The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis [and] Consistency-Proof for the Generalized Continuum-Hypothesis [and] The Independence of the Continuum Hypothesis, I-II in Proceedings of the National Academy of Sciences, Volume 24, (1938), pp.556-557 and Volume 25 (1939), 220-224 by Gödel, Kurt

Easton: National Academy of Sciences. 1st Edition. 1938 TWO VOLUME TRUE FIRST EDITION OF GÖDEL'S MILESTONE PAPERS PROVING THE CONSISTENCY OF THE AXIOM OF CHOICE AND THE GENERALIZED CONTINUUM HYPOTHESIS WITH THE AXIOM OF SET THEORY. Note that the 1940 publication by the same title is a collection of notes from Gödel's lectures published two years after these original documents.


In 1878, German mathematician Georg Cantor put forth a hypothesis that said any infinite subset of the set of all real numbers can be put into one-to-one correspondence either with the set of integers or with the set of all real numbers ("There is no set whose cardinality is strictly between that of the integers and that of the real numbers").

The continuum problem, as Cantor's problem came to be know, was the first in Hilbert's famous list of mathematical problems, presented in an address in 1900. All attempts to prove or disprove Cantor's conjecture failed until 1938, when Kurt Gödel, in these papers, showed it was impossible to disprove the continuum hypothesis. "Gödel studied the relationship of the continuum hypothesis and the axiom of choice to basic set theory as formulated by the mathematicians Ernst Zermelo and Abraham Fraenkel.

In 1940 Gödel showed that both the continuum hypothesis and the axiom of choice are consistent with the axioms of set theory. More precisely, he demonstrated that if the Zermelo-Fraenkel system without the axiom of choice is consistent, then the Zermelo-Fraenkel system with the axiom of choice is consistent, and that the continuum hypothesis is consistent with the Zermelo-Fraenkel system" (Ryan, Thinkers of the Twentieth Century, p. 212f). "Godel's result, joined to [Paul J.] Cohen's (1963-1964), set the stage for a whole new era in the theory of sets in which a host of problems of the consistency or independence of various conjectures in set theory relative to this or that set of axioms are being investigated by constructing models" (Shanker, Godel's Theorem in Focus, 66-67).

According to the Stanford Encyclopedia of Philosophy, "the principle of set theory known as the ‘Axiom of Choice' has been hailed as ‘probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago" (Stanford Portal; Fraenkel, 1973, II.4). CONDITION & DETAILS: Eaton: Mack Printing Company. 4to (10 x 7 inches; 250 x 175mm). Both volumes handsomely rebound in aged brown cloth (identical to original binding). Gilt-lettered and dated at the spine. Solidly and tightly bound, with both volumes set into a gilt-titled brown slipcase. New endpapers. Very good to near fine condition in every way.