Borel Sets of Finite Rank - MIMUW
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].
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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 v E T).When T is some tree, we subsequences i.e., satisfies Vu E T Vv (v C u 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"
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 v E T).When T is some tree, we subsequences i.e., satisfies Vu E T Vv (v C u 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"
