What is the general opinion on the Generalized Continuum Hypothesis?

Question

I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this.

From what I've seen, model theorists and logicians are mostly opposed to GCH, while on the other end of the spectrum, some functional analysis depends on GCH, so it is much better tolerated among functional analysts. In fact, I considered myself very much +GCH for a while, but Joel and Francois noted some interesting stuff about forcing axioms, (the more powerful ones directly contradict CH).

What is the general opinion on GCH in the mathematical community (replace GCH with CH where necessary)? Does it happen to be that CH/GCH doesn't often come up in algebra?

Please don't post just post "I agree with +-CH". I'd like your assessment of the mathematical community's opinion. Maybe your experiences with mathematicians you know, etc. Even your own experiences or opinion can work. I am just not interested in having 30 or 40 one line answers. Essentially, I'm not looking for a poll.

Edit: GCH=Generalized Continuum Hypothesis CH= Continuum Hypothesis

CH says that $\aleph_1=\mathfrak{c}$. That is, the successor cardinal of $\aleph_0$ is the continuum. The generalized form (GCH) says that for any infinite cardinal $\kappa$, we have $\kappa^+=2^\kappa$, that is, there are no cardinals strictly between $\kappa$ and $2^\kappa$.

Edit 2 (Harry): Changed the wording about FA. If it still isn't true, and you can improve it, feel free to edit the post yourself and change it.

Answer 1

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.

Answer 2

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

Answer 3

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!!!

Answer 4

Among the group of mathematicians that I know best, category theorists, a common attitude is as follows. (G)CH is neither true nor false. It's something that a model of (any given collection of) axioms for set theory might or might not satisfy --- and that's that. So being "pro" or "anti" doesn't make sense. If there were a God-given model of set theory then we could ask whether (G)CH was true in it, but there isn't.

(I'm probably projecting here, but even if this isn't a majority opinion among category theorists, I'm pretty sure it's a common one.)

I'm not aware of any work in category theory that depends on (say) a topos satisfying or violating (G)CH. My ignorance doesn't mean much, though.

Answer 5

I think it should be pointed out that, while many people working in set theory have a strong opinion about CH (with many feeling it is "false"), they generally do not have strong feelings against GCH above $\aleph_0$. That is, GCH is not such a controversial statement except that it implies CH.

Answer 6

My impression is that most mathematicians these days who work outside of mathematical logic would agree with the following statement: ``If at all possible, one should try to prove theorems within ZFC, or at least within ZFC plus some mild large cardinal axioms (e.g. the existence of inaccessible cardinals).''

I think this is even true in model theory, and I admit to having this bias myself -- partly because I generally want to prove things using as few hypotheses as possible, and partly just because ZFC is the system that I'm most used to. (I say this as someone who proved a result using Martin's Axiom in my thesis, and then was very happy when I later found a ZFC proof.)

To give an example of this attitude, in the 1970's pure model theory seemed to be getting more ''set theoretic,'' with natural statements such as Chang's Conjecture being proven to be independent of ZFC. I've heard that some model theorists were grateful to Shelah for demonstrating (in his 1978 book Classification Theory) that in fact there still were deep model-theoretic results that could be obtained in ZFC alone. The strategy was to define classes of well-behaved theories (e.g. the superstable theories) where one could prove within ZFC that the category of models is ''nice,'' and then show (again in ZFC) that the models of any theories not satisfying these tameness properties are ''as wild as possible.''

Answer 7

I am Formalist. And from a formal point of view, GCH is independent from ZF(C) - nothing more. To me, sets are not really "existing" (have you ever seen an infinite ordinal flying around somewhere? well, I didn't.) - but assuming GCH could make mathematics easier - and that is what is important to me. Without GCH, there must be at least one set (and therefore infinitely many sets) with an undecidable cardinality.

On the other hand, with GCH, if you can prove $|A| > \aleph_i$ and $|A| \le \aleph_{i+1}$ then you are already done proving $|A| = \aleph_{i+1}$.