets we need to consider another definition of the Wadge degree that put apart the self dual ones. DEFINITION 1. o dj(0) = d,(- 0) 1 o dA = sup{dw(B) + 1: B non self dual A B <w A} (for either A self dual or not, A >a 0)1. Thus the Wadge degree of A is a measure of the complexity of A, just as its Borel rank or its level in the Hausdorff-Kuratowski hierarchy. But the Wadge degree is a much finer measure. Computing the height of the Wadge hierarchy requires a hierarchy of fast increasing ordinal functions: the Veblen hierarchy [7]. DEFINITION 2. The Veblenhierarchy (V: )<,,, of continuousfunctions from i+ \ {O} - iii+ -. {O}is definedby: Vois thefunction that enumeratesordinals of cofinality X or 1 that are closed under ordinal addition, namely Vo(1) 1, Vo(a + 1) VoJ(a) X, Vo((A) = iA when cof((A)= X, Vo((A)= &A+'when co < cof((A) < K. For > 0, V, is thefunction that enumeratesordinals of cofinality K or 1 that are closed undereach function V(for any 4 < I. The Wadge degree of Borel sets of finite rank ranges from 1 to sup V0 o ... o Vo(2) n Coa n (denoted 'CEosince it is the first fixpoint of the exponentiation of base K) while the Borel rank only runs through integers. And things get even much worse for Borel sets of infinite ranks: the Wadge degree of sets of Borel rank co climbs up to V1(2) (characterized also as the K"th fixpoint of the exponentiation of base K, therefore denoted fej. And when dealing with all wol Borel ranks the Wadge hierarchy involves sup,,(,, V, (2) distinct degrees. As a consequence, it is quite a problem to provide for each Wadge degree af a description of a Borel set Q(ca) of degree precisely a. A solution for K = col was given essentially by Wadge [9], and confirmed after ten years of work culminating with Louveau and Saint-Raymond [2], [3]. We have given a new solution in the case af < V1(2), that is for AO sets (and we are writing an extension to all Borel ranks). This solution is canonical because its characteristic feature is to give a Borel definition of Q(a) whose is isomorphic to the canonical description of the ordinal Oaitself. Let us quote Wadge in the conclusion of his Ph. D. Thesis: One goes from the standard definition of the Wadge degree to ours by: o if A is non self dual, only three possibilities occur: (a) df A = 2.n with n integer, then d'cA = n + 1 (b) d' A = A + 2n with n integer and A limit (necessarily cof(A) = IKholds), then d'A = A + n (c) d WA = A+2n +I1 with n integer and A limit (necessarilycof(A) < IKholds), then dfA = A+n o if A is self dual, also there is only three cases: (a) d' A = 2.n + 1 with n integer, then d,?UA= n + 1 (b) diA = A + 2n with n integer and A limit (necessarily cof(X) < I%holds), then d?A = A + n (c) divA = A + 2n + 1 with n integer andA limit (necessarily cof(A)K = %holds), thenOA d A = X+n+ 1 58 J. DUPARC "It would also be very interesting to know of some "inductive" definition of the degrees, which allows a given degree to be constructed from those below it. This might, for example, allow properties of subsets of 0%oto be proved by induction on the degree of the set involved" ([9], p. 323) Wadge worked in the Baire space co;w he defined set theoretic counterparts of the sum and < wol-multiplicationso that on the basis of the empty set (which degree is 1) he was able to generate sets of degree al for any al < c<' but couldn't go any further. In particular, with his method, he couldn't reach a p-complete set (which degree is precisely <' ). We hope this work is an attempt to respond to the wish formulated by Wadge. ?2. Away with self-dual sets, the conciliating Wadge hierarchy. We say a set A is conciliating if it is a subset of A<', where AA is of cardinality less than K. And given two conciliating sets A, B, we define the conciliatory version C(A, B) of the Wadge game W(A, B), which is this game when the sequences a, ,8 produced by the players may be finite.

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