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First edition, complete journal issue in original printed wrappers, of Gödel's milestone paper proving the consistency of the Axion of Choice and the Generalized Continuum Hypothesis with the axions of set theory. In 1878, German mathematician Georg Cantor put forward the 'continuum hypothesis': 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 real numbers ("There is no set whose cardinality is strictly between that of the integers and that of the real numbers"). This was the first in Hilbert's famous list of mathematical problems, presented in an address to the International Congress of Mathematicians at Paris in 1900. All attempts to prove or disprove the hypothesis failed until 1938, when Kurt Gödel, in this paper, showed that 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 . . . 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 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). Note that Godel's booklet 'The Consistency of the Continuum Hypothesis' (Princeton, 1940) is a collection of notes from Gödel's lectures published two years after this original document. 8vo, pp. 525-572, vii (volume title and index) (small area of damage to upper outer corner of volume title due to adhesion to following leaf of index, tiny spot of dampstaining affecting upper margin of pp. 525-526 & 553-564, nowhere near text). Original printed wrappers (lightly creased, ends of spine a little worn).
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