Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned out to be undecidable based on the current set of the axioms of mathematics.

However, as Woodin stated in a panel discussion, this independence result may indicate that there are some missing mathematical principles that need to be discovered and added to $\text{ZFC}$ in order to give us enough mathematical tool to settle the question of the true size of the continuum in the "correct" way. So in some sense, Hilbert's first problem is still open and the search for an answer to the question of determining the true size of the set of real numbers is still ongoing.

It is worth mentioning that amongst various transfinite values that $|\mathbb{R}|$ can take up to consistency, set theorists have isolated one particularly special one, namely $\aleph_2$, as the most likely value of the continuum.

It is often said that both Goedel and Cohen favored $\aleph_2$ as the "true" size of the continuum as well but I have seen no direct reference to any original quote of them concerning this. In the case of Cohen, who had a strong background in analysis (mostly related to the Littlewood Conjcture), it is particularly important to know whether his choice of $\aleph_2$ (if real) was based on some possibly deep understanding of the mathematical machinery in analysis or not.

Question 1. What are references to some original works/interviews of Goedel and Cohen in which they clearly expressed their opinion about the true size of the continuum? What were their reasons to believe in such a specific value for $|\mathbb{R}|$?

Though, the story of considering $|\mathbb{R}|=\aleph_2$ goes far beyond mere speculations of Goedel and Cohen. There are actually some mathematical theorems which could be interpreted as a justification for such an assumption.

For instance, Woodin's $\Omega$-logic argument in favor of $2^{\aleph_0}=\aleph_2$ initially convinced him to believe that $\aleph_2$ is the true value of continuum, however, later he changed his mind in favor of the Ultimate $L$ principle which implies $2^{\aleph_0}=\aleph_1$. Also, some forcing axioms such as PFA and Martin's Maximum (which successfully settle many independent statements of ordinary mathematics) directly imply the value $\aleph_2$ for the continuum. Here the following natural question arises:

Question 2. What are some other examples of mathematical evidence backing $|\mathbb{R}|=\aleph_2$? Maybe some machinery in certain parts of mathematics which work more smoothly if one assumes $2^{\aleph_0}=\aleph_2$ rather than $\aleph_1$ or any other cardinal $\geq\aleph_3$? Maybe some mathematical objects (including the real line itself) start to behave "nicely" or demonstrate some "regularity properties" if the continuum is exactly $\aleph_2$?

Please provide references to the results (and quotes from mathematicians) if there is any. Evidence from mathematical disciplines other than set theory are especially welcome.

• Re: Woodin changing his mind, I believe Sargsyan's thesis showed that there was a serious flaw in that approach; maybe someone familiar with inner model theory can comment (or correct me)? Incidentally, my recollection is that Cohen said quite the opposite: that he tended towards the continuum being quite large, e.g. above the first weakly inaccessible. And re: question 2, strong forcing axioms imply $2^{\aleph_0}=\aleph_2$ and give nice properties of the continuum. – Noah Schweber Jul 30 '18 at 0:03
• @NoahSchweber As for Cohen, I also have heard two different stories (i.e. $|\mathbb{R}|=\aleph_2$ or very large). I am not sure which is historically accurate and that is why I asked for reliable references here! Maybe just like Woodin he also changed his mind at some point! Not quite sure anyway! – Morteza Azad Jul 30 '18 at 0:10
• Following up re: Woodin: from this answer and its attendant comments I believe that the issue was that part of Woodin's "$\Omega$-program" relied on the claim that from the $\Omega$-conjecture it follows that the set of $\Omega$-validities is definable in (or around?) $H(\mathfrak{c}^+)$. This claim, or rather the argument for it given, was flawed, and this was demonstrated in Sargsyan's thesis. Meanwhile, I think the answers over there say a bit about why $\aleph_2$ is attractive, too, especially Justin Moore's. – Noah Schweber Jul 30 '18 at 0:32
• From the answers to that question we also get Cohen on the value of the continuum. – Noah Schweber Jul 30 '18 at 0:35
• Justin Moore has an article on this very subject: What makes the continuum $\aleph_2$ – Iian Smythe Jul 30 '18 at 2:28

A partial answer to (2): my recollection is that the strongest viewpoint Godel ever put forth on the value of the continuum was that it should be $\aleph_2$, this being captured in his manuscript "Some considerations leading to the probable conclusion that the true power of the continuum is $\aleph_2$." He gave a rather involved argument (with various issues) which is analyzed in this paper of Brendle, Larson, and Todorcevic. (This was all mentioned in Moore's article which Iian referenced, which was what reminded me of it.) Godel's manuscript itself can be found in his collected works.
Besides the technical results, I think the relevant takeaways from the Brendle/Larson/Todorcevic paper are that (a) the additional axioms Godel postulated are poorly understood themselves at present and (b) there doesn't seem to be a tight connection between Godel's ideas in that manuscript and current approaches like ultimate $L$, generic large cardinals, forcing axioms, ... However, I'm far from an expert, so take this conclusion with a grain of salt. Ultimately, what I draw from it is that the program Godel outlined doesn't yet constitute "strong evidence for $2^{\aleph_0}=\aleph_2$" (that is, a strong argument in favor of adopting some additional axioms which happen to imply $2^{\aleph_0}=\aleph_2$; "evidence," of course, is a bit of an odd word to use here, but oh well).
Judah intensively discusses why the actual evidences accumulated by 30 years of forcing considerations suggest that the most reasonable size for the continuum is $\aleph_2$.