Reconstruction of non-$\aleph_0$-categorical theories

TL;DR

This paper extends the correspondence between theories and automorphism groups from countably categorical theories to all complete theories in classical logic and some in continuous logic, broadening the scope of model-theoretic classification.

Contribution

It generalizes the known correspondence to a wider class of theories, including non-$oldsymbol{eth}_0$-categorical theories, in both classical and continuous logic.

Findings

Established a generalized correspondence for classical logic theories.

Extended the framework to certain continuous logic theories.

Includes all $oldsymbol{eth}_0$-categorical theories in the generalization.

Abstract

We generalise the correspondence between $\aleph 0$-categorical theories and their automorphism groups to arbitrary complete theories in classical logic, and to some theories (including, in particular, all $\aleph 0$-categorical ones) in continuous logic.