Variants of {\L}o\'s's Theorem

TL;DR

This paper explores variants of Lo's theorem in a choice-free setting, demonstrating their equivalence to the original theorem despite appearing weaker, thus deepening understanding of its foundational aspects.

Contribution

It introduces and analyzes variants of Lo's theorem in a choiceless context, proving their equivalence to the original theorem.

Findings

Variants are weaker but equivalent to Lo's theorem

Establishes foundational understanding in a choice-free setting

Deepens theoretical insights into Lo's theorem

Abstract

We study \L o\'s's theorem in a choiceless context. We introduce some variants of \L o\'s's theorem. These variants seem weaker than \L o\'s's theorem, but we prove that these are equivalent to \L o\'s's theorem.