Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki.
Parts of the paper are above my head, but I understand it well enough for my own amateurish purposes - apart from one sentence.
The context is a quotation from Dieudonne's A Panorama of Pure Mathematics (1982):
The first axiomatic treatments (Dedekind-Peano arithmetic, Hilbert Euclid geometry) dealt with univalent theories , i.e. theories which are entirely determined by their complete system of axioms, unlike the theory of groups.
Mathias's commentary on this passage ends with the following sentence, which baffles me:
In saying that Peano arithmetic is univalent, Bourbaki probably has in mind some second-order characterisation of the standard model of arithmetic, which is, of course, to beg the question.
I can only imagine that he means that even the second-order axioms cannot stand on their own, because any such version of the Peano Axioms has two hidden prerequisites:
Reference to a particular (but unmentioned) version of set theory.
Reference to a "standard model of arithmetic", whose existence and uniqueness is silently taken for granted (thus "begging the question" in a simpler sense than item 1).
But neither of these ideas is really clear to me, nor do I have any idea whether the author is alluding to either of them, both of them, or neither.
(When one is confused, it is hard to explain the precise way in which one is confused!)
If the meaning of the quoted sentence is not obvious to others, I'll consider asking the author himself about it via e-mail, but I'm marginally less nervous about posting a question here - and perhaps an answer to the question here will also interest others.
The paper is about Bourbaki's blind spot in relation to developments in logic since 1929. As my own blind spots are incomparably more severe than any of Bourbaki's, it is entirely possible that I will fail to understand a perfectly good explanation of the meaning of the above sentence!
But I will be well enough satisfied by an answer that reduces my current bafflement, by one sentence in an otherwise intelligible paper, to a more familiar perplexity about mathematics itself.