## The Independence of the Continuum Hypothesis. Offprint from: Proceedings of the National Academy of Sciences, Vol. 50, No. 6, December 1963. [With:] The Independence of the Continuum Hypothesis II. Offprint from: Proceedings of the National Academy of Sciences, Vol. 51, No. 1, January 1964. [With:] Autograph letter signed from Cohen to Martin Davis, Stanford, CA, November 27, 1963. [With:] Set Theory and the Continuum Hypothesis.

New York: W. A. Benjamin, 1966.

First edition, extremely rare offprints, of Cohen’s proof that the continuum hypothesis (CH) and the axiom of choice (AC) cannot be proved from the generally accepted Zermelo-Fraenkel axioms (ZF) of set theory; the method of ‘forcing’ which Cohen devised for the purposes of his proof revolutionized the subsequent development of set theory. Kurt Gödel wrote in 1964 that this work “no doubt is the greatest advance in the foundations of set theory since its axiomatization” (Gödel, Works 2, p. 270). Gödel had proved in 1938 that CH and AC cannot be disproved from ZF – Cohen’s and Gödel’s results together showed that CH and AC are independent of ZF. Gödel had, in fact, also proved that AC could not be proved from ZF, although he never published this result, but his methods had failed to establish this for CH. 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. The ‘Axiom of Choice’, proposed by the German logician Ernst Zermelo in 1904, states that, given any collection of sets (even an infinite collection), each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set. “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 Encyclopedia of Philosophy). All attempts to prove or disprove CH or AC failed until the work of Gödel and Cohen. For this work, Cohen was in 1966 awarded a Fields Medal (the equivalent for mathematics of a Nobel Prize). Just as Gödel followed up his 1938 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. Cohen’s offprints and book are accompanied here by an autograph letter from Cohen to the mathematician Martin Davis, whom Cohen writes was “directly responsible for my looking once more at set theory”. RBH lists one other copy of each offprint. Not on OCLC.

Provenance: Martin Davis (1928-2023), American logician and computer scientist (ownership signature and his notes on Set Theory and the Continuum Hypothesis; ALS from Cohen to Davis).

Two problems had preoccupied workers in the field of set theory since its creation by Cantor beginning in the 1870s: the well-ordering principle and the cardinality of the continuum. An ‘ordering’ of a set X is a rule for deciding, given any two different elements of X, which one precedes the other (such as the usual ordering on the set of integers …, –2, –1, 0, 1, 2, …). A ‘well-ordering’ of X is an ordering with the property that every non-empty subset Y of X has a least element (an element that precedes all the other elements of Y). So the usual ordering of the integers is not a well-ordering (there is no least integer), but the same ordering on the set of positive integers is a well-ordering.

“Cantor had conjectured the proposition, now called the ‘well-ordering theorem,’ that every set can be well-ordered. In 1904 Zermelo gave a proof of this conjecture, using in an essential way the following mathematical principle: for every set X there is a ‘choice function, f, which is defined on the collection of non-empty subsets of X, such that for every subset A of X,f(A) is an element of A [so f ‘chooses’ an element of each subset of X]. Subsequently, in 1908, Zermelo presented an axiomatic version of set theory in which his proof of the well-ordering theorem could be carried out. One of the axioms was the principle just stated, which Zermelo referred to as the ‘Axiom der Auswahl’ (the Axiom of Choice, abbreviated AC).

“Zermelo’s proof was the subject of considerable controversy. The well-ordering theorem is quite remarkable, since, for example, there is no obvious way to define a well-ordering of the set of real numbers. Nor is such an explicit well-ordering provided by Zermelo’s proof. Thus, many people who thought Zermelo’s result implausible cast doubt upon the validity of AC. The other set-existence axioms all have the form that some collection of sets, explicitly definable from certain given parameters, is itself a set. The axiom of choice, on the other hand, asserts the existence of a choice function but does not provide an explicit definition of such a choice function. Zerrnelo was well aware that his axiom had this purely existential character, but many other mathematicians were uncomfortable with existence proofs that did not provide the construction of specific examples of what was asserted to exist” (Gödel, Works 2, pp. 1-2).

Beginning in 1874, Cantor introduced the concept of the ‘cardinality’ of a set. For a set with a finite number of elements, its cardinality is just the number of elements in the set. But the cardinality is also defined for infinite sets. Two sets have the same cardinality if and only if they can be put into one-to-one correspondence. If χ is the cardinal number of a set X, the cardinal number of the set of all subsets of X is denoted by 2χ(because if a finite set has n elements, the set of all its subsets has 2n elements). A set X has cardinality less than or equal to that of another set Y if X can be put into one-to-one-correspondence with a subset of Y. The smallest infinite cardinal number, denoted by χ0 (‘aleph zero’), is the cardinality of the set of natural numbers 0, 1, 2, 3, …. (Note that the set of integers also has cardinality χ0, because writing the integers in a list such as 0, –1, 1, –2, 2, … enables them to be put into one-to-one correspondence with the natural numbers). Cantor showed that, for every cardinal, there is a next-larger cardinal, so we get a sequence of cardinal numbers χ0, χ1, χ2, … .

“In his theory of infinite cardinals Cantor proved (making essential but implicit use of AC) that the totality of all infinite cardinal numbers is well-ordered. However, an important question was left open by Cantor’s work. Let c be the cardinal number of the set of real numbers (or, as this set is sometimes referred to, the continuum). Cantor showed that c is not the first infinite cardinal, but he was unable to determine its precise place in the hierarchy of infinite cardinals. He conjectured, however, that c is precisely equal to χ1, the second infinite cardinal. This conjecture became known as the continuum hypothesis (CH). It is easily shown that c = 2χ0, and so CH is equivalent to the statement 2χ0 = χ1. A natural generalization, called the generalized continuum hypothesis (GCH), asserts that for every a, 2χa = χa+1” (Gödel, Works 2, pp. 2-3).

“In an effort to work on the continuum hypothesis and other problems in the theory of infinite sets, Ernst Zermelo (1871-1956) came up with a set of axioms to try to avoid any concealed paradoxes. Other mathematicians had already run aground on some of the intricacies of dealing with the infinite, and Zermelo’s axiomatization (known in a slightly altered form as ZF) was designed both to avoid paradoxes and to make further progress possible. He had a particular interest in the status of what has become known as the axiom of choice, but his system of axioms for set theory proved to be useful in addressing other issues as well.

“The most important advance in the first half of the twentieth century with regard to the continuum hypothesis was the work of Kurt Gödel (1906-1978). He had already established some of the most important results in mathematical logic and assured the field of its status as an independent discipline. Then in 1938 he proved that the continuum hypothesis was consistent with the ZF axioms for set theory. This result suggested that there was good reason to keep working on the problem with the hope that the continuum hypothesis could be proved from the axioms. On the other hand, it did not establish that the continuum hypothesis was a consequence of the axioms, which would have finally answered the question posed by Hilbert.

“Paul J. Cohen (1934-2007) was a mathematician who did not specialize in mathematical logic when he arrived at Stanford University as an assistant professor of mathematics in 1961. He did, however, have an impressive mathematical background, and he was looking for a problem of some importance on which to work. His attention was directed to the continuum hypothesis, and he undertook a thorough study of the literature that surrounded the earlier attempts to establish it on the basis of the axioms of set theory. Over the next few years he managed to create an entirely new technique in mathematical logic that enabled him to provide a kind of answer to Hilbert's question.

“There are three possible relationships between a statement and a set of axioms: the statement can be provable from the axioms, its negation can be provable from the axioms, or the statement can be neither provable nor unprovable from the axioms. A good example is the parallel postulate included by Euclid (fl. 300 B.C.) in his list of axioms for geometry. For many years mathematicians tried to prove that statement on the basis of the other axioms provided by Euclid, but they were always unsuccessful. Not until the nineteenth century was it demonstrated that the parallel postulate could not be proved from the other axioms.

“The way in which this was demonstrated was to come up with one model for the other axioms in which the parallel postulate was true and another in which the parallel postulate was false. A model is a specific collection of objects to which the axioms apply. If two different models for a set of axioms can give two different answers for the question of the truth of a statement, that statement is said to be independent of the axioms. Specific geometric models for the Euclidean axioms without the parallel postulate showed that the parallel postulate was indeed independent of the other axioms.

“Gödel succeeded in showing that the continuum hypothesis was consistent with the axioms of set theory by constructing a model of set theory based on the axiom of constructibility. This is the assertion that every set is built up from other sets by certain well-defined processes. Within this model (called ‘the constructible universe’) the continuum hypothesis could be proved. As a result, the continuum hypothesis had been shown to be consistent with the other axioms of set theory – in other words, no contradiction could arise from including the continuum hypothesis with the other axioms. There was some disagreement about whether the axiom of constructibility adequately captured mathematical intuition about the realm of infinite sets.

“Cohen introduced the method of ‘forcing’ to try to show the other side of independence for the continuum hypothesis. He needed a model for the other axioms of set theory in which the continuum hypothesis was false. From that it would follow that the continuum hypothesis was independent of the axioms of set theory. Forcing involves specifying which statements (Cohen started by working with statements about the positive whole numbers) are supposed to be true in the model being constructed. In particular, one introduces a kind of relationship that determines the truth of compound statements by the truth of the component statements of which it is made up. The only statements true in the model are those that are forced to be true by the forcing conditions. This kind of case-by-case analysis had also appeared in Gödel’s proof that the continuum hypothesis was consistent with the axioms of set theory.

“When the news of Cohen’s result became known in the community of mathematical logicians, it was widely regarded as the most important development in set theory since it had first been axiomatized. He received the highest honor paid by the mathematical community when he received the Fields Medal at the International Mathematical Congress of 1966. The medal is given every four years to the outstanding mathematician under the age of 40. He was the first recipient of the medal for work in logic and helped to give the discipline an added boost in the judgment of the rest of the mathematical community.

“The technique of forcing became a standard part of the repertoire of logicians working in the area of mathematics known as recursion theory. This field studies the complexity of mathematical subsets of the positive whole numbers and leads to a hierarchy of sets. Varieties of forcing have continued to be introduced in an effort to achieve more and more sophisticated models of the axioms for set theory. The term ‘Cohen reals’ is used to refer to the numbers introduced by the stipulations of forcing conditions.

“In a philosophical sense the issue of the truth of the continuum hypothesis has been a matter for much speculation in light of Cohen’s result. If the continuum hypothesis is not implied by the standard axioms of set theory nor is its negation, then somehow the standard axioms of set theory leave open a rather fundamental issue about the relationship between the set of whole numbers and the set of the reals. Some philosophers of mathematics have urged the introduction of new axioms, especially those asserting the existence of extremely large infinite numbers, as a way of resolving the question. Others have suggested variants on the axiom of choice in which Zermelo was interested as more intuitive but capable of settling the truth of the continuum hypothesis. Still others have argued that there is no actual truth about statements of set theory, since the objects in question are so far removed from human intuition. The same kind of objection that had been raised to the axiom of constructibility was brought up with regard to the other proposed axioms as well. After the introduction by Cantor of the techniques that first led to the inclusion of infinite sets within the arsenal of mathematicians and not just philosophers, the demonstration by Cohen of the independence of the continuum hypothesis has taken the subject back into the domain for philosophers as well as mathematicians” (Thomas Drucker, Science and Its Times: Understanding the Social Significance of Scientific Discovery).

Paul Cohen recalled meetings with Gödel at the Institute for Advanced Study. “I visited Princeton again for several months and had many meetings with Gödel. I brought up the question of whether, as rumor had it, he had proved the independence of the axiom of choice. He replied that he had, evidently by a method related to my own, but he gave me no precise idea or explanation of why his method evidently failed to succeed with the CH. His main interest seemed to lie in discussing the truth or falsity of these questions, not merely their undecidability. He struck me as having an almost unshakable belief in this realist position that I found difficult to share. His ideas were grounded in a deep philosophical belief as to what the human mind could achieve” (Baaz et al., Kurt Gödel and the Foundations of Mathematics: Horizons of Truth (2011), p. 438).

Gödel himself wrote a letter to Cohen in June 1963, a draft of which stated, “Let me repeat that it is really a delight to read your proof of the ind[ependence] of the cont[inuum] hyp[othesis]. I think that in all essential respects you have given the best possible proof & this does not happen frequently. Reading your proof had a similarly pleasant effect on me as seeing a really good play” (Solomon Feferman, The Gödel Editorial Project: A Synopsis, p. 11). In a postscript to the revised edition (1964) of his article ‘What is the continuum hypothesis,’ Gödel notes: “Cohen's work, which no doubt is the greatest advance in the foundations of set theory since its axiomatization, has been used to settle several other important independence questions” (Gödel, Works 2, pp. 269-270).Gödel, however, suggested a number of revisions to Cohen’s paper that delayed publication, “to Cohen’s increasing discomfort” according to Feferman. Indeed, the letter offered here indicates that Gödel did not forward Cohen’s paper to the Proceedings of the National Academy of Sciences until November.

Cohen attended the University of Chicago (M.S., 1954; Ph.D., 1958). He held appointments at the University of Rochester, N.Y. (1957–58), and the Massachusetts Institute of Technology (1958–59) before joining the Institute for Advanced Study, Princeton, N.J. (1959–61). In 1961 he moved to Stanford University in California; he became professor emeritus in 2004. Cohen also made many valuable contributions to analysis. He was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper ‘On a conjecture by Littlewood and idempotent measures,’ and lends his name to the ‘Cohen–Hewitt factorization theorem.’ After proving his revolutionary result about the continuum hypothesis, Cohen returned to research in analysis.

The letter offered here is actually reproduced in the article ‘Interview with Martin Davis’ (Notices of the American Mathematical Society, Vol. 55, No. 5, May 2008, pp. 560-571). The letter indicates that Davis was responsible for Cohen’s turn from research in analysis to problems in set theory:

Dear Martin,

I have thought about writing you for some time, but never quite got around to it. First, I hope you received a preprint of my work. Gödel has now submitted the paper to the PNAS [Proceedings of the National Academy of Sciences] and it comes out in Dec. and Jan. issues.

I really should thank you for the encouragement you gave me in Stockholm [where the International Congress of Mathematicians was held, 15-22 August 1962, and at which Cohen was an invited speaker]. You were directly responsible for my looking once more at set theory. Previously I had felt rather outplayed & even humiliated by the logicians I had spoken to. Of course, the problem I solved had little to do with my original intent. In retrospect, though, the basic ideas I developed previously played a big role when I tried to think of throwing back a proof [of] the Axiom of Choice, as I had previously thought about throwing back a proof of a contradiction.

I have received 2 letters from A. Edelson of Technical Publishing Co. I have not decided what to do about a book and so I cannot say much at this point. What do you think I should attempt to cover in a book?

How is Newman? I will be in Miami in January & then in N.Y. so I will certainly see you then.

Regards to all.

Sincerely,

Paul Cohen

“Martin Davis [wa]s one of the world’s outstanding logicians. He was born in 1928 in New York City, where he attended City College and was influenced by Emil L. Post. Early on, Davis came under the spell of Hilbert’s Tenth Problem: Does there exist an algorithm that can, given an arbitrary Diophantine equation, decide whether that equation is solvable? Davis’s Ph.D. dissertation, written at Princeton University under the direction of Alonzo Church, contained a conjecture that, if true, would imply that Hilbert’s Tenth Problem is unsolvable. In rough terms, the conjecture said that any computer can be simulated by a Diophantine equation. The implications of this conjecture struck many as unbelievable, and it was greeted with a good deal of skepticism; for example, the conjecture implies that the primes are the positive part of the range of a Diophantine polynomial. Work during the 1950s and 1960s by Davis, Hilary Putnam, and Julia Robinson made a good deal of headway towards proving the conjecture. The final piece of the puzzle came with work of Yuri Matiyasevich, in 1970. The resulting theorem is usually called either DPRM or MRDP (Matiyasevich favors the former and Davis the latter). The unsolvability of Hilbert’s Tenth Problem follows immediately.

“Davis became one of the earliest computer programmers when he began programming on the ORDVAC computer at the University of Illinois in the early 1950s. His book Computability and Unsolvability first appeared in 1958 and has become a classic in theoretical computer science. After a peripatetic early career that included stints at Bell Labs in the era of Claude Shannon and at the RAND Corporation, Davis settled at New York University, where he spent thirty years on the faculty and helped to found the computer science department. He retired from NYU in 1996 and moved to Berkeley, California” (ibid.).

Offprints: Two vols., large 8vo (251 x 175 mm), pp. 1143-1148 & 105-110. Original printed wrappers. Book: 8vo (228 x 153 mm), pp. 154. Original printed stiff wrappers. Letter: Single sheet (215 x 140 mm), with Stanford University Department of Mathematics letterhead, written on recto and verso.

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