 The project of axiomatizing mathematical structures with second-order sentences was so successful in the first quarter of the twentieth century that Carnap even proposed that every mathematical structure is determined, up to isomorphism, by its second-order theory. There are trivial cardinality reasons why this could not possibly hold for all models of size \(2^\omega\) and larger. There are simply not enough second-order theories in comparison with the number of non-isomorphic models. However, the proposal of Carnap is trivially true for finite models, and not entirely unreasonable for countable models. In fact Ajtai (1979) showed that it is consistent with ZFC, provided ZFC itself is consistent, that any two countable structures in a finite vocabulary that satisfy the same second-order sentences are isomorphic. Ajtai showed that it is also consistent with ZFC, provided ZFC itself is consistent, that there are two countable structures in a finite vocabulary that satisfy the same second-order sentences without being isomorphic. Thus Carnap’s conjecture (for countable models) is independent of ZFC.