The volumes of G¨ odel’s collected papers under review consist almost entirely of a rich selection of his philosophical/scientific correspondence, including English translations face-to-face with the originals when the latter are in German. The residue consists of correspondence with editors (more amusing than of any scientific value) and five letters from G¨ odel to his mother, in which explains to her his religious views. The term “selection” is strongly operative here: The editors state the total number of items of personal (...) and scientific correspondence in G¨ odel’s Nachlass to be around thirty-five hundred. The correspondence selected involves fifty correspondents, and the editors list the most prominent of these: Paul Bernays, William Boone, Rudolph Carnap. Paul Cohen, Burton Dreben, Jacques Herbrand, Arend Heyting, Karl Menger, Ernest Nagel, Emil Post, Abraham Robinson, Alfred Tarski, Stanislaw Ulam, John von Neumann, Hao Wang, and Ernest Zermelo. The correspondence is arranged alphebetically, with A-G in Volume IV. The imbalance results from the disproportionate size of the Bernays correrspondence: 85 letters are included (almost all of them), spanning 234 pages) including the face-to-face originals and translations). Each volume contains a calendar of all the items included in the volume together with separate calendars listing all known correspondence (whether included or not) with the major correspondents (seven in Volume IV and ten in Volume V). Let me recommend to the reader the review of these same volumes by Paolo Mancosu in the Notre Dame Journal of Formal Logic 45 (2004):109- 125. This essay very nicely describes much of the correspondence in terms of broad themes relating, especially, to the incompleteness theorems—their origins in G¨ odel’s thought, their reception, their impact on Hilbert’s program. (shrink)
There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between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, G¨ odel’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 [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *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, P A; but starting in the Dialectica paper.. (shrink)
The last section of “Lecture at Zilsel’s” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s ﬁrst version of his consistency proof for P A , reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), ﬁll in the details (with some corrections) of Godel's reformulation, and discuss the relation between the two proofs.
Restricted to ﬁrst-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of ﬁrst-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x : A.F (x) is understood as disjoint union, are the projections, and these do not preserve ﬁrstorderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting’s system.
This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a matter (...) of recognizing trutlis wliich had previoasly been unrecognized, but of extending the domain of what is true. (shrink)