# Benacerraf's identification problem and PA categoricity

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 ?

• The Zermelo vs. von Neumann example doesn’t have to do with categoricity, since they are isomorphic models. Jun 2 at 14:56
• Oh yeah, my bad. Then from what I understand my question is not relevant in the end ? Jun 2 at 15:08
• I was inspired by your question to read the entry on mathematical structuralism in the Stanford Encyclopedia of Philosophy, plato.stanford.edu/entries/structuralism-mathematics. The issue you raise comes up repeatedly, with way too many subtleties to get into here. Jun 2 at 15:16
• I’m finding it hard to imagine an answer to your question that wouldn’t be mostly the writers’ opinions. Maybe try the philosophy stackexchange? They have a tag philosophy-of-mathematics. Jun 2 at 15:21
• Oh great yeah, I'll try that. Though after some thinking, both ordinals defined above are also models of second order PA, which is categorical, so I'd say that I missed the point Jun 2 at 19:39