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 ?

  • 2
    $\begingroup$ The Zermelo vs. von Neumann example doesn’t have to do with categoricity, since they are isomorphic models. $\endgroup$ Jun 2 at 14:56
  • $\begingroup$ Oh yeah, my bad. Then from what I understand my question is not relevant in the end ? $\endgroup$
    – vigoux
    Jun 2 at 15:08
  • $\begingroup$ 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. $\endgroup$ Jun 2 at 15:16
  • $\begingroup$ 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. $\endgroup$ Jun 2 at 15:21
  • $\begingroup$ 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 $\endgroup$
    – vigoux
    Jun 2 at 19:39


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