What is the general opinion on the Generalized Continuum Hypothesis?
There is definitely a not-CH tendency among set theorists with a strong Platonist bent, and my impression is that this is the most common view. Many of these set theorists believe that the large cardinal hierarchy and the accompanying uniformization consequences are pointing us towards the final, true set theory, and that the various forcing axioms, such as PFA, MM etc. are a part of it.
Another large group of set theorists working in the area of inner model theory have GCH in all the most important models that they study, and regard GCH as one of the attractive regularity features of those inner models.
There is a far smaller group of set theorists (among whom I count myself) with a multiverse perspective, who take the view that set theory is really about studying all the possible universes that we might live in, and studying their inter-relations. For this group, the CH question is largely settled by the fact that we understand in a very deep way how to move fom the CH universes to the not-CH universes and vice versa, by the method of forcing. They are each dense in a sense in the collection of all set-theoretic universes.
Here is a historical answer of sorts. I'm looking at a copy of a spirit-duplicated questionnaire, dated August 1, 1967, which was circulated at the AMS-ASL 1967 Summer Institute in Axiomatic Set Theory. The notation "80 ballots cast" is pencilled in, rather sloppily. The tally of votes for each answer is inked in by someone with neat handwriting. From the numbers, I surmise that IC was only answered by those who answered "meaningless" to IA.
It would be interesting to know if this survey has been published somewhere.
AMS-ASL Summer Institute
in
Axiomatic Set Theory
OFFICIAL BALLOT
[pencilled in: "80 ballots cast"]
I. A. I believe that the proposition
$\ \ \ \ \ \ \ \ \ $'The axiom MC of measurable cardinals is true in the real universe of sets'
is
$\quad$(38) meaningful $\quad$ (38) meaningless
$\ \ \ \ $B. (To be answered only if your answer to A is "meaningful")
$\ \ \ \ \ \ \ \ \ $(8) I think that MC is almost certainly true
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely true than false
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely false than true
$\ \ \ \ \ \ \ \ \ $(2) I think MC is almost certainly false
$\ \ \ \ \ \ \ $(14) I have no idea whether MC is true or false
$\ \ \ \ $C. Regarding the prediction that MC will someday be refuted in ZF,
$\ \ \ \ \ \ \ \ \ $(0) I think this prediction is almost certainly true
$\ \ \ \ \ \ \ \ \ $(2) I think this prediction is more likely true than false
$\ \ \ \ \ \ \ $(16) I think this prediction is more likely false than true
$\ \ \ \ \ \ \ \ \ $(8) I think this prediction is almost certainly false
$\ \ \ \ \ \ \ \ \ $(4) I have no idea whether this prediction is true or false
II. A. I believe that the proposition
'The continuum hypothesis CH is true in the real universe of sets'
is
$\quad$(42) meaningful $\quad$ (35) meaningless
$\ \ $B. (To be answered only if your answer to IIA is 'meaningful')
$\ \ \ \ \ \ $(2) I think CH is almost certainly true
$\ \ \ \ \ \ $(2) I think CH is more likely true than false
$\ \ \ \ $(12) I think CH is more likely false than true
$\ \ \ \ $(14) I think CH is almost certainly false
$\ \ \ \ $(12) I have no idea whether CH is true or false.
$\ \ $B'. (To be answered only if your answer to IIA is 'meaningless')
$\ \ \ \ $(1) My position on IIA
$\quad\quad$(2) does$\quad$(33) does not
$\ \ \ \ \ \ $cast doubt in my own mind on the value of set theory.
$\ \ \ \ $(2) I am inclined to think that set theory based on the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is destined to play in the long-range future develop-
$\ \ \ \ \ \ \ \ \ $ment of mathematics a
$\ \ \ \ \ \ \ \ \ $(11) more important role than
$\ \ \ \ \ \ \ \ \ $(13) role of equal importance with
$\ \ \ \ \ \ \ \ \ $(11) less important role than
$\ \ \ \ \ \ \ \ \ $set theory based on the denial of the continuum hypothesis.
$\ \ $ C. Assuming that human mathematicians still exist then, I believe that
$\ \ \ \ \ \ \ $in 2067 the prevailing opinion among them will be that the continuum
$\ \ \ \ \ \ \ $problem:
$\ \ \ \ $(4) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the continuum hypothesis is
$\ \ \ \ \ \ \ \ \ $a consequence
$\ \ $(18) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the denial of the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is a consequence
$\ \ $(37) has been settled by the general acceptance of the belief that
$\ \ \ \ \ \ \ \ \ $there is no one true set theory and that the continuum hypothesis
$\ \ \ \ \ \ \ \ \ $simply holds in some theories and fails in others
$\ \ $(11) is still unsettled
III. A. I believe that there is an absolute sense in which every sentence of
$\ \ \ \ \ \ \ \ \ $first-order number theory based on addition, multiplication, and
$\ \ \ \ \ \ \ \ \ $exponentiation is either true or false.
$\quad$(54) yes $\quad$ (26) no
$\ \ \ \ $B. I believe that there is an absolute sense in which every $\underline{\text{universal}}$
$\ \ \ \ \ \ \ \ \ $sentence of first-order number theory based on addition, multiplication,
$\ \ \ \ \ \ \ \ \ $and exponentiation is either true or false.
$\quad$(62) yes $\quad$ (18) no
Please do not sign your ballot.
August 1, 1967
University of California, Los Angeles
After oscillating furiously in the 1960's and 1970's, the Berkeley Continuum Meter settled on $2^{\aleph_0} = \aleph_2$ for a large part of the 1980's and 1990's with occasional dips to $2^{\aleph_0} = \aleph_1$. These dips started getting stronger in the last decade, I'm starting to suspect a Cardinal Shrinking Crisis... Somebody call Al Gore!!!