Elements of affine model theory

TL;DR

This paper introduces affine continuous logic, a fragment of continuous logic avoiding certain connectives, leading to new model-theoretic tools and properties such as ultramean ultraproducts and convex type spaces.

Contribution

It develops the foundations of affine continuous model theory, including new constructions like ultramean ultraproducts and explores properties of affine structures.

Findings

Ultramean ultraproducts replace ultrafilters with finitely additive measures.

Compact structures with multiple elements have proper elementary extensions.

Type spaces are compact convex sets with extreme types playing a key role.

Abstract

By Lindstr\"{o}m's theorems, the expressive power of first order logic (and similarly continuous logic) is not strengthened without losing some interesting property. Weakening it, is however less harmless and has been payed attention by some authors. Affine continuous logic is the fragment of continuous logic obtained by avoiding the connectives $\wedge,\vee$. This reduction leads to the affinization of most basic tools and technics of continuous logic such as the ultraproduct construction, compactness theorem, type, saturation etc. The affine variant of the ultraproduct construction is the ultramean construction where ultrafilters are replaced with maximal finitely additive probability measures. A consequence of this relaxation is that compact structures with at least two elements have now proper elementary extensions. In particular, they have non-categorical theories in the new setting. Thus, a model theoretic framework for study of such structures is provided. A more remarkable aspect of this logic is that the type spaces are compact convex sets. The extreme types then play a crucial role in the study of affine theories. In this text, we present the foundations of affine continuous model theory.