Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic. [...] The original (second-order) Peano axioms... have ONLY ONE MODEL, up to isomorphism." (my emphasis)
Would this be an example of the incompleteness of the first-order Peano Axioms?
Update 5 months later
Though I don't fully understand Noah's thoughtful answer and am at a loss to formulate any more follow-up question(s), as helpfully suggested, I did find his comment that Gödel's Incompleteness Theorem has "nothing whatsoever" to do with 2nd order Peano arithmetic to be very interesting. On this basis, I have accepted his answer.