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arXiv · 2503.00310

Cardinalities of Ultraproducts of Finite Sets in ZF + DC

Jacob Kowalczyk
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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.

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Jacob Kowalczyk. 2025-03-01. Cardinalities of Ultraproducts of Finite Sets in ZF + DC. https://arxiv.org/abs/2503.00310

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