Extensions of First-Order Logic

Abstract

We have seen that the structure n of natural numbers cannot be characterized in the first-order language corresponding to n. The same situation holds for the field of real numbers and the class of torsion groups. As we showed in Chapter VII, one can, at least in principle, overcome this weakness by a set-theoretical formulation: One introduces a system of axioms for set theory in a first-order language, e.g. ZFC, which is sufficient for mathematics, and then, in this system, carries out the arguments which are required, say, for a definition and characterization of n. However, this approach necessitates an explicit use of set theory to an extent not usual in ordinary mathematical practice.