Extreme types and extremal models

Abstract

In the affine fragment of continuous logic, type spaces are compact convex sets. I study some model theoretic properties of extreme types. It is proved that every complete theory $T$ has an extremal model, i.e. a model which realizes only extreme types. Extremal models form an elementary class in the full continuous logic sense if and only if the set of extreme $n$-types is closed in $S_n(T)$ for each $n$. Also, some applications are given in the special cases where the theory has a compact or first order model.

Theorems & Definitions (61)

• Theorem 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• Theorem 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• proof