GÖDEL, KURT.

Über Vollständigkeit und Widerspruchsfreiheit (+) Eine Eigenschaft der Realisierungen des Aussagenkalküls.

Leipzig & Berlin, B.G. Teubner, 1932.

8vo. In the original wrappers. In "Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling, herausgegeben von Karl Menger, Heft 3". A near mint copy, Pp. 12-13; Pp. 20-21. [Entire volume: 26 pp].

First printing of these two important papers both closely related to Gödel landmark paper Über formal unentscheidbare Sätze". Here Gödel applies the extensions of the incompleteness theorems to a wider class of formal systems : "is already the more modern first-order Peano arithmetic, the system in which Godel in his abstract described his incompleteness results. The passage [in the present paper] envisages the introduction of higher-type variables, which would have the effect of re-establishing the system P, but as one proceeds to higher and higher types, that "all the [unprovable] propositions constructed are expressible in Z (hence are number-theoretic propositions)" is an important point about incompleteness. The last sentence of the [1932] passage is Godel's first remark on set theory of substance, and significantly, his example of an "axiom of cardinality" to take the place of type extensions is essentially the one that both Abraham Fraenkel [1922] and Thoralf Skolem [1923] had pointed out as unprovable in Ernst Zermelo's [1908] axiomatization of set theory and used by them to motivate the axiom of Replacement. " (Kanamori, Gödel and Set Theory).

"By invitation, in October 1929 Gödel began at tending Menger's mathematics colloquium, which was modeled on the Vienna Circle. There in May 1930 he presented his dissertation results, which he had discussed with Alfred Tarski three months ear lier, during the latter's visit to Vienna. From 1932 to 1936 he published numerous short articles in the proceedings of that colloquium (including his only collaborative work) and was coeditor of seven of its volumes. Gödel attended the colloquium quite regularly and participated actively in many discus sions, confining his comments to brief remarks that were always stated with the greatest precision." (DSB)

Order-nr.: 49345