Spaces of Types in Positive Model Theory

TL;DR

This paper develops a new topological framework for positive model theory using Stone duality, enabling the extension of classical results from traditional model theory to the positive setting.

Contribution

It introduces a novel space of types in positive model theory based on Stone duality, bridging the gap with classical first-order model theory.

Findings

The space of types in positive model theory mirrors the classical Stone space.

Generalizes classical results on countable models to positive model theory.

Establishes a topological foundation for positive model theory.

Abstract

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.