Beschreibung:

GÖDEL, Kurt. Autograph letter signed in German to Thoralf Skolem ("Sehr geehrter Herr Professor"). Vienna, November 1, 1931; boxed. 2½ pages, on single folded sheet. Provenance : Given by Skolem to the mathematician Ernst Straus (1922-83), who co-authored papers with Einstein, and was also a friend and collaborator of the mathematician Paul Erdös (1913-96). "Thank you very much for your friendly card and the offprints of your papers. I am concurrently sending you reprints of my two essays regarding the fundamentals; several passages therein relate to the results that you obtained. For example, my paper entitled "Formally unsolvable theorems" also provides a contribution to the set-theoretical relativism held by you. This is because, as shown by a simple calculation*, the consistent, but not \Kw\k-consistent systems examined on page 190 indicate that there exist realizations for axiom systems in set theory in which certain quantities that are infinite from an absolute standpoint are "finite" within the system. In other words, that which you showed for the term "uncountable quantity" also holds true for the term "finite quantity," namely that it cannot be axiomatically characterized (expressed by a number). Since you made a suggestion in your paper "On several set functions in arithmetic" which points in this direction, I think you will find this particularly interesting." An historic autograph letter written shortly after Gödel's publication of his paper on "Formally unsolvable theorems" to the Scandinavian logician Thoralf Skolem in which Gödel discusses logical concepts set forth in his celebrated paper, "Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I" ( Monatshefte füür Mathematik und Physik 38 [1931]: 173-98). This work permanently altered the foundations of logic and mathematics. Addressing himself in this paper to the first two of David Hilbert's three questions about mathematics posed in 1928, Gödel came up with his two famous theorems establishing the incompleteness and inconsistency of arithmetic, and by extension any formal mathematical system rich enough to include the theory of numbers. At the time our bibliography was written, this was the only recorded letter between Gödel and Skolem. The letter is of special significance because Gödel carefully followed Skolem's work and applied it in some instances. The letter will appear in a forthcoming volume of Gödel's Collected Works (1986- ), edited by Solomon Feferman. Autograph letters by Gödel are unusually scarce. OOC 306.