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CONSTRUCTING WADGE CLASSES

Part of: Connections with other structures, applications Set theory

Published online by Cambridge University Press:  26 January 2022

RAPHAËL CARROY ,
ANDREA MEDINI  and
SANDRA MÜLLER Open the ORCID record for SANDRA MÜLLER [Opens in a new window]
RAPHAËL CARROY
Affiliation:
DIPARTIMENTO DI MATEMATICA “GIUSEPPE PEANO” PALAZZO CAMPANA, UNIVERSITÁ DI TORINO VIA CARLO ALBERTO 10 10123 TURIN, ITALY E-mail: raphael.carroy@unito.it
ANDREA MEDINI
Affiliation:
INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE TECHNISCHE UNIVERSITÄT WIEN WIEDNER HAUPTSTRASSE 8-10/104 1040 VIENNA, AUSTRIA E-mail: andrea.medini@tuwien.ac.at E-mail: sandra.mueller@tuwien.ac.at URL: http://muellersandra.github.io
SANDRA MÜLLER
Affiliation:
INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE TECHNISCHE UNIVERSITÄT WIEN WIEDNER HAUPTSTRASSE 8-10/104 1040 VIENNA, AUSTRIA E-mail: andrea.medini@tuwien.ac.at E-mail: sandra.mueller@tuwien.ac.at URL: http://muellersandra.github.io
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Abstract

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.

Keywords

Wadge theorylevelexpansionseparated differencesdeterminacyHausdorff operationω-ary Boolean operation

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 03E60: Determinacy principles

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Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 2 , June 2022 , pp. 207 - 257
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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