Gödel on Intuition and on Hilbert’s finitism

There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, PA; but starting in the Dialectica paper [Gödel, 1958], he expressed in writing the view that 0 is an upper bound on the finitist ordinals, and that, therefore, the consistency of PA, cannot be finitistically proved. Although I do not understand the “therefore” (see §8 below), here was a genuine change in his views. But I am unaware of any writings in which he retracted this new position. Incidentally, the analysis he gives of what should count as a finitist ordinal in [Gödel, 1958; Gödel, 1972] should in fact lead to the bound ω, the ordinal of primitive recursive arithmetic, PRA. (Again, see §8 below.) The one area of double-reversal in the development of his ideas concerns the view, expressed in letters to Bernays in the early 196o’s, about whether or not 0 is the least upper bound on the finitist ordinals. (See §1 below.) ∗I have had valuable comments from Michael Friedman, Wilfried Sieg, Daniel Sutherlandand members of the editorial board on earlier versions of this paper, and I thank all of them.