Theory of series and Gödel's theorems sections 17–22 [1948e]

Hans Reichenbach

pp. 409-428

in: Maria Reichenbach, Robert S. Cohen (eds), Selected writings 1909–1953, Berlin, Springer, 1978

Abstract

Since the time of their first publication, Gödel's results have been considered as a serious blow against Hilbert's program of proving the consistency of mathematics. If the consistency of a system cannot be proved within the system itself but only in the metalanguage, a proof of consistency seems valueless because we do not know whether the language within which the proof is given is consistent. It was Hilbert's program to secure the reliability of ordinary mathematical proofs by giving a proof of the consistency of the mathematical language. If it turns out with Gödel's second theorem that this proof can only be given in another language whose consistency is not proved, this seems to show that Hilbert's program cannot be carried through because it leads into an infinite regress.