Advanced Class Logic

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\omega$-categorical theories always have a core companion, which is also $\omega$-categorical.

In the first part of this talk, we show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved when moving to the core companion of a complete theory.

In the second part of this talk, we study the notion of core interpretability, which arises by taking the core companions of structures interpretable in a given structure. We show that there are structures which are core interpretable but not interpretable in $(\mathbb{N};=)$ or $(\mathbb{Q};<)$. We conjecture that the class of structures which are core interpretable in $(\mathbb{N};=)$ equals the class of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We present some partial results in this direction, including the answer a question of Walsberg.

This is joint work with Manuel Bodirsky and Bertalan Bodor.