COHEN, Paul (1934-2007). ‘The Independence of the Continuum Hypothesis.’ Offprint from: Proceedings of the National Academy of Sciences, Vol.50, No 6, pp. 1143-1148. December, 1963. [With:] ‘The Independence of the Continuum Hypothesis II.’ Offprint from: Proceedings of the National Academy of Sciences, Vol.51, No 1, pp.105-110. January, 1964.

Extremely rare offprints announcing that the Continuum Hypothesis cannot be proved, and revolutionising set theory in mathematics. Georg Cantor (1845-1918) first introduced the Continuum Hypothesis in 1878, advocating that there is no set whose cardinality is strictly between that of the integers and the real numbers. Cantor believed the Continuum Hypothesis to be true but the proof eluded him. Its importance was such that David Hilbert listed it as the first of his 23 problems in mathematics, presented in 1900 to the Paris conference of the International Congress of Mathematicians. Hilbert also believed it to be true but failed to prove it. With the discovery of the Russell paradox, existing set theory was realised to be too naïve. An axiom system subsequently suggested by Ernst Zermelo and Abraham Fraenkel became commonly accepted.

Kurt Gödel (1906-1978) announced in 1939 through these same Proceedings (Vol.25, pp.220-224) that the Continuum Hypothesis cannot be refuted within that commonly accepted axiom system. He did that by constructing a model of the Zermelo - Fraenkel axioms in which the Continuum Hypothesis holds. He failed, however, to make any progress with actually proving the Continuum Hypothesis.

In the winter of 1963-1964, Paul Cohen announced that the Continuum Hypothesis cannot be proved using that commonly accepted axiom system. He constructed a model of that axiom system in which the Continuum Hypothesis fails. Before Cohen’s work, progress was stymied by the lack of tools for constructing models of the Zermelo - Fraenkel axioms. He introduced a totally new technique called forcing for obtaining such models. This utterly revolutionised the subject of set theory. For this work, he was in 1966 awarded a Fields Medal (the equivalent for mathematics of a Nöbel Prize). Combined with Gödel’s earlier result he had shown that the Continuum Hypothesis is independent of the Zermelo - Fraenkel axioms, hence the title.

Just as Gödel followed up his 1939 announcement with a monograph based on a series of lectures, published in 1940 by the Princeton University Press, so Cohen followed up his announcement with a monograph entitled Set Theory and the Continuum Hypothesis, which acted as a very readable introduction to both Set Theory and his remarkable results. It was first published in 1966, and the slightly augmented version, published in 2008 after Cohen’s death, is still available today.

2 offprints, large octavo (251 x 175mm). (Second part with some light marginal creasing, both parts with some very minor marginal finger-soiling.) Original buff wrappers, stapled (part 2 with covers very lightly creased, some faint soiling to both parts, staples rusted).

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