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

### Volume 174 / 2002

#### Abstract

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.