Consistency of the Silver dichotomy in generalised Baire space

Volume 227 / 2014

Sy-David Friedman Fundamenta Mathematicae 227 (2014), 179-186 MSC: 03E15, 03E35, 03E45, 03E55. DOI: 10.4064/fm227-2-4

Abstract

Silver's fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $\kappa ^\kappa $ for a regular uncountable $\kappa $ fails in Gödel's $L$, even for $\kappa $-Borel equivalence relations. We show here that Silver's dichotomy for $\kappa $-Borel equivalence relations in $\kappa ^\kappa $ for uncountable regular $\kappa $ is however consistent (with GCH), assuming the existence of $0^\#$.

Authors

  • Sy-David FriedmanKurt Gödel Research Center
    University of Vienna
    1090 Wien, Austria
    e-mail

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