Cardinalities of Ultraproducts of Finite Sets in ZF + DC

2025/03/01 by Jacob Kowalczyk, Kowalczyk, Jacob
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2503.00310

openalex publication_date 2025/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Related

• A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis
• One Repeat Point Gives a Closed, Unbounded Ultrafilter on \ω₁
• The combinatorics of reasonable ultrafilters
• On topological properties of ultraproducts of finite sets
• Some combinatorial properties of Ultimate L and V