COHEN, Paul.

The Independence of the Continuum Hypothesis [I–II]. [With:] Set Theory and the Continuum Hypothesis. [And:] autograph letter signed from Cohen to Martin Davis.

Washington, D.C.; Stanford; New York: National Academy of Sciences; W. A. Benjamin, 1963; 1964; 1966.

First edition, in the rare author’s offprints, of the two papers in which Cohen proved that the continuum hypothesis and the axiom of choice cannot be derived from the standard axioms of set theory — and devised, in order to prove it, the method of forcing that has governed the subject ever since. Gödel, who communicated the two papers to the Academy, wrote that Cohen’s work was no doubt the greatest advance in the foundations of set theory since the subject was axiomatised, and for it Cohen received the Fields Medal in 1966, the first ever awarded for mathematical logic. The offprints are here joined by the first edition of Cohen’s book and an autograph letter signed in which Cohen tells the logician Martin Davis that he was ‘directly responsible for my looking once more at Set Theory’. The whole comes from Davis’s own library, with his ownership signature and his working annotations in the book.

Provenance: Martin Davis (1928–2023), with his ownership signature in the book and his annotations in ink worked through the proof of the central theorem. Davis was himself one of the principal logicians of the age. A student at City College of Emil Post and a doctoral pupil of Alonzo Church at Princeton, he spent his career on the unsolvability of Hilbert’s tenth problem — the question whether an algorithm can decide the solvability of Diophantine equations — framing the bold conjecture, proved with Hilary Putnam and Julia Robinson and completed by Yuri Matiyasevich in 1970, that the computably enumerable sets are exactly the Diophantine ones; the resulting theorem carries his initial. His Computability and Unsolvability (1958) became a foundation text of theoretical computer science, and he was among the earliest programmers of an electronic computer. Cohen, casting about for a problem and stung by his reception among logicians, was turned back to set theory by Davis’s encouragement, and said so in writing within months of the breakthrough; the annotations are the marks of one founder of modern logic reading another at first hand.

The letter is the centre of the group. Written from the Stanford mathematics department on 27 November 1963, two pages in Cohen’s hand, it is addressed to Martin Davis, and in it Cohen traces the work back to him. He thanks Davis for the encouragement he had given at Stockholm — the International Congress of Mathematicians of August 1962, where Cohen had spoken — and tells him plainly that he was directly responsible for Cohen’s looking once more at set theory, after Cohen had felt, in his own words, outplayed and even humiliated by the logicians he had spoken to. He reports that Gödel had submitted the paper to the Academy and that it would appear in the December and January issues; he asks Davis’s advice about whether to write a book, and what it should cover. The book that resulted is the one beside the letter. The exchange is recorded in the interview Davis gave to the Notices of the American Mathematical Society in May 2008.

The question Cohen answered was the oldest in the theory of the infinite. In the 1870s Georg Cantor had shown that the infinity of the real numbers strictly exceeds that of the integers, and conjectured that no infinity lies between the two. Cantor measured the sizes of infinite sets by their cardinal numbers, the smallest infinite cardinal being that of the integers; he proved that for every cardinal there is a next-larger one, so that the infinite cardinals form an ascending sequence, and he showed that the cardinal of the continuum is not the smallest but could not fix its place. His conjecture, the continuum hypothesis, was that the continuum is the very next cardinal after the countable — in the modern notation, that two raised to the power of the first infinite cardinal equals the second. He could neither prove it nor abandon it, and when David Hilbert set out the twenty-three problems he laid before the new century in 1900, the continuum hypothesis stood first on the list. A second question shadowed it: the axiom of choice, isolated by Ernst Zermelo in 1904 in the course of proving that every set can be well-ordered, which asserts that from any collection of non-empty sets one may select a member of each. The principle is indispensable to much of mathematics, yet it sat uneasily with many of Cantor’s successors, for unlike the other set-existence axioms it asserts that a choice function exists without giving any rule to construct one; Zermelo’s own proof drew immediate controversy on exactly this ground. His axiomatisation of set theory, refined into the system now called Zermelo–Fraenkel (ZF), was built in part to discipline these questions. For half a century neither yielded.

The first breach was Kurt Gödel’s. In 1938 he proved that the continuum hypothesis and the axiom of choice are consistent with ZF — that they cannot be disproved from the axioms — by building, inside any model of set theory, the ‘constructible universe’, the inner model of those sets reachable by definable stages, in which both propositions hold. Gödel’s result left the harder half open: it did not show that the hypothesis followed from the axioms, only that it could not be refuted. To settle Hilbert’s problem one needed a model of set theory in which the continuum hypothesis fails. None of Gödel’s methods could build one, and for twenty-five years none could.

Paul Cohen (1934–2007) was an analyst, not a set-theorist, when he arrived at Stanford in 1961; he came to the continuum problem looking for a result of the first importance and taught himself the field to reach it. Early in 1963 he found his way through. The instrument was forcing. Beginning from a countable model of set theory, Cohen adjoined to it a new set of integers — a ‘generic’ set, designed to share none of the special properties the model could define, and specified not all at once but through a partially ordered system of finite conditions, each giving only incomplete information about it. The truth of every statement about the new object, and so about the enlarged model, was settled by what the conditions force: a definable forcing relation fixed the meaning of the extension without the generic set ever being fully given, and a category argument guaranteed that such a set existed. The ordinals of the model were left untouched, so the construction stayed inside set theory rather than appealing to anything beyond it. By adjoining one such ‘Cohen real’ Cohen could obtain a model in which Gödel’s axiom of constructibility fails; by adjoining a sufficiently large family of them, a model in which the continuum hypothesis fails outright, the continuum forced past the second cardinal. With Gödel’s consistency result, this closed the question: the continuum hypothesis and the axiom of choice are independent of the axioms of set theory, neither provable nor refutable from them. Hilbert’s first problem had its answer, and the answer was that the axioms do not decide it.

The two notes offered here are that announcement in its first printed form. Cohen had circulated a mimeographed manuscript at Stanford in April 1963 and lectured on the method at the Institute for Advanced Study on 3 May 1963; the full statement appeared as two communications to the Proceedings of the National Academy of Sciences, ‘The Independence of the Continuum Hypothesis’ in the volume for December 1963 and its sequel in January 1964. Each bears, beneath Cohen’s name, the line Communicated by Kurt Gödel — the first dated 30 September 1963, the second 27 November 1963, the day of the letter to Davis: the Academy accepted papers from non-members only on a member’s sponsorship, and it was Gödel — the author of the theorem Cohen had completed — who carried the younger man’s work into print.

Gödel’s response to the proof was written in a draft letter to Cohen, and he set the judgement in print. In a draft letter to Cohen of June 1963 he wrote that reading the proof had given him the pleasure of seeing a really good play, and that in all essential respects it was the best possible; a fortnight later he told Cohen that he had achieved the most important progress in set theory since its axiomatisation. In a postscript of September 1966 to the revised edition of his essay on Cantor’s problem he set the judgement in print: Cohen’s work, he wrote, was no doubt the greatest advance in the foundations of set theory since the subject was axiomatised. Yet the same admiration produced friction. Gödel, who communicated the papers, pressed a succession of revisions on Cohen through the summer of 1963 that delayed their appearance, to Cohen’s mounting discomfort — the cost, as Solomon Feferman has described it, of having the founder of the field as one’s editor.

The forcing method did not stay with the continuum problem. It became at once the standard instrument of set theory, the means by which the independence of one proposition after another was established and by which the relative strength of competing axioms could be calibrated; the ‘Cohen reals’ his conditions introduce are now a fixed term of art, and the technique runs through every later development of the subject, from large-cardinal theory to the fine structure of inner models. Cohen set out the whole of it in Set Theory and the Continuum Hypothesis (New York: W. A. Benjamin, 1966), the book present here in first edition. It remains the readable introduction at once to set theory and to forcing; its account of the construction is the one most later expositions follow.

The separately-issued offprints are rare: struck in small numbers for the author’s own distribution, in their printed wrappers, they are distinct from the bound journal volumes of the Academy through which the result reached its readers. The continuum hypothesis that these pages left open has not been settled since. Cohen and Gödel between them showed that the standard axioms leave the size of the continuum open, and the question has passed back to the philosophers of mathematics: whether new axioms — the large cardinals, the determinacy hypotheses — should be sought to decide it, or whether there is no fact of the matter to decide. Gödel had built the inner model in which the hypothesis holds; Cohen built the outer models in which it fails; and the method he invented to build them, set down first in these two notes and explained in the book beside them, is the one with which set theory has done its work ever since — refined a generation later by Saharon Shelah into the general theory of Proper Forcing (1982), and standing today behind every independence result the subject has produced.

References: Gödel, Collected Works, vol. II: Publications 1938–1974, ed. S. Feferman et al. (Oxford, 1990), pp. 269–270 (Gödel’s assessment of Cohen) and vol. IV (Oxford, 2003), Gödel–Cohen correspondence — A. Kanamori, ‘Cohen and Set Theory’, Bulletin of Symbolic Logic 14 (2008), pp. 351–378 — S. Feferman, The Gödel Editorial Project: A Synopsis (2005) — P. Cohen, ‘The Discovery of Forcing’, Rocky Mountain Journal of Mathematics 32 (2002), pp. 1071–1100 — G. H. Moore, ‘The Origins of Forcing’, in Logic Colloquium ’86 (1988) — A. Jackson, ‘Interview with Martin Davis’, Notices of the American Mathematical Society 55 (2008), pp. 560–571 — T. Drucker, ‘Independence of the Continuum Hypothesis’, in Science and Its Times.

Three printed items and an autograph letter. Offprints from the Proceedings of the National Academy of Sciences (December 1963 and January 1964): two vols., large 8vo (251 × 175 mm), pp. 1143–1148 and 105–110, in the original printed wrappers, each titled The Independence of the Continuum Hypothesis (and … II) and bearing the communication line of Kurt Gödel, dated 30 September 1963 and 27 November 1963 respectively. Book: 8vo (228 × 153 mm), pp. 154, original printed stiff wrappers, with the ownership signature of Martin Davis and his autograph working annotations in ink in the text. Letter: a single sheet (215 × 140 mm) on Stanford University Department of Mathematics letterhead, written in ink on recto and verso, signed by Paul Cohen, dated 27 November 1963. Light handling wear; very good throughout.

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Item #6158

Price: $50,000.00