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Gödel, Kurt. "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis", in Proceedings of the National Academy of Sciences, vol 24, pp.556-557, 1938, in the volume of 572pp.__+__ Offered with: _____. "Consistency-proof for the Generalized Continuum-Hypothesis", in Proceedings of the National Academy of Sciences, vol 25 pp. 220-224, 1939, in the volume of 661pp. Both first editions, and uniformly bound in brown cloth, nicely imprinted in gilt (with the full name of the journal, which is unusual given its length). There are some rubbed places on the covers and along spine bottom, making those areas a light shade of brown. Small 6x8mm perforated stamp on title; the text is in very fine condition. Overall, a solid near-Fine on both volumes.__+__ These works are arguably Gödel 's next most important contributions next to the incompleteness theorems, which are arguably mathematical logic's most important achievement in the 20th century. Hilbert famously put forward a list of 23 important questions in mathematics, 10 of which were presented at the International Congress of Mathematics in Paris in 1900. The very first of these questions was the continuum hypothesis, a problem simply stated: .how many points on a line are there? Strangely enough, this simple question turns out to be deeply intertwined with most of the interesting open problems in set theory, a field of mathematics with a very general focus, so general that all other mathematics can be seen as part of it, a kind of foundation on which the house of mathematics rests. Most objects in mathematics are infinite, and set theory is indeed just a theory of the infinite. --Juliette Kennedy, Can the Continuum Hypothesis Be Solved? , 2011, on the IAS website.__+__ "In 1935 Godel had made the first breakthrough in his new area of research: set theory. During May and June 1937 he lectured at Vienna on his striking result that the axiom of choice is relatively consistent. That summer he obtained the much stronger result that the generalized continuum hypothesis is relatively consistent; and in September 1937 John von Neumann, an editor of the Princeton journal Annals of Mathematics, urged him to publish his new discoveries there. Yet Gödel did not announce them until November 1938, and then not in the Annals but in a brief summary communicated to the Proceedings of the National Academy of Sciences. --Complete Dictionary of Scientific Biography.__+__ On the continuum Hypothesis: Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain (Dauben 1990). It became the first on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated. Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. The second half of the independence of the continuum hypothesis i.e., unprovability of the nonexistence of an intermediate-sized set was proved in 1963 by Paul Cohen. --Wikipedia __+__ On Gödel , in general: "The character and achievement of Gödel, the most important logician since Aristotle, bear comparison with those of the most eminent mathematicians. His aversion to controversy is reminiscent of Newton s, while his relatively small number of publications each quite precise and almost all making a major contribution echo Gauss s motto of few but ripe. Like Newton, he revolutionized a branch of mathematics in this case, mathematical logic giving it a structure and a Kuhnian paradigm for research. His notebooks, like those of Gauss, show him to be well in advance of his contemporaries. While Newton made substantial efforts in theology as well as in mathematical physics." --Complete Dictionary of Scientific Biography.
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