Full Idea
If we did not know that the second-order axioms characterise the natural numbers up to isomorphism, we should have no reason to suppose, a priori, that first-order Peano Arithmetic should be complete.
Gist of Idea
It is only 2nd-order isomorphism which suggested first-order PA completeness
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)
Book Reference
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.412