Model theory of the regularity and reflection schemes

This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme, and its strengthening REF(L) (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here L is a language with a distinguished linear order <, and REF(L) consists of the universal closure of formulas of the form ∃x∀y1 < x · · · ∀yn < x φ(y1, · · ·, yn) ↔ φ(y1, · · ·, yn), where φ(y1, · · ·, yn) is an L-formula, φ is the 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: Theorem. The following five conditions are equivalent for a complete first order theory T in a countable language L with a distinguished linear order : (1) Some model of T has an elementary end extension with a first new element. (2) T ` REF(L). (3) T has an ω1-like model that continuously embeds ω1. (4) For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ. (5) For some regular uncountable cardinal κ, T has a κ-like model M that has an elementary extension in which the supremum of M exists. Moreover, if κ is a regular cardinal satisfying κ = κ, then each of the above conditions is equivalent to: (6) T has a κ-like model that continuously embeds a stationary subset of κ. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ 2000 Mathematics Subject Classification: Primary 03C64, 03C80; Secondary 03C62. Date: March 19, 2008.