Introduction to Bernays Text No. 6, “Appendix to Hilbert’s Lecture ‘The Foundations of Mathematics’”

Hilbert’s 1928 article, to which this piece is an appendix, was presented in July 1927 to the Hamburg Mathematical Seminar. Hilbert had first introduced his program for the foundations of mathematics in the same venue in a series of talks in 1921 (Hilbert, 1922). In the 1927 talk, he presented a mature version of his program, including technical details of the axiomatization of mathematics based on the ε-calculus, Hilbert’s ε-substitution method, as well as a discussion of the finitary standpoint. Bernays’s appendix concerns the ε-substitution method. The ε-calculus is a version of first-order logic which contains the ε-operator instead of quantifiers. Given a formula A(a) with free variable a, the ε-operator can be used to form a term εaA(a) (an “ε-functional”). Intuitively (if A contains no free variables other than a), it provides a witness for A(a), if one exists, and an arbitrary object otherwise. The ε-calculus is the quantifier-free first-order calculus with identity in a language including the ε-operator, and the additional transfinite axiom, A(a)→ A(εaA(a)).