ABSTRACT
For some time now, it has been a virtual constant of the literature on Hilbert’s Program (HP) to maintain that Gijdel’s work demonstrates its untenability. The “demonstration” typically given is one which proceeds from Giidel’s Second Incompleteness Theorem (G2) and the claim that HP requires the sort of consistency proofs that it (i.e. G2) rules out. However, more recently (cf. Kreisel 1976; Prawitz 1981; Simpson 1988; Smorynski 1977, 1985, 1988) it has become increasingly common to claim that Gbdel’s First
Incompleteness Theorem (Gl) affords a refutation of HP, and that this refutation is at least as good as (and perhaps even better than) that based on G2. Thus, one finds such claims as that “the First Incompleteness Theorem . . . effectively kills Hilbert’s programme; the Second Incompleteness Theorem is merely a refinement” (cf. the introduction to Smorynski 1988), and “it was the Firstand not the Second Incompleteness Theorem that killed Hilbert’s Programme” (cf. 1985: 10).
