What rubs me the wrong way about Arnon Avron’s What Numbers Really Cannot Be and What They Plausibly Are, aside from the set theory triumphalism, is that it refuses to accept that the phrase “natural numbers” is genuinely ambiguous. In Avron’s scenario, the two children disagree on the number of solutions to an equation, which should clue him in that they’re using the same natural-language terms with truly different mathematical meanings! The question, structurally, is “when defining addition on the natural numbers (defined as Peano), should the starting element be an additive identity?” He acts as though structuralism requires the two definitions of “natural numbers” to be equivalent, as though a different definition of addition doesn’t give a different structure!
On the other hand, in the Benacerraf paper he’s criticizing, the two set-theoretic notions of natural number are equivalent on any legitimate mathematical question about the natural numbers, but disagree when you take them overly literally and start asking questions like “is 3 a member of 17”. Benacerraf’s argument derives its force both from the disagreement between the two models (which Avron claims is irrelevant because one is simply correct) and from this being a nonsense question that nobody would seriously ask. If the numbers really were sets this would be a perfectly reasonable question, so they can’t be.
The more serious component of Avron’s argument, in my opinion, is the claim that, if we can identify “the natural numbers containing 0” and “the natural numbers not containing 0” as distinct, then 0 must be an independently existing mathematical object which may or may not be part of a structure. I see where he’s coming from but I would say that the numerals are defined with respect to the structure of addition. If you’re just using the natural numbers as indices for sequences it literally doesn’t matter, you can start at 22 if you want, you just need a progression. But 0 always denotes the additive identity if you have addition.
There’s a definite flaw in some structuralisms where they’re too focused on the minimal definition of a structure on its own rather than its relationships with other structures. The “starting element and successor” definition of Peano nats leaves out most of the structure of the natural numbers as used — you can define all the usual operations and relations on top of it, but those definitions are additional structure, and they control how the structure relates to others (e.g. embedding of the nats into the integers, rationals, reals, cardinalities, etc.)
He quotes Resnik as saying that “Intuitively, one would like to say that the natural numbers with addition and multiplication exemplify the same structure as the natural numbers with addition, multiplication, and less-than” — I would say Resnik is simply wrong here. We can put different less-thans on N(+x), and for any ordering we can “forget” it to recover N(+x). This is not, and shouldn’t be, an equivalence relation. One or two orderings being particularly well-behaved doesn’t change this.