Cardinalities of Ultraproducts of Finite Sets in ZF + DC

Abstract

It is consistent with ZF + DC that there exists an ultrafilter $U$ on $\omega$ such that two infinite ultraproducts of finite sets, $\prod A_n / U$ and $\prod B_n / U$, have the same cardinality if and only if $0 < \lim_U |A_n|/|B_n| < \infty$. In particular, this holds in $W[U]$, where $W$ is the Solovay Model and $U$ is $[\omega]^\omega$-generic.