Lately I have been interested in mathematical philosophy, and especially structuralism.
In this setting, Benacerraf's famous paper is a classic and works as follows: Take the Zermelo ordinals ($x \to \{x\}$) and the Von Neumann ordinals ($x \to x \cup \{x\}$), those two are models of First order Peano Arithmetic $PA$, but they do not give the same answer to the sentence $1 \in 3$ (respectively false and true).
Benacerraf uses this to then conclude that when talking about numbers, one cannot do anything appart from talking of a given number in relation with the others, as a so-called structure.
On the other hand, $PA$ is not categorical: there exists multiple classes of models up to isomorphism.
The question is thus the following: is Benacerraf's identification problem specific to the fact that $PA$ is not categorical ?