A survey of proof theory

Abstract

One might fairly say that the very meaning of our subject has changed since Hilbert introduced it under the name Beweistheorie (it was meant to be the principal tool for formulating Hubert's general conception of how to analyze mathematical reasoning). Specifically, the roles of the two principal elements of proof theory, namely the intuitive proofs accepted and the formal proofs (or derivations) studied, have turned out to be quite different from what Hilbert thought. In his view the hard work had been done in the discovery of formalization, and what remained was the study of certain given formal systems.