The dilemma has led to skepticism about whether mathematical theories can coherently be taken to refer to mathematical objects at all and to attempts to claim that the subject matter of mathematics needs no independent characterization. No such drastic reactions are necessary, since the dilemma is only apparent. Skolem and his heirs have missed a third possibility, one that solves the problem posed. To compare two models of, for example, Peano arithmetic with full schematic induction, consider a structure that includes both models. The full induction schemas will have instances in the combined language that are not in the language of either model taken separately, and those instances suffice to show that the two models are isomorphic.
Analogous results hold for standard axiomatizations of set theory, the real numbers, and so forth.
I argue first, that the full induction schema is a codification of our intentions in defining the natural numbers, second, that the possibility of combining two models of Peano arithmetic with full schematic induction into one structure cannot coherently be denied so long as one is willing to raise the question whether our conception of the natural numbers suitably characterizes them, and finally, that the notion of a full schema is not subject to the problems posed for higher-order and infinitary logics. Related proposals by Parsons, McGee, and Shapiro will be discussed, as well as criticisms by Field of the present proposal.