Gödel’s Second Incompleteness Theorem

Abstract

One of the main goals of research in the foundations of mathematics in the 1920’s was to find a consistency proof for number theory. Not that the consistency of number theory was considered to be very dubious, but the consistency of set theory was considered a partially open question and finding a convincing consistency proof for number theory seemed a natural first step in that direction. The possibility of such a proof does not seem too unlikely since we can formalize the statement of consistency in a rather simple form: (x) — Bew(x, ‘0 = 1’). It is true that one would have to use some number theoretic principles or their equivalents to prove a statement of this form, but there was reason to hope that the proof could be carried out in a relatively ‘small’ subsystem, for example, using induction only for quantifier free formulas. Gödel showed, however, in 1931 that no such proof was possible.