On equivalence relations second order definable over $H(\kappa)$

Saharon Shelah ¹   ; Pauli Vaisanen ²

¹ Institute of Mathematics The Hebrew University Jerusalem, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A.
² Department of Mathematics P.O. Box 4 00014 University of Helsinki, Finland

Fundamenta Mathematicae, Tome 174 (2002) no. 1, pp. 1-21

Résumé

Let $\kappa$ be an uncountable regular cardinal. Call an equivalence relation on functions from $\kappa$ into $2$ second order definable over $H(\kappa)$ if there exists a second order sentence $\phi$ and a parameter $P \subseteq H(\kappa)$ such that functions $f$ and $g$ from $\kappa$ into $2$ are equivalent iff the structure $\langle H(\kappa), \in, P, f, g \rangle$ satisfies $\phi$. The possible numbers of equivalence classes of second order definable equivalence relations include all the nonzero cardinals at most $\kappa^+$. Additionally, the possibilities are closed under unions and products of at most $\kappa$ cardinals. We prove that these are the only restrictions: Assuming that ${\rm{GCH}}$ holds and $\lambda$ is a cardinal with $\lambda^\kappa = \lambda$, there exists a generic extension where all the cardinals are preserved, there are no new subsets of cardinality $ \kappa$, $2^\kappa = \lambda$, and for all cardinals $\mu$, the number of equivalence classes of some second order definable equivalence relation on functions from $\kappa$ into $2$ is $\mu$ iff $\mu$ is in ${\mit\Omega}$, where ${\mit\Omega}$ is any prearranged subset of $\lambda$ such that $0 \not\in {\mit\Omega}$, ${\mit\Omega}$ contains all the nonzero cardinals $\leq \kappa^+$, and ${\mit\Omega}$ is closed under unions and products of at most $\kappa$ cardinals.

Détail

DOI : 10.4064/fm174-1-1

Keywords: kappa uncountable regular cardinal call equivalence relation functions kappa second order definable kappa there exists second order sentence phi parameter subseteq kappa functions kappa equivalent structure langle kappa rangle satisfies phi possible numbers equivalence classes second order definable equivalence relations include nonzero cardinals kappa additionally possibilities closed under unions products kappa cardinals prove these only restrictions assuming gch holds lambda cardinal lambda kappa lambda there exists generic extension where cardinals preserved there subsets cardinality kappa kappa lambda cardinals number equivalence classes second order definable equivalence relation functions kappa mit omega where mit omega prearranged subset lambda mit omega mit omega contains nonzero cardinals leq kappa mit omega closed under unions products kappa cardinals