Someone drew my attention to the Russian translation of Rademacher and Toeplitz's Von Zahlen und Figuren (The Enjoyment of Mathematics in the English translation). In the chapter on set theory the Russian editor, Isaak Yaglom, added a footnote saying

"Research in mathematical logic (especially by K. Goedel) led to a conviction that (at least in most axiomatic set theories) the problem of continuum is undecidable, meaning that it cannot be deduced from the basic assumptions of set theory."

However, the Russian translation was published in 1962, and Cohen's results appeared in 1963 (nor is Cohen named in the footnote). The footnote evidently references Goedel's work on CH, which is not mentioned in the text of the chapter because the original is from 1930.

Goedel showed that the axioms of set theory cannot refute the continuum hypothesis, but Yaglom seems to be indicating that before Cohen's result people were already convinced that the CH was not provable either. Was that really the case, and what was the evidence thought to be for or against the independence of CH before Cohen?

  • 1
    $\begingroup$ Although not directly related to the question, it may be of some interest to note that efforts to "solve" the CH continue to this day. The Institute for Advance Study site has an interesting paper on the subject : Can the Continuum Hypothesis be Solved?. $\endgroup$
    – nwr
    Commented Jun 12, 2019 at 13:09

1 Answer 1


According to Cohen himself, no. Most people, apparently, did not bother to believe one way or the other, and those who did (perhaps with notable exceptions of Lusin and Sierpinski) relied on Gödel's semi-philosophical remarks in a popular paper, and word of mouth that he already partially solved the problem (originating from him, presumably). This is probably what Yaglom refers to by "research in mathematical logic (especially by K. Gödel)". The paper was What is Cantor's Continuum Problem? (1947) written for the American Mathematical Monthly, where he wrote:

"There are (assuming the consistency of the axioms) a priori three possibilities for Cantor's conjecture: It may be either demonstrable or disprovable or undecidable. The third alternative (which is only a precise formulation of the conjecture stated above that the difficulties of the problem are perhaps not purely mathematical) is the most likely, and to seek a proof for it is at present one of the most promising ways of attacking the problem.

He proceeds to give two arguments in favor of this opinion. One is that the standard axioms do not seem to settle whether the sets talked about should be taken platonistically, or as "extensions of definable properties" (a reference to Gödel's own constructible universe). And investigations of cardinal arithmetic since Cantor suggested that settling questions about it would require additional axioms based on a choice between the two.

The second argument (for which Gödel credits Lusin and Sierpinski) is the existence of "certain facts (not known or not existing at Cantor's time) which seem to indicate that Cantor's conjecture will turn out to be wrong", while proving it wrong, as Gödel already showed, was impossible from the accepted axioms. The facts cited are "meager" sets (first category on every perfect set, coverable by infinitely many intervals of any given length, etc.) that are provably uncountable, but with no hope of showing that they have the cardinality continuum; and some "paradoxical" consequences (e.g. a set of reals of the cardinality continuum such that any dense open set covers all but countably many of its points).

He also makes a famous remark about accepting new axioms based on the "fruitfulness in consequences and in particular in "verifiable" consequences, i.e., consequences demonstrable without the new axiom, whose proofs by means of the new axiom, however, are considerably simpler and easier to discover", and sketches what came to be known as the large cardinal program. Much later (1970s) Gödel claimed to have "proved" that the continuum is no more than aleph two, and speculated that it is exactly that, see Kurt Gödel: Essays for his Centennial, p.175. Gödel's reputation was such that his name attached to a conjecture might have been enough evidence for a "conviction". But, in hindsight, this "evidence" turned out to be pretty thin. From Cohen's The Discovery of Forcing:

"After the publication of Gödel’s result in 1939 and the appearance about a year later of a detailed exposition as lecture notes, there were a total of four papers to my knowledge which in any way dealt with his construction, until my own work in 1963. One of these, actually a series of three papers, was used by Shepherdson who showed that the method of inner models used by Godel and von Neumann could never show the consistency of the negation of AC or CH. This result evidently received insufficient attention because, when I rediscovered them in 1962, I was urged to publish them despite some reservations I had.

[...] From general impressions I had of the proof, there was a finality to it, an impression that somehow Gödel had mathematicized a philosophical concept, i.e., constructibility, and there seemed no possibility of doing this again, especially because the negation of CH and AC were regarded as pathological. I repeat that these hazy and even self-contradictory impressions I had were strictly my own, but, nevertheless, I think that it is very possible that others had similar impressions. For example, as a graduate student I had looked at Kleene’s large book, and there was very little emphasis or even discussion of the entire matter.... Furthermore, there was little mention of the problem of showing the consistency of the negation of CH. One reference was an expository article of Gödel, in which he refers to it as a likely outcome, but hardly seems to refer to it as a pressing problem for research.

[...] A rumor had circulated, very well known in all circles of logicians, that Gödel had actually partially solved the problem, specifically as I heard it, for AC and only for the theory of types (years later, after my own proof of the independence of CH, AC, etc., I asked Godel directly about this and he confirmed that he had found such a method, specifically contradicted the idea that type theory was involved, but would tell me absolutely nothing of what he had done)... It seems that from 1941 to 1946 he devoted himself to attempts to prove the independence. In 1967 in a letter he wrote that he had indeed obtained some results in 1942 but could only reconstruct the proof of the independence of the axiom of constructibility, not that of AC, and in type theory (contradicting what he had told me in 1966). After 1946 he seems to have devoted himself entirely to philosophy."

  • $\begingroup$ Now THAT'S the way to write an answer ! $\endgroup$
    – Clive Long
    Commented Jan 25, 2022 at 10:07

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