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.

For those of us who don't know what GCH is, can you unacronymize it once so we know? :-) — Kevin H. Lin, Feb 06 '10 at 01:28
I'm told that Erdos believed that GCH was a fact and anything else was blasphemy. — François G. Dorais, Feb 06 '10 at 01:34
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 06 '10 at 01:35
I think I started to like set theory! So much room for imagination (: — Hailong Dao, Feb 06 '10 at 01:38
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 06 '10 at 02:05
Could somebody please elaborate on "a lot of functional analysis depends on GCH"? — algori, Feb 07 '10 at 00:54
@algori: To say that "functional analysis depends on CH" is a misstatement. CH has been used to construct C* algebras and related structures. A survey was published by Nik Weaver in the Bulletin of Symbolic Logic 13 (2007). Recent work Ilijas Farah and others shows that some important and sophisticated set theoretic principles are closely tied to the theory of C* algebras. — François G. Dorais, Feb 07 '10 at 01:32
@Martin: For some reason there is also a tag for [tag:gch] which fits better here. — Asaf Karagila, Dec 07 '17 at 08:57
@AsafKaragila Thanks for letting me know, I completely missed that. Now that the character limit is increased to 35, maybe a synonym (generalized-continuum-hypothesis) $\to$ could be created. I guess the tag would be easier to find in the longer form. — Martin Sleziak, Dec 07 '17 at 10:33
See also https://math.stackexchange.com/questions/499560/why-is-the-continuum-hypothesis-believed-to-be-false-by-the-majority-of-modern-s/499947. — user76284, May 19 '20 at 09:46

score 47 · Accepted Answer · answered Feb 06 '10 at 03:05

47

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.

answered Feb 06 '10 at 03:05

Joel David Hamkins

224,022

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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 06 '10 at 04:46
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@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 06 '10 at 05:06
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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 06 '10 at 05:23
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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 06 '10 at 05:47
Thanks: that's helpful. I've heard of Maddy's work before, and will go and have a look. – Tom Leinster Feb 06 '10 at 05:55
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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 06 '10 at 14:03
@JoelDavidHamkins "Many set theorists take this as evidence that these large cardinal axioms put us on the right path towards the ultimately correct set theory." But how can anything get us closers to an ultimate set theory, if there is none (at least no effective ones)? For any effective set theory, there will be things true statements that are left out of it. – Christopher King Oct 10 '16 at 19:33
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@PyRulez Well, the large cardinal theories and the forcing axioms are computably axiomatizable, and they are held to be true for the philosophical reasons I mentioned. So the view is that they are a larger fragment of the truth than having, say, only the ZFC axioms. But of course, there is no computably axiomatizable complete theory. Your argument is something like: how can any food be delicious, if there is no perfectly delicious food? – Joel David Hamkins Oct 10 '16 at 21:24
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@JoelDavidHamkins Well you did say "ultimately correct set theory", not "more correct set theory" or "delicious set theory". I get what your mean though. – Christopher King Oct 10 '16 at 21:37
I had meant that those theories are taken to be approaching the true theory, not that they are equal to it. – Joel David Hamkins Oct 10 '16 at 21:38
@JoelDavidHamkins I don't think there is a computable sequence of effective theories approaching any complete theory, since then you could compute the axioms of your complete theory by dovetailing. – Christopher King Oct 11 '16 at 19:16
Yes, that is obvious. And? All I'm saying is that some set theorists think that the large cardinal axioms and axioms like PFA, MM and so on are really true. – Joel David Hamkins Oct 11 '16 at 19:32
Isn't the "ultimately correct set theory" (for us or for any denizen of any 'universe' of the set-theoreic multiverse) the minimal (in the sense of reverse mathematics) set theory needed to provide a foundation for mathematics at its current state of development? The question now becomes: "How does one advance mathematics so as to make the current set-theoretic foundation inadequate to the task?" . Why would this not be the case? – Thomas Benjamin Nov 19 '17 at 13:51
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No, the "ultimately correct set theory," according to the Platonists I had mentioned in my first paragraph, would view it as something closer to a maximal theory than a minimal theory. The universist, who holds that there is ultimately a unique set-theoretic universe, might believe that the ultimately true set theory is a complete theory, that every set-theoretic assertion has a truth value, even if it is difficult for us to determine. – Joel David Hamkins Nov 19 '17 at 13:58
@JoelDavidHamkins: That may be, but wouldn't these Platonists' "True $ZFC$" be in the same boat (so to speak) as "True Arithmetic", i.e., nonaxiomatizable? And since "True Arithmetic" can have nonstandard models, so can "True $ZFC$". Are these Platonists willing to count nonstandard integers and infinitesimals (and whatever other nonstandard sets the nonstandard "True $$ZFC$" might offer) as being in their Ideal realm? – Thomas Benjamin Nov 19 '17 at 14:11
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@ThomasBenjamin Saying that Platonists believe in a true theory of ZFC does not imply that's the only thing they believe in. As Joel mentioned, the "universist" view is that there is one unique real set-theoretic universe. Of course whatever axiomatic theory this universe satisfies, there exist other models of the same set of axioms, but, e.g., just because there may be other models of our laws of physics does not imply that those models are as real as our universe. – Will Sawin Dec 07 '17 at 07:33
@WillSawin: So the "universist" view does not require "True $ZFC$" to be categorical? How does Categoricity fit (if at all) into the universist point of view ("True $ZFC$" would still be nonaxiomatizable, though....)? – Thomas Benjamin Dec 07 '17 at 09:22
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@ThomasBenjamin I don't see why it would require the theory to be categorical. Again, the theory just says that there is some set-theoretic universe, satisfying the axioms of ZFC, which is real, and all other universes are either not real, or contain only some of the really-existing sets. The first-order statements true in this universe form a theory, "True ZFC", which as you say is non-axiomatizable. This theory might or might not gave any other number of models, but the model is the primary thing, not the theory. – Will Sawin Dec 07 '17 at 10:04
@WillSawin: Wasn't the Universist view that there is a single 'true' universe of sets based on the categoricity of full second-order $PA$ and the partial categoricity of full second-order $ZFC$ and as such, isn't the notion of categoricity (external or internal) an integral part of the Universist point of view? – Thomas Benjamin Dec 10 '17 at 18:02
@ThomasBenjamin See https://plato.stanford.edu/entries/philosophy-mathematics/#Cat. – user76284 May 19 '20 at 09:51

bof · Answer 2 · 2017-12-07T11:28:23.283

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

Wow, that's pretty awesome. Now I want to do that in the next conference I attend. — Asaf Karagila, Jul 07 '14 at 08:16
@Gerry: Considering that the next conference I attended after that post was over a year later, no. I hadn't even remembered this more than a couple of months after posting, I guess. Certainly not until today. — Asaf Karagila, Oct 18 '17 at 23:04
Apparently that 1967 survey was never published? Should I expand my answer to include the rest of that old survey, i.e., the part about measurable cardinals and the part about number theory? — bof, Oct 19 '17 at 03:16
@bof At least one person (me) would be interested in reading the extended survey. That might not be sufficient reason to derail the thread. If you decide not to add in the rest of the survey, could you send me an email with the survey at asving@umich.edu? Thanks. — Asvin, Nov 19 '17 at 12:30
I would also like to see the full survey, and I suggest that you post it. — Joel David Hamkins, Nov 19 '17 at 13:32
Thanks! It is really great. I wonder if we might be able to find out an attendee list, to get some idea of who were the likely respondents. I am surprised by the level of opposition to the absolute universist view of truth, in 1967. — Joel David Hamkins, Dec 07 '17 at 03:28
@JoelDavidHamkins I believe there was a conference book, and it must have contained a list of participants. — bof, Dec 07 '17 at 03:30
We have the list of ~50 people who spoke at the institute, a cast well-known even 50 years later. While the proceedings are not readily available online, the end matter is: http://www.ams.org/books/pspum/013.1/pspum013.1-endmatter.pdf (vol 1) says that the organizers were Paul Cohen, Abraham Robinson and Dana Scott; http://www.ams.org/books/pspum/013.2/pspum013.2-endmatter.pdf (vol 2) has the list of lectures on pp. 11-14. — , Dec 10 '17 at 01:03
+1 for the informative and meaningful question II B' (2). I wonder who was there that would have written that survey question so well -- googling the unusual phrase "I am inclined" (https://www.google.com/search?q="i+am+inclined"+mathematician), would it have been Halmos? — , Dec 10 '17 at 01:31
@MattF. Hmm. 125 people attended, and 80 people voted, so not just the invited speakers. — bof, Dec 10 '17 at 02:40
@MattF. Wow, I didn't know that the phrase "I am inclined" has become unusual in these diminished times. I don't know who constructed the survey, but I'm inclined to suspect John Addison, for no reason that I can put my finger on. Well, maybe because it is so carefully written. — bof, Dec 10 '17 at 02:42
@bof, I too use that phrase, but I'd never seen it in a survey! — , Dec 10 '17 at 03:02

score 24 · Answer 3 · answered Feb 06 '10 at 01:48

24

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!!!

answered Feb 06 '10 at 01:48

François G. Dorais

43,723

1

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 06 '10 at 01:50
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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 06 '10 at 02:04
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 06 '10 at 02:09
A solution to what? – François G. Dorais Feb 06 '10 at 02:11
I don't know, something like a "best of both worlds" sort of scenario? – Harry Gindi Feb 06 '10 at 02:28
I thought that Woodin's recent work has moved the Berkeley continuum meter away from Aleph_1, but I may be wrong. This kind of stuff is not very popular in my place. :( – Haim Feb 06 '10 at 21:57
@Haim: I don't have much of a personal opinion. My answer should not be taken too seriously, but it does match what I've been hearing (from Berkeley and elsewhere) over time... – François G. Dorais Feb 07 '10 at 02:07
@francois: PFA implies the singular cardinal hypothesis, which is saying it has pretty drastic effects even above the continuum – Justin Palumbo Feb 07 '10 at 03:05
.. GCH => SCH... – Harry Gindi Feb 07 '10 at 03:39
@Justin: Wouldn't the failure of SCH be a drastic event by itself? (I always thought Viale's result was easier to understand in the contrapositive.) – François G. Dorais Feb 07 '10 at 14:02
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@Justin: However, you're completely right that I shouldn't have said that PFA and MM are "very local" since at the heart of their local nature is a very global reflection principle. – François G. Dorais Feb 07 '10 at 17:39
@François: If we take ZFC+MM, at what point does GCH "come into effect", as it were? – Harry Gindi Feb 11 '10 at 17:15
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@Harry: There is no fixed point. The usual way to get these forcing principles will preserve GCH from some point on. – François G. Dorais Feb 11 '10 at 17:45

score 22 · Answer 4 · answered Feb 06 '10 at 01:46

22

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.

answered Feb 06 '10 at 01:46

Tom Leinster

27,167

3

Would it be fair to say that counting, beyond large/small/countable/finite, is mostly irrelevant in category theory? – François G. Dorais Feb 06 '10 at 02:20
Well, Universe successors are important. – Harry Gindi Feb 06 '10 at 02:29
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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 06 '10 at 02:46
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 06 '10 at 03:10
Sorry, François, maybe I misunderstood you. Were you interested in the relevance of counting infinite sets in category theory? – Tom Leinster Feb 06 '10 at 04:13
I think we misunderstood each other. My use of finite includes all finite cardinals, but I see how that was unclear in my original question. I then understood from your answer that combinatorial species were also used to count infinite objects. Anyway, I have seen infinite cardinals (even large cardinals!) appear in works on accessible categories. I don't know if this is very exceptional or not. – François G. Dorais Feb 06 '10 at 04:23
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Yes, the accessible category stuff does seem to reach deep into issues about large cardinals. I don't know much about it. Maybe Mike Shulman will turn up and shed some light. Anyway, I think this is an exception. – Tom Leinster Feb 06 '10 at 05:00
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In a general topos I don't think the question of GCH is particularly meaningful. Without the axiom of choice, there's no reason for 2^c to be an aleph, so it doesn't even make sense to ask which aleph it is. I'd be surprised to find many toposes that violate AC but in which GCH is "true" in the sense that P(N) is isomorphic to the set of well-orderings on N. – Mike Shulman Feb 06 '10 at 16:52
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Large cardinals do turn up in accessible categories and even occasionally some other places in topos theory, like whether there can be an inaccessible lex endofunctor of Set. I don't know to what extent these sorts of things happen without AC or even intuitionistically. But I'm not sure whether any of that is very related to GCH. No large cardinal hypotheses have been found that decide GCH one way or the other, have they? – Mike Shulman Feb 06 '10 at 16:54
The problem with this viewpoint in my opinion, Tom, is that a topos is meant to have sets for homsets, and those sets are meant to live in a universe of sets, and that universe either satisfies $\mathrm{GCH}$, or it doesn't. But perhaps this can be gotten around somewhat by considering each topos as canonically self-enriched. – goblin GONE Jun 27 '15 at 17:19
2

@goblin a topos is meant to have sets for homsets <-- toposes are described by some first-order theory, and don't have to rely on any kind of ambient sets. They of course are cartesian closed, so have an "internal" homsets, but the "real" collection of arrows $a\to b$ doesn't have to be anything like a set... – David Roberts Oct 19 '17 at 03:14

score 14 · Answer 5 · answered Feb 20 '11 at 16:31

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.

score 6 · Answer 6 · answered Feb 06 '10 at 21:25

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.''

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

score 2 · Answer 7 · edited Nov 19 '17 at 12:10

2

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

edited Nov 19 '17 at 12:10

Glorfindel

2,743

answered Feb 08 '10 at 10:38

Christoph-Simon Senjak

757

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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 08 '10 at 13:58
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I think this is too naive to be useful. If you adopt $\mathsf{GCH}$ because of how it "simplifies mathematics", it would be desirable that you explain how to reconcile this with the many known examples of classification problems that are much harder under this assumption than under natural alternatives. – Andrés E. Caicedo Nov 19 '17 at 14:11

What is the general opinion on the Generalized Continuum Hypothesis?

7 Answers7

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