Lieber Herr Bernays! Lieber Herr Gödel! Gödel on Finitism, Constructivity, and Hilbert's Program

Abstract

The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel's collected works devoted to his letters of (primarily) scientific, philosophical and historical interest.It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post-Kantian philosophy, and contains Gödel's thoughts on many topics that are not expressed elsewhere.In addition, it testifies to their life-long warm personal relationship.I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol.IV of Gödel's Collected Works, pp.41-79. 1 My purpose here is to focus on only one group of interrelated topics from these exchanges, namely the light that ittogether with assorted published and unpublished articles and lectures by Gödelthrows on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program. 2 In that connection, this piece has an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career.1 The five volumes of Gödel's Collected Works (1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003) are referred to below, respectively, as CW I, II, III, IV and V. CW I consists of the publications 1929-1936, CW II of the publications 1938-1974, CW III of unpublished essays and letters, CW IV of correspondence A-G, and CW V of correspondence H-Z.References to individual items by Gödel follow the system of these volumes, which are either of the form Gödel 19xx or of the form *Gödel 19xx with possible further addition of a letter in the case of multiple publications within a given year; the former are from CW I or CW II, while the latter are from CW III.Thus, for example, Gödel 1931 is the famous incompleteness paper, while Gödel 1931c is a review that Gödel wrote of an article by Hilbert, both in CW I; Gödel °1933o is notes for a lecture, "The present situation in the foundations of mathematics," to be found in CW III.Pagination is by reference to these volumes, e.g.Gödel 1931, CW I, p. 181, or simply, CW I, p. 181.In the case of correspondence, reference is by letter number and/or date within a given body of correspondence, as e.g.(Gödel to Bernays) letter #56, or equivalently 2 Dec. 1965, under Bernays in CW IV.When an item in question was originally written in German, my quotation from it is taken from the facing English translation.Finally, reference