By 'ambiguous', I meant to suggest the existence of multiple non-isomorphic models.

The thing that puzzled cousin_it was that the axioms of first-order Peano arithmetic can be satisfied by non-standard models of arithmetic, and that there is no way to add additional first-order axioms to exclude these unwanted models.

The solution I proposed was to use a second-order axiom of induction - working with properties (i.e.sets) of numbers rather than the first-order predicates over numbers. This approach successfully excludes all the non-standard models of arithmetic, leaving only the desired standard model of cardinality aleph nought. But it extends the domain of discourse from simply numbers to both numbers and sets of numbers. And now we are left with the ambiguity of what model of sets of numbers we want to use.

It is mildly amusing that in the case of arithmetic, the unwanted non-standard models were all too big, but in the case of set theory, people seem to prefer to think of the large models as standard and dismiss Godel's constructive set theory as an aberation.