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Gödel was the first to show that ~CH was not provable from ZFC. However, he also thought CH was false in his view of the "Platonic" reality of set theory. It seems this view was also somewhat common among set theorists of a Platonist bent, until Cohen's later forcing result.

Does anyone know what Gödel's reasoning was for CH being false? Did he ever write anything about it, addressing his views on the subject?

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  • $\begingroup$ Are you asking for a source for the statement that Gödel though CH was false? $\endgroup$ – Lee Mosher Apr 6 at 16:13
  • $\begingroup$ One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability." $\endgroup$ – spaceisdarkgreen Apr 6 at 18:46
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    $\begingroup$ I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $\mathfrak c =\aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $\mathfrak c =\aleph_2,$ and even CH, more recently.) $\endgroup$ – spaceisdarkgreen Apr 6 at 19:16
  • $\begingroup$ Cohen also thought CH was false. $\endgroup$ – user4894 Apr 9 at 9:11
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There is a classical survey of Gödel about the continuum hypothesis:

"What is Cantor's Continuum Problem", K. Gödel, The American Mathematical Monthly, Vol. 54, No. 9 (Nov., 1947), pp. 515-525

In section 4, he discusses "in what sense and in which direction a solution of the continuum problem may be expected". While this is of course just a survey, it still represents some of Gödel's individual thoughts about the subject at the time.

A barrier free link is right now e.g. this.

Edit: (by David Richerby) He says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523), though there was never really any wide agreement with Gödel that these were so implausible to be worth singling out.

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    $\begingroup$ Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful. $\endgroup$ – David Richerby Apr 6 at 20:00
  • $\begingroup$ @DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $\mathfrak c=\aleph_2,$ but that too was considered a failure. $\endgroup$ – spaceisdarkgreen Apr 6 at 22:09
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    $\begingroup$ There's nothing at all wrong with this answer. The post has two interelated questions including "Did he ever write anything about it, addressing his views on the subject?", and this is a fine answer to that one. $\endgroup$ – Lee Mosher Apr 7 at 3:14
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    $\begingroup$ I first apologize to all the people waiting for my edit. I agree that the answer is thin as to almost just giving a link and a appropriate edit would be helpful. Although I agree with the diplomacy regarding the made and revoked edit on my post, I will this time implement the addition suggested by @DavidRicherby as it is a really nice summary of the main point and I myself don't want to go over into a copycat state. $\endgroup$ – blub Apr 7 at 9:38
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    $\begingroup$ @Quid: I agree, my comment was a response to another comment (now deleted) that was calling for this answer to be deleted. I'll probably therefore delete my comment at some point. $\endgroup$ – Lee Mosher Apr 7 at 14:36
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Gödel's view on CH changed over his lifetime, so it is hard to give a comprehensive answer to the question about his reasoning. It evolved over the years, and toward the end of his life he even came to believe that the CH may be true (although he still believed the GCH was false).

Fortunately, there is a three-volume collected works of Gödel, and most of what I say here is gleaned from the commentary in there, as well as some secondary sources I gave in the comments below the questions.

First off, I should say that while many of Gödel's philosophical ideas on set theory from the mid 40s onward (i.e. after his development of the $L$ hierarchy and proof of the consistency of AC and GCH) are regarded as important, even if they weren't all super influential at the time, his ideas on the specific question of the absolute truth of CH are mostly considered dead ends.

With that said, the natural place to start is his proof of the consistency of GCH in the late 30s. He did this by defining the constructible sets $L,$ and showing that they form a model of ZFC + GCH. In his initial development, Gödel believed that the great clarification of the set concept given by his axiom of constructibility was perhaps the missing piece needed to decide our set theoretical questions. This, of course, would amount to a belief that CH is true.

However he quickly reversed this position and came to what has since been the dominant view among Platonists that the axiom of constructibility is obviously false. In his 1947 expository paper What is Cantor's Continuum Problem?, he likens the constructible sets to a model of non-Euclidean geometry constructed within Euclidean geometry: while this establishes the consistency of non-Euclidean geometry, it has no bearing on the "true" Euclidean universe. The axiom does clarify the notion of a set, but it does so by placing severe restrictions on what a set is, saying they all need to be obtained from transfinite iteration of simple constructive operations. This, to Gödel and the majority of set theorists after him, seemed to be the exact opposite of what a principle guiding the concept of an arbitrary set should do.

In the same passage, Gödel argued that the CH was probably not provable from ZFC. Essentially, although it may be the the wrong clarification, the axiom of constructibility does seem to be a very strong clarification of what sets there are, and it would be odd if a question like CH did not require this clarification (or one of similar magnitude) in its solution. (Of course on this point, Gödel was resoundingly correct.)

Now to finally touch on the issue in your question. As a secondary argument that CH is not provable, he asserts that it is probably false. His argument is fairly thin: he states without much elaboration that he finds several descriptive set theory consequences of CH to be implausible. (For instance the existence of uncountable absolute measure zero sets and Sierpinski sets.) My descriptive set theory is pretty weak, so I don't know quite what to make of this, but eminent set theorist Donald Martin has said

While Gödel's intuitions should never be taken lightly, it is very hard to see that the situation is different from that of Peano curves, and it is even hard for some of us to see why the examples Gödel's cites are implausible at all.

(Peano curves are a counterintuitive construction that does not require CH that Gödel claims without much substantiation that the situation is different for.)

So although much of this article was insightful (including a lot of stuff I didn't touch on about the direction forward in finding new axioms), Gödel's arguments for the falsity of CH were not taken up by the mathematical community.

Gödel didn't have much output between 1947 and the advent of forcing in the early 60s (although he had attempted with some progress to establish the consistency of the negation of choice). Cohen's proof was more than just a confirmation that ZFC could not prove CH: it showed that $2^{\aleph_0}$ could consistently take arbitrarily large values, and that the meager facts that we already knew about the size of the continuum were essentially all that ZFC could tell us. This intensified what was already a suspicion in the set theory community that the continuum was probably very large.

While he was rightfully in awe of Cohen's work, Gödel had of course long believed that CH was not provable and had been looking in other directions. He had expressed hope that large cardinal axioms would decide the CH, but shortly after the advent of forcing it was discovered by Levy and Solovay that this would not work. (Despite this, the large cardinal program has been very fruitful in general, and did strongly refute the axiom of constructibility.) Meanwhile, he had hit upon an old idea of Hausdorff that seemed to him to produce tractable conjectures that were intuitively true and informative on the continuum.

This work is the subjuct of an unpublished handwritten note in 1970s in which Gödel claims to have a convincing argument that $2^{\aleph_0}=\aleph_2.$ Details can be found in the collected works or Kanamori's Gödel and Set Theory. Interestingly, once he had formulated his axioms (known as "rectangle axioms") it was discovered that, amongst other issues, they actually implied the CH rather than $2^{\aleph_0}=\aleph_2.$ Undeterred, he came around to the belief that the CH was probably true after all, and that in any event this approach would give strong evidence that the continuum was small (no larger than $\aleph_2$).

Although this was considered a failure (an interesting one by some), Kanamori notes that it rhymes with the broader history of set theory post-Cohen. After years of believing the continuum was large, set theorists began to seriously consider some deep principles that would imply (of all things) $2^{\aleph_0}=\aleph_2.$ And sure enough, shortly after Kanamori wrote this, another principle came into vogue that would imply CH. Which goes to show that thinking about the truth of CH itself has largely given way to thinking about what deeper and more general principles should hold, and then accepting whatever they imply about the CH.

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