Keisler's Theorem and Cardinal Invariants

TL;DR

This paper explores variants of Keisler's isomorphism theorem, linking them to cardinal invariants and set-theoretic hypotheses, and analyzes their implications for models of different sizes.

Contribution

It characterizes saturation hypotheses stronger than Keisler's theorem using set-theoretic assumptions and examines their consistency and implications.

Findings

Keisler's theorem for size  models implies  = 

Keisler's theorem for size  models implies  

Keisler's theorem for size  fails in the random model

Abstract

We consider several variants of Keisler's isomorphism theorem. We separate these variants by showing implications between them and cardinal invariants hypotheses. We characterize saturation hypotheses that are stronger than Keisler's theorem with respect to models of size $\aleph_1$ and $\aleph_0$ by $\mathrm{CH}$ and $\operatorname{cov}(\mathsf{meager}) = \mathfrak{c} \land 2^{<\mathfrak{c}} = \mathfrak{c}$ respectively. We prove that Keisler's theorem for models of size $\aleph_1$ and $\aleph_0$ implies $\mathfrak{b} = \aleph_1$ and $\operatorname{cov}(\mathsf{null}) \le \mathfrak{d}$ respectively. As a consequence, Keisler's theorem for models of size $\aleph_0$ fails in the random model. We also show that for Keisler's theorem for models of size $\aleph_1$ to hold it is not necessary that $\operatorname{cov}(\mathsf{meager})$ equals $\mathfrak{c}$.