The Independence of the Continuum Hypothesis, Part I-II [and] Set Theory and the Continuum Hypothesis in Proceedings of the National Academy of Sciences, Volume 50 (1963) and Volume 51 (1964) by Cohen, Paul Joseph

TWO MILESTONE PAPERS ON THE FOUNDATIONS OF MATHEMATICS. Two volume (in original wraps) first edition of Paul Cohen's seminal contribution to set theory wherein he established the independence of the axiom of choice and the independence of the continuum hypothesis. Cohen's efforts rank among the most important achievements in 20th century mathematics. Establishing the truth or falsehood of the continuum 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 showed that it was impossible to disprove the continuum hypothesis. Building upon Godel's landmark work, Cohen developed the mathematical technique of forcing in order to prove consistency and independence in results. These two papers are the inaugural examples of the new technique of forcing. "If Godel's construction of L had launched set theory as a distinctive field of mathematics, then Cohen's forcing began its transformation into a modern, sophisticated one" (Kanamori, "Cohen and Set Theory," Bulletin of Symbolic Logic, 14, 3, Sept. 2008, 352-374). Cohen's new method was so helpful that within a decade, forcing grew to play "a crucial role in the transformation of set theory into a modern, sophisticated field of mathematics, one tremendously successful in the investigations of the continuum, transfinite combinatorics, and strong propositions and their consistency strength. In all these directions, forcing became integral to the investigation and became part of their very sense, to the extent that issues about the method became central and postulations in its terms, "forcing axioms", became pivotal... The extent and breadth of the expansion of set theory henceforth dwarfed all that came before, both in terms of the numbers of people involved and the results established. With clear intimations of a new and concrete way of building models, set theorists rushed in and with forcing were soon establishing a cornucopia of relative consistency results, truths in a wider sense, with some illuminating classical problems of mathematics. CONDITION & DETAILS: Easton: The Proceedings of the National Academy of Sciences. 4to (10 x 6.75 inches; 250 x 169mm). Both issues in original wraps, rebacked at the spine with minor wear. Very slight and light toning to the front wrap of Volume 51. Both issues housed in a handsome brown clamshell case gilt-lettered at the spine. The interiors are in near-fine condition.