Hilbert's Programme and Gödel's Theorems

Abstract

In this paper, we attempt to show that a weak version of Hilberťs metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates. Defining first 4T proves the consistency of a theory S indirectly in one step, we subsequently prove (i) PA proves its own consistency indirectly in one step and sketch the proof for (ii) If S is a recursively enumerable extension of (QF-IA), S proves its own consistency indirectly in one step. The formalizations of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. 1. Goders Incompleteness Theorems and Consistency Proofs The main goal of Hilberťs foundational project was to vindicate all of classical mathematics by means of a finitist metamathematical consistency proof'. Hilbert considered classical mathematics to be the paradigm of unassailable truth and believed that finitist means, as conceived by him, were absolutely reliable.1 For decades it has been widely held that Godel's Second Incompleteness Theorem put an end to Hilberťs original proof-theoretic programme. On the face of it, this view seems plausible: if we succeeded in carrying out a consistency proof for all of mathematics in metamathematics, mathematics would prove its own consistency, given that metamathematics is only a small fragment of mathematics in its entirety. Yet the very possibility that mathematics proves its own consistency is ruled out by Godel's Second