Model Theory of the Reflection Scheme

This paper develops the model theory of ordered structures that satisfy the analogue, REF (L), of the reflection principle of ZermeloFraenkel 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 · · · ∀y1 < 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 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.