I will report on a couple of projects investigating the "structural complexity" of models of first-order theories with foundational character. In a project with Antonio Montalbán, we performed a Scott analysis of models of Peano arithmetic and showed, in layperson's terms, that nonstandard models of arithmetic cannot be simple. More formally, our main result shows that every completion of PA has models of Scott rank \(\alpha\) for every infinite Scott rank \(\alpha\). However, the standard model is the unique model of PA with finite Scott rank.

In other work with Uri Andrews and Steffen Lempp, we give a characterization of first-order theories that have a \(\pmb \Pi^0_\omega\) complete set of models. As a corollary, we obtain that all sequential theories have a \(\Pi^0_\omega\) complete set of models.

At last, I will talk about a new project with Darius Kalociński and Mateusz Łełyk that aims to generalize and improve the results obtained with Montalbán.