Cardinalities of Ultraproducts of Finite Sets in ZF + DC
Jacob Kowalczyk
TL;DR
The paper shows that in ZF+DC it is consistent to have an ultrafilter $U$ on $\omega$ such that two infinite ultraproducts of finite sets have the same cardinality exactly when the ultrafilter limit $0<\lim_U |A_n|/|B_n|<\infty$, and it does so in the Solovay-model extension $W[U]$ with $U$ generic over $[\omega]^\omega$. It provides a precise equivalence: in this setting, two ultraproducts are isomorphic if and only if they share the same cardinality, with the main technical work delivered via a forcing construction $\mathbb{P}_{F,f,g}$ that preserves ultrafilters and supports continuous/Lipschitz reading of names. The approach combines order-structure analysis of ultraproducts, a detailed forcing analysis over $[\omega]^\omega$, and Shelah–Zapletal results to control reals and submeasures, yielding a thorough characterization of when ultraproducts of finite sets are isomorphic or have equal cardinality. This advances the understanding of choiceless models (notably $W[U]$) for Ramsey-like ultrafilters and their impact on the cardinality behavior of ultraproducts.
Abstract
It is consistent with ZF + DC that there exists an ultrafilter $U$ on $ω$ 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 $[ω]^ω$-generic.