Categoricity-like properties in the first order realm

By classical results of Dedekind and Zermelo, Peano Arithmetic and Zermelo-Fraenkel set theory exhibit categorical features when formulated within second order logic. However, as shown by Skolem, categoricity disappears from the first order formulations–commonly known as PA and ZF– of these theories. I will focus on certain categoricity-like properties (including the notions of tightness, solidity, and internal categoricity) exhibited by a distinguished class of first order theories that include PA and ZF. The aforementioned categoricity-like properties are all formulated within the framework of (relative) interpretability theory. As we shall see, viewing foundational first order theories through the lens of interpretability theory sheds light on what is special about canonical foundational first order theories.