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.
