Regular Ultrapowers at Regular Cardinals

TL;DR

This paper proves that the finite square principle square^fin_{lambda,D} holds at regular cardinals for certain regular filters without assuming GCH, advancing understanding of model theoretic properties in set theory.

Contribution

It establishes in ZFC that doubly^+ regular filters satisfy square^fin_{lambda,D} at regular cardinals, providing new solutions to open problems in model theory.

Findings

square^fin_{lambda,D} holds for doubly^+ regular filters at regular cardinals in ZFC

Provides new positive answers to open problems in model theory

Extends previous results assuming GCH to ZFC without additional assumptions

Abstract

In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{lambda,D} holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in the book "Model Theory" by Chang and Keisler.