Model theory of the regularity and reflection schemes

Abstract

This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF $${(\mathcal{L})}$$ (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here $${\mathcal{L}}$$ is a language with a distinguished linear order <, and REF $${(\mathcal {L})}$$ consists of formulas of the form $$\exists x \forall y_{1} < x \ldots \forall y_{n} < x \varphi (y_{1},\ldots ,y_{n})\leftrightarrow \varphi^{ < x}(y_1, \ldots ,y_n),$$ where φ is an $${\mathcal{L}}$$ -formula, φ <x is the $${\mathcal{L}}$$ -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include: The following five conditions are equivalent for a complete first order theory T in a countable language $${\mathcal{L}}$$ with a distinguished linear order: Moreover, if κ is a regular cardinal satisfying κ = κ <κ , then each of the above conditions is equivalent to: