INCOMPLETENESS FROM THE STANDPOINT OF FIRST-ORDER LOGIC

GÖDEL, KURT.

Ein Spezialfall des Entscheidungsproblems der theoretischen Logik.

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

8vo. A mint copy, in the original wrappers, still contained in the original plastic protection. In "Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling, herausgegeben von Karl Manger, Heft 2". Pp. 27-28. [Entire volume: 38 pp].

First edition of Gödel's important paper, which constitutes a supplement to his "Die Vollständigkeit der Axiome des logischen Funktionskalküls" (1930). In the present paper, Gödel seminally formulates his earlier incompleteness results from the standpoint of first-order logic, thereby contributing substantially to modern mathematical logic.

"In 1932 Godel published his formulation of the incompleteness results from the standpoint of first order logic. If number theory is regarded as a formal system in first-order logic, then the above results about incompleteness and unprovability of consistency apply to S. If, however, S is extended by variables for sets of numbers, for sets of sets of numbers, and so on (together with the corresponding comprehension axioms), then we obtain a sequence of systems S; the consistency of each system is provable in all subsequent systems. But in each subsequent system there are undecidable propositions. Going up in type in this way, he noted, corresponds in a type-free system of set theory to adding axioms that postulate the existence of larger and larger infinite cardinalities. This was the beginning of Gödel's interest in large cardinal axioms, an interest that he elaborated in 1947 in regard to the continuum problem." (DSB).

The issue contains several papers by Karl Menger.

Order-nr.: 46285