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I could not find a question that seemed to answer my specific query, despite lots of material on the Continuum Hypothesis (CH) on MSE and MO. If there is already a question on this, I'd greatly appreciate a link.

My question has to do with the resilience of CH to being resolved. I realize that it is independent of ZFC and "most" large cardinal axioms, and that CH can fail as "badly as you want" (see this question).

My question is: why is this the case, and is there intuition as to why "ZFC + any large cardinal axiom" cannot seem to anchor this problem? I realize the "multiverse" approach is simply to study CH in whatever universe you are in. But my question whether there is intuition as to why CH has such an indeterminate status no matter what axioms you throw at it. Thanks!

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    There are axioms that determine the truth of $\mathsf{CH}$. For example, $\diamondsuit$ and $\mathsf{PFA}$. – William Aug 20 '14 at 00:19
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    @William: Sure, as well as CH, GCH, and Woodin's work. But my question is about the context within large cardinal axioms. Since large cardinals don't settle CH, some set theorists are comfortable saying that this unsettled state is the natural order (multiverse). I'm interested in the intuition as to why we should accept the unsettled large cardinal state of affairs. –  Aug 20 '14 at 00:25
  • Didn't Woodin suggest Ultimate L to incorporate all large cardinals, and it happens to imply CH? – Conifold Aug 20 '14 at 00:31
  • @Conifold: Yes, per this entry on MO: http://mathoverflow.net/questions/23829/solutions-to-the-continuum-hypothesis. But since we still have the multiverse view per Hamkins, my question still stands. Though, as you imply, Woodin seemed to back off his ~CH stance a bit. –  Aug 20 '14 at 00:43
  • Large cardinals have been historically useful for measuring consistency strength, but I do not think there is any reason to believe they are special in actually proving things. Moreover, it is hard to say what is a large cardinal. $0^\sharp$ is often regarded as a large cardinal axiom. Should $\mathsf{PFA}$ be regarded as a large cardinal axiom. It implies the consistency of Woodin cardinals. Then you would have $2^{\aleph_0} = \aleph_2$. – William Aug 20 '14 at 00:45
  • @William: Agreed, so my question is why are such axioms not accepted as "correct"? The large cardinal framework seems to dominate and not settle CH. –  Aug 20 '14 at 00:52
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    @trb456: There's a lot about history, tradition, the fact that we are interested in both CH holding and failing since both have interesting consequences, and of course, we aim to have a minimal axiomatic setting. So once the dust settled and $\sf ZFC$ seemed like a good candidate, it's hard to add more axioms to the "hardcore of mathematics" without a good reason. – Asaf Karagila Aug 20 '14 at 00:57
  • What do you mean by "dominate"? I do not think anyone believe that mathematics or set theory should be done is something like $\mathsf{ZFC} +$ there exists a Woodin cardinals. There is a difference between studying consistency strength and actually proving things. Even though a supercompact cardinal implies there is a model of $\mathsf{PFA}$, people still work with $\mathsf{PFA}$ since there are things that $\mathsf{PFA}$ can prove that a supercompact cardinal can not. – William Aug 20 '14 at 01:00
  • @Asaf Karagila: Yes, exactly. So what is the intuition, if any, as to why this is? Why is there no good reason? –  Aug 20 '14 at 01:02
  • @William: I realize in set theory that all sorts of non-axioms are assumed and then reasoned from. My question is why none of these have been embraced as the answer. Asaf Karagila's comment encapsulates this view. –  Aug 20 '14 at 01:06
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    Large cardinal axioms are not accepted as "correct" either. They are popular because rich consequences can be derived from them with current methods. Before that forcing axioms were popular, some still are. Before that GCH itself was popular, etc. The reason why they weren't permanently adopted is not specific to CH: none of them are needed in most of mathematical practice. – Conifold Aug 20 '14 at 01:08
  • @trb456: I don't know why my previous comment is not a good enough reason. It's a combination of tradition, the fact that adding new axioms removes interesting problems from the mathematical world, and the fact that we are interested in "less axioms" rather than "more axioms". – Asaf Karagila Aug 20 '14 at 01:14
  • @Conifold: This is a great comment that I wish I could upvote more (others, please upvote!). Your last comment is key: there is no non-CH application that otherwise warrants adoption as an axiom. But then why is resolution of CH not on the same level as "most of mathematical practice"? This is the intuition I am seeking. –  Aug 20 '14 at 01:15
  • @Asaf Karagila: So your answer seems to be for "as big as possible set theory". I get this, so my question is why no hypothesis has caused most set theorists to want to narrow their focus. As you say, this has not happened, so my question is why CH fails to narrow the focus. Why is there no axiom which is considered "natural" in deciding CH? There may be no good intuition, but this is what I am asking. –  Aug 20 '14 at 01:20
  • Your original question was completely different though: why CH isn't 'close enough' to large cardinals. It is not on the same level because it is not as intuitive as AC, doesn't provide highly useful closure properties in classical mathematics like AC, and AC is already suspicious. – Conifold Aug 20 '14 at 01:23
  • I think that you're not asking the right question. Note that "natural axiom" is something which depends a lot on the context, and set theory has many different contexts. $\sf PFA$ can be very natural in some contexts, and having a real-measurable continuum can be natural in a different context. Not to mention choiceless contexts of $\sf AD$ where the continuum has properties both of $\sf CH$ and of its failure (in more than one way). This is like asking why aren't all groups abelian, or why aren't all rings commutative/PID/etc. -- because there's more to it than just this one property. – Asaf Karagila Aug 20 '14 at 01:24
  • I do not know what you mean by "non-axiom"? I do think large cardinals or even $\mathsf{ZFC}$ are natural in any particular way? ZFC is likely only widely accepted because it was useful in formalizing the mathematics historically known at that time. I like to think of set theoretic axioms as the tools for an exercise in recording what can be proved from what; rather than a search for a panacea to prove everything you want. – William Aug 20 '14 at 01:32
  • @AsafKaragila: I think you are getting at my question. How does your group analogy apply to set theory re. CH? I think we all agree that all groups are not abelian. So does this help explain why CH cannot be settled? To be clear, I agree that CH cannot be settled! I'm asking if there is any intuition as to why this is the case. Perhaps the answer is no. –  Aug 20 '14 at 01:33
  • Yes, not all groups are abelian, but not all groups are not abelian either. Both things have uses and generate interest. – Asaf Karagila Aug 20 '14 at 01:40
  • @trb456 I think the point that is being driven home here is that the only way to 'decide' it is for a wide array of mathematicians to accept it as useful and necessary. But no one cares about it outside of researchers on this topic, so such a decision is impossible. – zibadawa timmy Aug 20 '14 at 01:42
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    @zibadaw: No, that's not quite right. The continuum hypothesis and its negations have implications to operator theory and they continue from there. All the way to physics. To say that no one but set theorists care for its value is plain false; in fact I don't think that many set theorists care for the value of the continuum, and perhaps the emerging school of multiverse approach to set theory (rather than formalist or Platonist) will end up as the dominant one in the future. Precisely because there are so many conflicting interests. – Asaf Karagila Aug 20 '14 at 01:47
  • @AsafKaragila I was being hyperbolic with the "no one" bit. Obviously someone cares by sheer number of mathematicians (and physicists). The conflicting interests parts is something I didn't gather from the comments. Did I miss explanation of that, or can you explain? Are there research areas that make assumptions on CH (or a related matter) that are at logical odds with other assumptions of the same used elsewhere (perhaps even on a similar problem)? Or do I completely misunderstand what you mean by conflicting interests? – zibadawa timmy Aug 20 '14 at 01:56
  • @zibadawa: In most non-set theoretical areas, I think, most researchers wouldn't dare writing something like "Now, assuming the continuum hypothesis ...". But it doesn't mean there is no research towards that, or that it has no implications. People just try to circumvent these issues, or take steps closer to where $\sf CH$ has little to no effect. But things like Kaplansky conjecture, automorphisms of the Calkin algebra, and so on, end up being dependent on set theoretic assumptions, in particular $\sf CH$. And it trickles down the implications ladder, slowly but surely. – Asaf Karagila Aug 20 '14 at 02:00
  • @zibadawa: As to conflicting interests, I meant that set theorists are interested both in models where $\sf CH$ holds and in models where it is violated. So there is a conflicting interest as to whether or not $\sf CH$ should even have a decided truth value, since once it has, many of these interesting problems disappear from our attention. And mathematics is inclusive rather than exclusive, at least in this aspect. – Asaf Karagila Aug 20 '14 at 02:02
  • @Asaf Karagila Interesting problems related to $\neg$AC did not disappear after AC got into ZFC. And there are niche issues outside of set theory where $\neg$AC matters, coloring the plane for example. The difference between CH and AC has to be elsewhere. – Conifold Aug 20 '14 at 02:22
  • @Conifold: Choice is by far more useful as an axiom, as it helps tame infinite objects. Even things which are seemingly "natural" to us (through the widely use, but hardly noticable, Principle of Dependent Choice). There are still many interesting problems regarding the axiom of choice, and people in set theory still do that. The axiom of choice, however, has a much greater effect on "working mathematics" (read: non-set theory), than the continuum hypothesis has. Which is why it was settled quite quickly into the axioms of set theory and mathematics. – Asaf Karagila Aug 20 '14 at 02:27
  • @Conifold: If I had to pin down axioms which extend the usability of the axiom of choice outside of set theory, I'd not choose the continuum hypothesis. The obvious candidates would be either diamond axioms (which tame down uncountable cardinalities by ensuring that "when things happen, they happen often") or Martin's axiom which is more difficult to use, but has immediate topological implications, and it too helps to bring a little bit of structure into infinite cardinals. Between these two, I'd probably use diamonds if you'd force me to make a choice. – Asaf Karagila Aug 20 '14 at 02:34
  • @Asaf Karagila Well, GCH would give us all the usability of choice along with CH and plenty more of taming infinite objects. What it lacks, like CH, is intuitive basis, and like you said, much of practical effect beyond AC. And AC already has strange enough consequences to go any further. These seem to be the main reasons and not just against CH. – Conifold Aug 20 '14 at 02:49
  • @Conifold: The "strange consequences" that people like to object to so much, are in reality effects of the axiom of infinity, and all those weird things infinite objects have. People who object to the Banach-Tarski paradox clearly don't know that in the models we know that BT fails you can partition the line into more non-empty parts than points. To me, this is far more boggling than the BT paradox. Or that without the axiom of choice obvious "truths" need not be true anymore. E.g. $\Bbb N$ or $\Bbb R$ need not be Lindelof, both may seem obvious to us; or the regularity of $\omega_1$. – Asaf Karagila Aug 20 '14 at 03:07
  • @Asaf Karagila But you can not do strange things provably. The point of conservatism is to have some core where provable things are in line with intuition as much as possible, ZF+DC is probably that classically, and ZFC is a minimal compomise for more abstract mathematics. By the way, conservatism is exactly the inclusiveness that you praised, the smaller the core the more inclusive the theory. And intuition is the ultimate reason for believing a theory is consistent. – Conifold Aug 20 '14 at 03:16
  • @Conifold: No, you can't. But this is the whole reason we add axioms, we want to ensure something happens. So far, to my knowledge, if you want to ensure that BT fails, you have ensure that an even worse paradox befalls the real numbers. And $\sf DC$ is certainly not enough to stop that. As for smaller core theory, of course we can work with the empty theory and just investigate validities. It's not that interesting as it turns out. There's a silver lining here, and essentially this is what foundational work is all about. But the axiom of choice has more obvious merits than the rest. – Asaf Karagila Aug 20 '14 at 03:20
  • @Conifold: And please excuse me for not continuing this chat any further. I feel that I have to repeat myself once too many. And I have other things to do, which are currently much more important than internet discussions. Have a nice day! – Asaf Karagila Aug 20 '14 at 03:21

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Let me address the question as stated first.

Why does $\sf CH$ has no determinate provability from any of the axioms we throw at it? This is false. As remarked in the comments. Plenty of axioms prove $\sf CH$ or disprove it. Things like $V=L$ or $\lozenge$ imply $\sf CH$ whereas things like $\sf PFA$ and similar forcing axioms imply its negation (these in particular imply that $2^{\aleph_0}=\aleph_2$ of all values).

Specifically you might be asking about large cardinals. Well, this is because large cardinals are large, and the continuum is small. Historically, I believe, the motivation for the thought that large cardinals might settle the continuum hypothesis came from the fact that we can prove the continuum hypothesis for Borel sets. Namely every uncountable Borel set has size continuum, and we can push this to analytic sets, but not further.

However, if there exists a measurable cardinal then we can prove the continuum hypothesis for co-analytic sets as well. Which is a push forward. So perhaps by having "enough" large cardinals, or a strong enough large cardinal axiom we can push this proof and go through all the possible sets of reals?

Well, no, not really. While these things definitely can occur (in the presence of Woodin cardinals we can push the continuum hypothesis higher up the projective hierarchy), there are more sets of real numbers than that. And at some point the large cardinals exhaust their power.

In particular we have the Levy-Solovay phenomenon:

Suppose that $\kappa$ is an inaccessible cardinal, and $\Bbb P$ is a forcing such that $|\Bbb P|<\kappa$. Then $\Vdash_\Bbb P\check\kappa\text{ is inaccessible}$.

Namely, small forcings cannot change the largeness of an inaccessible cardinal. The same is true for many other types of large cardinals, e.g. weakly compact cardinals, measurable cardinals, Woodin cardinals, etc.

On the other hand we always have a forcing of size $\aleph_2$ which violates the continuum hypothesis: adding $\aleph_2$ Cohen reals; and a forcing of size continuum which restores the continuum hypothesis: Levy collapse of $2^{\aleph_0}$ to $\aleph_1$ (and it doesn't add real numbers too!)

So with particularly small forcings we can switch the truth value of the continuum hypothesis on and off as we like. Therefore large cardinals, which are unaffected by small forcings, cannot possibly decide the truth value of $\sf CH$.

(That been said, there are large cardinals properties which can be destroyed by small forcings, but so far these were shown consistent with $\sf CH$ and with its negation by various means, e.g. by showing that we can perhaps destroy the large property, but then we can have another forcing which restores it and doesn't change the value of the continuum.)

Asaf Karagila
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  • Therefore large cardinals, which are unaffected by small forcings, cannot possibly decide the truth value of CH. I think this is the key statement. The fact that we cannot decide which combination of large cardinal and small forcing is "correct" is a major factor in not being able to resolve CH. –  Aug 20 '14 at 01:44
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    It's like you don't even read what I wrote in all those comments and this long answer. :-) We're not even trying to decide what is "correct". You asked why large cardinals didn't decide the continuum hypothesis, this is why. Moreover saying that a "combination of large cardinal and small forcing is "correct" is a major factor ..." shows complete lack of understanding what forcing is used for and how it is used for. – Asaf Karagila Aug 20 '14 at 01:50
  • We use forcing to introduce new objects to a model of set theory, and thus show that a certain statement is consistent. We don't use it (often) to decide a particular model of set theory is "the one true model of set theory". Philosophically this is hardly makes any sense. If you insist that $\sf CH$ is necessary true, then you can't really use any type of forcing which changes the value of the continuum. In fact, just adding real numbers seems morally objectionable once you decide that $\sf CH$ is true. Forcing is a force majeure in how little we work towards settling the truth value of CH. – Asaf Karagila Aug 20 '14 at 01:53
  • Look, I'm not a professional mathematician, and I did read what you wrote, so please don't be insulting. I have an advanced, but not professional, background in mathematics. And I highly regard your work! I'm trying to narrow down to why we cannot resolve CH beyond that it eliminates lots of interesting set theory. As you noted, we could decide all groups are abelian, but we don't, for good reasons. So given that there is lots of interesting set theoretical exploration, why does none of this settle the problem? And this is where I acknowledge that there may be no good intuition as to why. –  Aug 20 '14 at 02:04
  • I'm sorry if it came off as insulting. I even put a little smiley there to make sure it reads with a smile, as a joke. As for the rest of the comment, "can't prove it from the axioms we regard as standard + either truth value has interesting results + the technique which revolutionized modern set theory will be handicapped" are three excellent reasons not to want to settle the problem "once and for all". – Asaf Karagila Aug 20 '14 at 02:20
  • I think you are getting at the issue. There is simply no set of assumptions that have narrowed the domain of discourse on this topic. So this does not get at my "intuition" question, but it suggests we don't yet have such intuition. Thanks! –  Aug 20 '14 at 02:25