The emergence of first-order logic

To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on first-order logic, it was a radical and unprecedented proposal. The radical nature of what Skolem proposed resulted, above all, from its effect on the notion of categoricity. During the 1870s, as part of what became known as the arithmetization of analysis, Cantor and Dedekind characterized the set n of real numbers (up to isomorphism) and thereby found a categorical axiomatization for n. Likewise, during the 1880s Dedekind and Peano categorically axiomatized the set M of natural numbers by means of the Peano Postulates. Yet in 1923, when Skolem insisted that set theory be treated within first-order logic, he knew (by the recently discovered Lowenheim-Skolem Theorem) that in first-order logic neither set theory nor the real numbers could be given a categorical axiomatization, since each would have both a countable model and an uncountable model. A decade later, Skolem (1933, 1934) also succeeded in proving, by the construction of a countable nonstandard model, that the Peano Postulates do not uniquely characterize the natural numbers within first-order logic. The Upward Lowenheim-Skolem Theorem of Tarski, the first version of which was published as an appendix to (Skolem 1934), made it clear that no axiom system having an infinite model is categorical in first-order logic. The aim of the present article is to describe how first-order logic gradually emerged from the rest of logic and then became accepted by mathematical logicians as the proper basis for mathematics—despite the opposition of Zermelo and others. Consequently, I have pointed out where