On Second-order Characterizability

TL;DR

This paper extends Shelah's Main Gap to second order logic, characterizing models of certain theories using infinitary logic, under specific stability and cardinal assumptions.

Contribution

It establishes a precise link between model-theoretic stability properties and second order characterizability of models, extending classical results.

Findings

All large models of certain theories are second order characterizable under stability conditions.

Characterizability depends on the theory being shallow superstable with NDOP and NOTOP.

Results vary with cardinal arithmetic assumptions and semantics of second order logic.

Abstract

We investigate the extent of second order characterizable structures by extending Shelah's Main Gap dichotomy to second order logic. For this end we consider a countable complete first order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second order logic with Henkin semantics.