Benacerraf famously pointed out that there are infinitely many isomorphic mathematical structures that could equally well be the referrent of “the natural numbers”. Mathematicians are generally not bothered by this underdetermination of the concept of “the natural numbers”, precisely because the different structures are isomorphic.
What is more worrying are the infinitely many elementarily inequivalent mathematical structures that, it seems, could count as “the natural numbers”. (This becomes particularly worrying given that we’ve learned from Goedel that these structures give rise to different notions of provability.)
(I suppose this is a kind of instance of the Kripke-Wittgenstein puzzles.)
In response, here is a start of a story. Those claims about the natural numbers that differ in truth value between models are vague. We can then understand this vagueness epistemically or in some more beefy way.
An attractive second step is to understand it epistemically, and then say that God connects us up with his preferred class of equivalent models of the naturals.
A problem with the vagueness solution is that it probably then becomes vague whether the axioms of set theory are consistent. For we know from Goedel's Second Incompleteness Theorem that there is a model of set theory according to which the axioms of set theory are inconsistent. But, probably, the axioms of set theory are true and hence consistent. So, on the vagueness story, it is vaguely true that the axioms of set theory are consistent.
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