Variants of Łoś's Theorem

TL;DR

The work investigates Łoś's theorem in choiceless set theory, introducing several variants and proving their equivalence to the standard theorem in $ZF$, thereby clarifying the role of Choice in ultrapower phenomena. It characterizes bounded and Σ-elementary forms of Łoś-type statements via $U$-$AC_I$ and analyzes when ultrapower embeddings yield elementary equivalence, showing that certain seemingly weaker conditions are in fact equivalent to Łoś's theorem. The paper then develops a generic framework using forcing to form generic ultrapowers and ultraproducts, establishing a generic Łoś's theorem under Choice and demonstrating that, in $ZF$, AC is equivalent to the full strength of the generic versions. Overall, the results illuminate the exact axiomatic strength necessary for Łoś-type transfer principles and reveal deep connections between ultraproduct behavior and the Axiom of Choice across both classical and generic settings.

Abstract

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

Theorems & Definitions (10)

• proof : Proof of (1) $\iff$ (2)
• proof : Proof of (1) $\iff$ (3)
• proof
• proof : Proof of (1) $\iff$ (4)
• proof
• proof
• proof
• proof : Proof of (1) $\iff$ (5) $\iff$ (6)
• proof
• proof