Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman¹

¹ Kurt Gödel Research Center University of Vienna 1090 Wien, Austria

Fundamenta Mathematicae, Tome 227 (2014) no. 2, pp. 179-186.

Résumé

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 $0^\#$.

Détail

DOI : 10.4064/fm227-2-4

Keywords: silvers fundamental dichotomy classical theory borel reducibility states borel even co analytic equivalence relation uncountably many classes has perfect set classes natural generalisation generalised baire space kappa kappa regular uncountable kappa fails dels even kappa borel equivalence relations here silvers dichotomy kappa borel equivalence relations kappa kappa uncountable regular kappa however consistent gch assuming existence