On the Model Theory of Second-Order Objects

TL;DR

This paper develops a model-theoretic framework for second-order objects, generalizing existing notions and applying it to logic and categoricity transfer in existential second-order logic.

Contribution

It introduces abstract elementary team categories, generalizes abstract elementary classes, and applies the framework to Lindström's theorem and categoricity transfer.

Findings

Established that abstract elementary team categories are accessible categories.

Proved a version of Lindström's theorem for the logic $\

Demonstrated downwards and upwards categoricity transfer results for existential second-order logic.

Abstract

Motivated by team semantics and existential second-order logic, we develop a model-theoretic framework for studying second-order objects such as sets and relations. We introduce a notion of abstract elementary team categories that generalizes the standard notion of abstract elementary class, and show that it is an example of an accessible category. We apply our framework to show that the logic $\mathsf{FOT}$ introduced by Kontinen and Yang satisfies a version of Lindstr\"om's Theorem. Finally, we consider the problem of transferring categoricity between different cardinalities for complete theories in existential second-order logic (or independence logic) and prove both a downwards and an upwards categoricity transfer result.