# What is the general opinion on the Generalized Continuum Hypothesis?

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.

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For those of us who don't know what GCH is, can you unacronymize it once so we know? :-) –  Kevin H. Lin Feb 6 '10 at 1:28
I'm told that Erdos believed that GCH was a fact and anything else was blasphemy. –  François G. Dorais Feb 6 '10 at 1:34
GCH is shorthand for Generalized Continuum Hypothesis. I don't think I've ever seen anyone mention it in our neck of the woods, though I might have missed it. –  Charles Siegel Feb 6 '10 at 1:35
I think I started to like set theory! So much room for imagination (: –  Hailong Dao Feb 6 '10 at 1:38
I am not sure that "a lot" of functional analysis depends on GCH, although I am prepared to believe that several natural questions will depend on it unless one restricts to separable spaces (or separable preduals) –  Yemon Choi Feb 6 '10 at 2:05

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.

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Joel, my mind is totally boggled by talk of "the final, true set theory". I can't even begin to imagine what anyone could mean by this. And it's not the first time I've heard such talk. I understand that you wouldn't use such a phrase yourself, but can you point me to something easy-read that would help me understand what people mean by this? –  Tom Leinster Feb 6 '10 at 4:46
@Tom: We know that the Incompleteness phenomenon means that we will never have a complete axiomatization of set theory. Our every proposed theory will be ultimately inadequate because of the incompleteness theorem. Nevertheless, the large cardinal hierarchy appears to be providing an increasingly regular and coherent picture of the set theoretic universe, particularly for sets of reals and other issues down low. Many set theorists take this as evidence that these large cardinal axioms put us on the right path towards the ultimately correct set theory. –  Joel David Hamkins Feb 6 '10 at 5:06
Joel, thanks for that. So let's see if I understand correctly. When you (or rather, others) say "ultimately correct set theory", the correctness is judged by aesthetic criteria. I see a picture of the world of sets that looks nice and (to borrow your words) regular and coherent, and gives me results that seem satisfying; so this is the one I choose to call correct. Have I got the right end of the stick? –  Tom Leinster Feb 6 '10 at 5:23
Yes, I personally agree with you, that it is not ultimately a matter of proof, but one of aesthetics. But my impression is also that many of the set theorists with this view would not characterize it as a mere aesthetic choice, but rather hold that their view is getting at some kind of ultimate set-theoretic truth. Some of the math philosophers, such as Penelope Maddy, are trying to explain what this view means. –  Joel David Hamkins Feb 6 '10 at 5:47
I would also suggest the philosophical work of Peter Koellner, as well as Hugh Woodin and Kai Hauser as explaining this view. –  Joel David Hamkins Feb 6 '10 at 14:03

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!!!

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Are the forcing axioms I noted above consistent with any weakened form of the Generalized Continuum Hypothesis? Like, maybe something silly happens for higher cardinals, and it implies GCH for all cardinals not equal to $\aleph_0$ –  Harry Gindi Feb 6 '10 at 1:50
Forcing axioms such as MA, PFA, and even MM are very local and they don't do all that much above the continuum. The usual way of getting them will preserve GCH from some point on, though that does not appear to be necessary. –  François G. Dorais Feb 6 '10 at 2:04
Why doesn't it appear to be necessary? Isn't having MM+(GCH-S), where S is some bounded set of a cardinals an optimal solution? –  Harry Gindi Feb 6 '10 at 2:09
A solution to what? –  François G. Dorais Feb 6 '10 at 2:11
I don't know, something like a "best of both worlds" sort of scenario? –  Harry Gindi Feb 6 '10 at 2:28

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.

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Would it be fair to say that counting, beyond large/small/countable/finite, is mostly irrelevant in category theory? –  François G. Dorais Feb 6 '10 at 2:20
Well, Universe successors are important. –  Harry Gindi Feb 6 '10 at 2:29
Francois: no, I don't think it would be fair. For instance, the theory of combinatorial species (especes de structures) is all about counting, and it's a fundamentally categorical theory, created by Joyal - even though when combinatorialists use it, the category theory often gets hidden. –  Tom Leinster Feb 6 '10 at 2:46
I've only seen combinatorial species used to count finite things; I would love to see them in action in the infinite. –  François G. Dorais Feb 6 '10 at 3:10
Sorry, François, maybe I misunderstood you. Were you interested in the relevance of counting infinite sets in category theory? –  Tom Leinster Feb 6 '10 at 4:13

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. The questionnaire is in three parts. Part I is about measurable cardinals, and part III is about first-order number theory. I will quote part II, which is about the continuum hypothesis. (It would be interesting to know if this survey has been published somewhere.)

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

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Wow, that's pretty awesome. Now I want to do that in the next conference I attend. –  Asaf Karagila Jul 7 '14 at 8:16

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.

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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.''

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Who said there isn't a "God given" model of set theory? We know ZFC doesn't axiomatize it (and that we'll never be able to). But that doesn't mean there isn't such a thing and that we can't attempt to discover its properties. –  Justin Moore Feb 20 '11 at 16:25
I think many people working in set theory would agree that it is better to prove something in ZFC than to prove an independence result. But in many cases, proving an independence result is helpful in formulating ZFC theorems and proving them. Shelah has certainly made statements to this effect. Also, is it better to prove a consistency result classifying a collection of mathematical structures or to prove that a mess of counterexamples exists in ZFC? –  Justin Moore Feb 20 '11 at 16:28
Also, another goal of set theory is also to develop a modest number of extensions of the axioms of ZFC which are sufficient to settle most questions which ZFC leaves unresolved. So even if a non set theorist asks "the wrong question" --- something which ends up being independent of ZFC, there is at least an apparatus in place to settle the question, even if this means showing it is independent. –  Justin Moore Feb 20 '11 at 16:29

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 didnt.) - 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 $|A| = \aleph_{i+1}$.

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Your last sentence works even without GCH. I think you mean to say |A| \leq 2^\aleph_i instead. –  Joel David Hamkins Feb 8 '10 at 13:58