Solutions to the Continuum Hypothesis

Background

The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $$\aleph_1=2^{\aleph_0}$$. In other words, it asserts that every subset of the set of real numbers that contains the natural numbers has either the cardinality of the natural numbers or the cardinality of the real numbers. It was the first problem on the 1900 Hilbert's list of problems. The generalized continuum hypothesis asserts that there are no intermediate cardinals between every infinite set X and its power set.

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explanation of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper entitled "The Generalized Continuum Hypothesis revisited "). See also the paper entitled "You can enter Cantor paradise" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's answer is by Lucca Belloti (also suggested by Haim).

The proposed answer $$2^{\aleph_0}=\aleph_2$$ goes back according to François to Goedel. It is (perhaps) mentioned in Foreman's presentation. (I heard also from Menachem Magidor that this answer might have some advantages.)

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuum with the First Two Uncountable Cardinals".

There is also a very rich theory (PCF theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

There are by now a couple of long detailed excellent answers (that I still have to digest) by Joel David Hamkins and by Andres Caicedo and several other useful answers. (Unfortunately, I can accept only one answer.)

Update (February 2011): A new detailed answer was contributed by Justin Moore.

Update (Oct 2013) A user 'none' gave a link to an article by Peter Koellner about the current status of CH:

Update (Jan 2014) A related popular article in "Quanta:" To settle infinity dispute a new law of logic

(belated) update(Jan 2014) Joel David Hamkins links in a comment from 2012 a very interesting paper Is the dream solution to the continuum hypothesis attainable written by him about the possibility of a "dream solution to CH." A link to the paper and a short post can be found here.

(belated) update (Sept 2015) Here is a link to an interesting article: Can the Continuum Hypothesis be Solved? By Juliette Kennedy

Update A videotaped lecture The Continuum Hypothesis and the search for Mathematical Infinity by Woodin from January 2015, with reference also to his changed opinion. (added May 2017)

Update (Dec '15): A very nice answer was added (but unfortunately deleted by owner, (2019) now replaced by a new answer) by Grigor. Let me quote its beginning (hopefully it will come back to life):

"One probably should add that the continuum hypothesis depends a lot on how you ask it.

1. $$2^{\omega}=\omega_1$$
2. Every set of reals is either countable or has the same size as the continuum.

To me, 1 is a completely meaningless question, how do you even experiment it?

If I am not mistaken, Cantor actually asked 2..."

Update A 2011 videotaped lecture by Menachem Magidor: Can the Continuum Problem be Solved? (I will try to add slides for more recent versions.)

Update (July 2019) Here are slides of 2019 Woodin's lecture explaining his current view on the problem. (See the answer of Mohammad Golshani.)

Update (Sept 19, 2019) Here are videos of the three 2016 Bernay's lectures by Hugh Woodin on the continuum hypothesis and also the videos of the three 2012 Bernay's lectures on the continuum hypothesis and related topics by Solomon Feferman.

Update (Sept '20) Here are videos of the three 2020 Bernays' lectures by Saharon Shelah on the continuum hypothesis.

Update (May '21) In a new answer, Ralf Schindler gave a link to his 2021 videotaped lecture in Wuhan, describing a result with David Asperó that shows a relation between two well-known axioms. It turns out that Martin's Maximum$$^{++}$$ implies Woodin's ℙ$$_{max}$$ axiom. Both these axioms were known to imply the $$\aleph_2$$ answer to CH. A link to the paper: https://doi.org/10.4007/annals.2021.193.3.3

• Another noteworthy reference is Gödel's unpublished note 1970a where he collects evidence in favor of $2^{\aleph_0} = \aleph_2$ - books.google.com/… Commented May 7, 2010 at 8:00
• Another important paper is Todorcevic's Comparing the Continuuum with the First Two Uncountable Cardinals - math.toronto.edu/stevo Commented May 7, 2010 at 8:02
• Can you please edit your second sentence starting by "In other words" ? "containing of real numbers" does not seem very clear. Also small typos like "now intermediate" -> "no intermediate" "Eralier"=>"Earlier" Another pedantic remark is that one does not "solve" an hypothesis. One adopts it or one rejects it, eventually replacing it by another. Perhaps should you slightly rephrase your question. Commented May 8, 2010 at 12:40
• Gil, you're absolutely right that using "solution" in this and related contexts is common usage, but IMHO it's bad English. One solves a problem, answers a question, and proves or disproves a hypothesis or conjecture. Whenever I hear that a "conjecture" has been "solved" I don't know whether the speaker means the conjecture is now known to be true, false, or undecidable (and I've seen examples in which each of the three was meant). But I'm grumpier about such things than most. Commented May 10, 2010 at 13:36
• There is an honorable precedent for calling it Cantor's Continuum Problem.
– bof
Commented Jan 3, 2014 at 12:43

12 Answers

Since you have already linked to some of the contemporary primary sources, where of course the full accounts of those views can be found, let me interpret your question as a request for summary accounts of the various views on CH. I'll just describe in a few sentences each of what I find to be the main issues surrounding CH, beginning with some historical views. Please forgive the necessary simplifications.

Cantor. Cantor introduced the Continuum Hypothesis when he discovered the transfinite numbers and proved that the reals are uncountable. It was quite natural to inquire whether the continuum was the same as the first uncountable cardinal. He became obsessed with this question, working on it from various angles and sometimes switching opinion as to the likely outcome. Giving birth to the field of descriptive set theory, he settled the CH question for closed sets of reals, by proving (the Cantor-Bendixon theorem) that every closed set is the union of a countable set and a perfect set. Sets with this perfect set property cannot be counterexamples to CH, and Cantor hoped to extend this method to additional larger classes of sets.

Hilbert. Hilbert thought the CH question so important that he listed it as the first on his famous list of problems at the opening of the 20th century.

Goedel. Goedel proved that CH holds in the constructible universe $$L$$, and so is relatively consistent with ZFC. Goedel viewed $$L$$ as a device for establishing consistency, rather than as a description of our (Platonic) mathematical world, and so he did not take this result to settle CH. He hoped that the emerging large cardinal concepts, such as measurable cardinals, would settle the CH question, and as you mentioned, favored a solution of the form $$2^\omega=\aleph_2$$.

Cohen. Cohen introduced the method of forcing and used it to prove that $$\neg$$CH is relatively consistent with ZFC. Every model of ZFC has a forcing extension with $$\neg$$CH. Thus, the CH question is independent of ZFC, neither provable nor refutable. Solovay observed that CH also is forceable over any model of ZFC.

Large cardinals. Goedel's expectation that large cardinals might settle CH was decisively refuted by the Levy-Solovay theorem, which showed that one can force either CH or $$\neg$$CH while preserving all known large cardinals. Thus, there can be no direct implication from large cardinals to either CH or $$\neg$$CH. At the same time, Solovay extended Cantor's original strategy by proving that if there are large cardinals, then increasing levels of the projective hierarchy have the perfect set property, and therefore do not admit counterexamples to CH. All of the strongest large cardinal axioms considered today imply that there are no projective counterexamples to CH. This can be seen as a complete affirmation of Cantor's original strategy.

Basic Platonic position. This is the realist view that there is Platonic universe of sets that our axioms are attempting to describe, in which every set-theoretic question such as CH has a truth value. In my experience, this is the most common or orthodox view in the set-theoretic community. Several of the later more subtle views rest solidly upon the idea that there is a fact of the matter to be determined.

Old-school dream solution of CH. The hope was that we might settle CH by finding a new set-theoretic principle that we all agreed was obviously true for the intended interpretation of sets (in the way that many find AC to be obviously true, for example) and which also settled the CH question. Then, we would extend ZFC to include this new principle and thereby have an answer to CH. Unfortunately, no such conclusive principles were found, although there have been some proposals in this vein, such as Freilings axiom of symmetry.

Formalist view. Rarely held by mathematicians, although occasionally held by philosophers, this is the anti-realist view that there is no truth of the matter of CH, and that mathematics consists of (perhaps meaningless) manipulations of strings of symbols in a formal system. The formalist view can be taken to hold that the independence result itself settles CH, since CH is neither provable nor refutable in ZFC. One can have either CH or $$\neg$$CH as axioms and form the new formal systems ZFC+CH or ZFC+$$\neg$$CH. This view is often mocked in straw-man form, suggesting that the formalist can have no preference for CH or $$\neg$$CH, but philosophers defend more subtle versions, where there can be reason to prefer one formal system to another.

Pragmatic view. This is the view one finds in practice, where mathematicians do not take a position on CH, but feel free to use CH or $$\neg$$CH if it helps their argument, keeping careful track of where it is used. Usually, when either CH or $$\neg$$CH is used, then one naturally inquires about the situation under the alternative hypothesis, and this leads to numerous consistency or independence results.

Cardinal invariants. Exemplifying the pragmatic view, this is a very rich subject studying various cardinal characteristics of the continuum, such as the size of the smallest unbounded family of functions $$f:\omega\to\omega$$, the additivity of the ideal of measure-zero sets, or the smallest size family of functions $$f:\omega\to\omega$$ that dominate all other such functions. Since these characteristics are all uncountable and at most the continuum, the entire theory trivializes under CH, but under $$\neg$$CH is a rich, fascinating subject.

Canonical Inner models. The paradigmatic canonical inner model is Goedel's constructible universe $$L$$, which satisfies CH and indeed, the Generalized Continuum Hypothesis, as well as many other regularity properties. Larger but still canonical inner models have been built by Silver, Jensen, Mitchell, Steel and others that share the GCH and these regularity properties, while also satisfying larger large cardinal axioms than are possible in $$L$$. Most set-theorists do not view these inner models as likely to be the "real" universe, for similar reasons that they reject $$V=L$$, but as the models accommodate larger and larger large cardinals, it becomes increasingly difficult to make this case. Even $$V=L$$ is compatible with the existence of transitive set models of the very largest large cardinals (since the assertion that such sets exist is $$\Sigma^1_2$$ and hence absolute to $$L$$). In this sense, the canonical inner models are fundamentally compatible with whatever kind of set theory we are imagining.

Woodin. In contrast to the Old-School Dream Solution, Woodin has advanced a more technical argument in favor of $$\neg$$CH. The main concepts include $$\Omega$$-logic and the $$\Omega$$-conjecture, concerning the limits of forcing-invariant assertions, particularly those expressible in the structure $$H_{\omega_2}$$, where CH is expressible. Woodin's is a decidedly Platonist position, but from what I have seen, he has remained guarded in his presentations, describing the argument as a proposal or possible solution, despite the fact that others sometimes characterize his position as more definitive.

Foreman. Foreman, who also comes from a strong Platonist position, argues against Woodin's view. He writes supremely well, and I recommend following the links to his articles.

Multiverse view. This is the view, offered in opposition to the Basic Platonist Position above, that we do not have just one concept of set leading to a unique set-theoretic universe, but rather a complex variety of set concepts leading to many different set-theoretic worlds. Indeed, the view is that much of set-theoretic research in the past half-century has been about constructing these various alternative worlds. Many of the alternative set concepts, such as those arising by forcing or by large cardinal embeddings are closely enough related to each other that they can be compared from the perspective of each other. The multiverse view of CH is that the CH question is largely settled by the fact that we know precisely how to build CH or $$\neg$$CH worlds close to any given set-theoretic universe---the CH and $$\neg$$CH worlds are in a sense dense among the set-theoretic universes. The multiverse view is realist as opposed to formalist, since it affirms the real nature of the set-theoretic worlds to which the various set concepts give rise. On the Multiverse view, the Old-School Dream Solution is impossible, since our experience in the CH and $$\neg$$CH worlds will prevent us from accepting any principle $$\Phi$$ that settles CH as "obviously true". Rather, on the multiverse view we are to study all the possible set-theoretic worlds and especially how they relate to each other.

I should stop now, and I apologize for the length of this answer.

• it seems that the multiverse view is the beginning of plurality in set theory. This is analogous to how there used to be only one geometry -- Euclidean -- but the investigation of PP and not PP led to a multiverse of geometries.
– user2529
Commented May 19, 2010 at 8:32
• I agree, Colin; the analogy with geometry is very strong and extends to many facets of how we think about the various geometries. Commented May 19, 2010 at 16:18
• "Formalist view. Rarely held by mathematicians, although occasionally held by philosophers ..." I recently read the following, written by a distinguished philosopher of mathematics. "The formalist's central thought is that arithmetic is not ultimately concerned with an extralinguistic domain of things. Rather, insofar as arithmetic has a proper subject matter, it is the language of arithmetic itself and certain formal relations among its sentences." This was accompanied by a footnote: "The view has few contemporary adherents among philosophers, though mathematicians often find it congenial." Commented Feb 6, 2011 at 17:28
• For what it's worth, I find it congenial myself. In particular, I incline to the view that there is no fact of the matter about whether CH is true. Commented Feb 6, 2011 at 17:29
• "Formalist view. Rarely held by mathematicians"? Do you have any evidence for your claim? I would've said formalism was the prevalent view among mathematicians (exept for the adherents to that strange religion called "platonism"), but I have no evidence to substantiate this: mine is probably just wishful thinking... Commented Dec 16, 2015 at 20:50

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain there.

You can find the slides here, under "Recent results about the Continuum Hypothesis, after Woodin". (In true Bourbaki fashion, I heard that the talk was not well received.)

Roughly, Woodin's approach shows that in a sense, the theory of $H(\omega_2)$ decided by the usual set of axioms, ZFC and large cardinals, can be "finitely completed" in a way that would make it reasonable to expect to settle all its properties. However, any such completion implies the negation of CH.

It is a conditional result, depending on a highly non-trivial problem, the $\Omega$-conjecture. If true, this conjecture gives us that Cohen's technique of forcing is in a sense the only method (in the presence of large cardinals) required to establish consistency. (The precise statement is more technical.)

$H(\omega_2)$, that Dehornoy calls $H_2$, is the structure obtained by considering only those sets $X$ such that $X\cup\bigcup X\cup\bigcup\bigcup X\cup\dots$ has size strictly less than $\aleph_2$, the second uncountable cardinal.

Replacing $\aleph_2$ with $\aleph_1$, we have $H(\omega_1)$, whose theory is completely settled in a sense, in the presence of large cardinals. If nothing else, one can think of Woodin's approach as trying to build an analogy with this situation, but "one level up."

Whether or not one considers that settling the $\Omega$-conjecture in a positive fashion actually refutes CH in some sense, is a delicate matter. In any case (and I was happy to see that Dehornoy emphasizes this), Woodin's approach gives strength to the position that the question of CH is meaningful (as opposed to simply saying that, since it is independent, there is nothing to decide).

(2) There is another approach to the problem, also pioneered by Hugh Woodin. It is the matter of "conditional absoluteness." CH is a $\Sigma^2_1$ statement. Roughly, this means that it has the form: "There is a set of reals such that $\phi$", where $\phi$ can be described by quantifying only over reals and natural numbers. In the presence of large cardinals, Woodin proved the following remarkable property: If $A$ is a $\Sigma^2_1$ statement, and we can force $A$, then $A$ holds in any model of CH obtained by forcing.

Recall that forcing is essentially the only tool we have to establish consistency of statements. Also, there is a "trivial" forcing that does not do anything, so the result is essentially saying that any statement of the same complexity as CH, if it is consistent (with large cardinals), then it is actually a consequence of CH.

This would seem a highly desirable ''maximality'' property that would make CH a good candidate to be adopted.

However, recent results (by Aspero, Larson, and Moore) suggest that $\Sigma^2_1$ is close to being the highest complexity for which a result of this kind holds, which perhaps weakens the argument for CH that one could do based on Hugh's result.

A good presentation of this theorem is available in Larson's book "The stationary tower. Notes on a Course by W. Hugh Woodin." Unfortunately, the book is technical.

(3) Foreman's approach is perhaps the strongest opponent to the approach suggested by Woodin in (1). Again, it is based in the technique of forcing, now looking at small cardinal analogues of large cardinal properties.

Many large cardinal properties are expressed in terms of the existence of elementary embeddings of the universe of sets. These embeddings tend to be "based" at cardinals much much larger than the size of the reals. With forcing, one can produce such embeddings "based" at the size of the reals, or nearby. Analyzing a large class of such forcing notions, Foreman shows that they must imply CH. If one were to adopt the consequences of performing these forcing constructions as additional axioms one would then be required to also adopt CH.

I had to cut my answer short last time. I would like now to say a few details about a particular approach.

(4) Forcing axioms imply that $2^{\aleph_0}=\aleph_2$, and (it may be argued) strongly suggest that this should be the right answer.

Now, before I add anything, note that Woodin's approach (1) uses forcing axioms to prove that there are "finite completions" of the theory of $H(\omega_2)$ (and the reals have $\aleph_2$). However, this does not mean that all such completions would be compatible in any particular sense, or that all would decide the size of the reals. What Woodin proves is that all completions negate CH, and forcing axioms show that there is at least one such completion.

I believe there has been some explanation of forcing axioms in the answer to the related question on GCH. Briefly, the intuition is this: ZFC seems to capture the basic properties of the universe of sets, but fails to account for its width and its height. (What one means by this is: how big should power sets be, and how many ordinals there are.)

Our current understanding suggests that the universe should indeed be very tall, meaning there should be many many large cardinals. As Joel indicated, there was originally some hope that large cardinals would determine the size of the reals, but just about immediately after forcing was introduced, it was proved that this was not the case. (Technically, small forcing preserves large cardinals.)

However, large cardinals settle many questions about the structure of the reals (all first order, or projective statements, in fact). CH, however, is "just" beyond what large cardinals can settle. One could say that, as far as large cardinals are concerned, CH is true. What I mean is that, in the presence of large cardinals, any definable set of reals (for any reasonable notion of definability) is either countable or contains a perfect subset. However, this may simply mean that there is certain intrinsic non-canonicity in the sets of reals that would disprove CH, if this is the case.

(A word of caution is in order here, and there are candidates for large cardinal axioms [presented by Hugh Woodin in his work on suitable extender sequences] for which preservation under small forcing is not clear. Perhaps the solution to CH will actually come, unexpectedly, from studying these cardinals. But this is too speculative at the moment.)

I have avoided above saying much about forcing. It is a massive machinery, and any short description is bound to be very inaccurate, so I'll be more than brief.

An ordinary algebraic structure (a group, he universe of sets) can be seen as a bi-valued model. Just as well, one can define, for any complete Boolean algebra ${\mathbb B}$, the notion of a structure being ${\mathbb B}$-vaued. If you wish, "fuzzy set theory" is an approximation to this, as are many of the ways we model the world by using a probability density to decide the likelihood of events. For any complete Boolean algebra ${\mathbb B}$, we can define a ${\mathbb B}$-valued model $V^{\mathbb B}$ of set theory. In it, rather than having for sets $x$ and $y$ that either $x\in y$ or it doesn't, we assign to the statement $x\in y$ a value $[x\in y]\in{\mathbb B}$. The way the construction is performed, $[\phi]=1$ for each axiom $\phi$ of ZFC. Also, for each element $x$ of the actual universe of sets, there is a copy $\check x$ in the ${\mathbb B}$-valued model, so that the universe $V$ is a submodel of $V^{\mathbb B}$. If it happens that for some statement $\psi$ we have $[\psi]>0$, we have established that $\psi$ is consistent with ZFC. By carefully choosing ${\mathbb B}$, we can do this for many $\psi$. This is the technique of forcing, and one can add many wrinkles to the approach just outlined. One refers to ${\mathbb B}$ as a forcing notion.

Now, the intuition that the universe should be very fat is harder to capture than the idea of largeness of the ordinals. One way of expressing it is that the universe is somehow "saturated": If the existence of some object is consistent in some sense, then in fact such object should exist. Formalizing this, one is led to forcing axioms. A typical forcing axiom says that relatively simple (under some measure of complexity) statements that can be shown consistent using the technique of forcing via a Boolean algebra ${\mathbb B}$ that is not too pathological, should in fact hold.

The seminal Martin's Maximum paper of Foreman-Magidor-Shelah identified the most generous notion of "not too pathological", it corresponds to the class of "stationary set preserving" forcing notions. The corresponding forcing axiom is Martin's Maximum, MM. In that paper, it was shown that MM implies that the size of the reals is $\aleph_2$.

The hypothesis of MM has been significantly weakened, through a series of results by different people, culminating in the Mapping Reflection Principle paper of Justin Moore. Besides this line of work, many natural consequences of forcing axioms (commonly termed reflection principles) have been identified, and shown to be independent of one another. Remarkably, just about all these principles either imply that the size of the reals is $\aleph_2$, or give $\aleph_2$ as an upper bound.

Even if one finds that forcing axioms are too blunt a way of capturing the intuition of "the universe is wide", many of its consequences are considered very natural. (For example, the singular cardinal hypothesis, but this is another story.) Just as most of the set theoretic community now understands that large cardinals are part of what we accept about the universe of sets (and therefore, so is determinacy of reasonably definable sets of reals, and its consequences such us the perfect set property), it is perhaps not completely off the mark to expect that as our understanding of reflection principles grow, we will adopt them (or a reasonable variant) as the right way of formulating "wideness". Once/if that happens, the size of the reals will be taken as $\aleph_2$ and therefore CH will be settled as false.

The point here is that this would be a solution to the problem of CH that does not attack CH directly. Rather, it turns out that the negation of CH is a common consequence of many principles that it may be reasonable to adapt in light of the naturalness of some of their best known consequences, and of their intrinsic motivation coming from the "wide universe" picture.

(Apologies for the long post.)

Edit, Nov. 22/10: I have recently learned about Woodin's "Ultimate L" which, essentially, advances a view that theories are "equally good" if they are mutually interpretable, and identifies a theory ("ultimate L") that, modulo large cardinals, would work as a universal theory from which to interpret all extensions. This theory postulates an $L$-like structure for the universe and in particular implies CH, see this answer. But, again, the theory is not advocated on grounds that it ought to be true, whatever this means, but rather, that it is "richest" possible in that it allows us to interpret all possible "natural" extensions of ZFC. In particular, under this approach, only large cardinals are relevant if we want to strengthen the theory, while "width" considerations, such as those supporting forcing axioms, are no longer relevant.

Since the approach I numbered (1) above implies the negation of CH, I feel I should add that one of the main reasons for it being advanced originally depended on the fact that the set of $\Omega$-validities can be defined "locally", at the level of $H({\mathfrak c}^+)$, at least if the $\Omega$-conjecture holds.

However, recent results of Sargsyan uncovered a mistake in the argument giving this local definability. From what I understand, Woodin feels that this weakens the case he was making for not-CH significantly.

Added link: slides of a 2016 lecture by Woodin on Ultimate L.

• Thanks very much for this answer, Andres. I had heard that Woodin proved an extremal property of CH, but I didn't know what it was. It is presumably your item (2). Commented May 18, 2010 at 23:47
• Hi Simon. I am not too certain of all the details; I believe that the issue is this: In the context of $AD^+$, say that $\alpha$ is a "local $\Theta$" if it is the $\Theta$ of a hod-model. Woodin had an argument showing that no such $\alpha$ could be overlapped by a strong cardinal. This put serious limitations on the strength of large cardinals that hod-models could contain. In particular, this was the reason why it was expected that "CH + there is an $\omega_1$-dense ideal on $\omega_1$" and "$AD_{\mathbb R}+\Theta$ regular" were expected to have really high consistency strength (continued) Commented Nov 23, 2010 at 15:33
• (2) Woodin's local definability argument depended on this limitation of hod models. (I am not sure of the details here.) Grigor's analysis in the context of the core model induction (there are slides of a talk at Boise on his website, and I can email you his thesis, let me know) showed that these "local overlaps" are possible, and deduced as a corollary that "$AD_{\mathbb R}+\Theta$ regular has much lower consistency strength than expected. Grigor in fact has a very detailed analysis of hod models. Without the overlap limitation, the set of $\Omega$-validities ends up being harder to define. Commented Nov 23, 2010 at 15:42
• (3) It can be shown that it is $H(\delta_0^+)$-definable, where $\delta_0$ is the smallest Woodin of $V$. But the argument pro-not-CH went by a level by level analysis of the $H(\kappa)$-levels, and this jump (from ${\mathfrak c}$ to a Woodin) is too high to overlook. As far as I understand, this is the nature of the issue. Commented Nov 23, 2010 at 15:45
• @Joel Thanks. I am aware of this (see here), and I knew there was a post of mine where I was using the incorrect terminology, but could never find it. :-) I'll update a bit later. Commented Mar 27, 2018 at 19:46

First I'll say a few words about forcing axioms and then I'll answer your question. Forcing axioms were developed to provide a unified framework to establish the consistency of a number of combinatorial statements, particularly about the first uncountable cardinal. They began with Solovay and Tennenbaum's proof of the consistency of Souslin's Hypothesis (that every linear order in which there are no uncountable families of pairwise disjoint intervals is necessarily separable). Many of the consequences of forcing axioms, particularly the stronger Proper Forcing Axiom and Martin's Maximum, had the form of classification results: Baumgartner's classification of the isomorphism types of $$\aleph_1$$-dense sets of reals, Abraham and Shelah's classification of the Aronszajn trees up to club isomorphism, Todorcevic's classification of linear gaps in $$\mathcal{P}(\mathbb{N})/\mathrm{fin}$$, and Todorcevic's classification of transitive relations on $$\omega_1$$. A survey of these results (plus many references) can be found in both Stevo Todorcevic's ICM article and my own (the later can be found here1). These are accessible to a general audience.

What does all this have to do with the Continuum Problem? It was noticed early on that forcing axioms imply that the continuum is $$\aleph_2$$. The first proof of this is, I believe, in Foreman, Magidor, and Shelah's seminal paper in Martin's Maximum. Other quite different proofs were given by Caicedo, Todorcevic, Velickovic, and myself. An elementary proof which is purely Ramsey-theoretic in nature is in my article " Open colorings, the continuum, and the second uncountable cardinal" (PAMS, 2002).2

Since it is often the case that the combinatorial essence of the proofs of these classification results and that the continuum is $$\aleph_2$$ are similar, one is left to speculate that perhaps there may some day be a classification result concerning structures of cardinality $$\aleph_1$$ which already implies that the continuum is $$\aleph_2$$. There is a candidate for such a classification: the assertion that there are five uncountable linear orders such that every other contains an isomorphic copy of one of these five. Another related candidate for such a classification is the assertion that the Aronszajn lines are well quasi-ordered by embeddability (if $$L_i$$ $$(i < \infty)$$ is a sequence of Aronszajn lines, then there is an $$i < j$$ such that $$L_i$$ embeds into $$L_j$$). These are due to myself and Carlos Martinez, respectively. See a discussion of this (with references) in my ICM paper1.

1 The Proper Forcing Axiom, Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010, Wayback Machine
2 Open colorings, the continuum and the second uncountable cardinal, Wayback Machine

Regarding Shelah's approach, I believe that the following paper should be quite accessible to non-professionals: YOU CAN ENTER CANTOR’S PARADISE!

Now, I have no idea how to explain Woodin's approach to CH without relying on some cryptic terminology, but I believe that the following paper by Luca Bellotti might be useful: Woodin on the Continuum Problem: an overview and some objections.

• Thanks Haim, I think I vaguely had this paper of Shelah also in mind but did not find it when I wrote the question. Commented May 8, 2010 at 8:54
• The second link is now broken. (I added the nice title for the first link.) Commented Apr 30, 2019 at 19:09
• @Gil I fixed the link. Commented Apr 30, 2019 at 22:26

The question of whether or not $2^{\aleph_{0}} = \aleph_{1}$ is not even considered in Shelah's approach. In fact, this question is regarded as part of the "white noise" which has distracted the attention of set theorists from some striking $ZFC$-results about cardinal exponentiation $\kappa^{\lambda}$ when you consider relatively small exponents $\lambda$ and relatively large bases $\kappa$.

Let me add in short details, views of three famous set theorists about the problem:

Shelah’s answer: The question was wrong. The right question should be about other combinatorial objects. There we can prove the “revised GCH” (Sh460). PCF Theory.

Foreman’s answer: Large cardinals can’t help, but “generic large cardinals” might.

Woodin’s answer: Instead of looking at the statements of new axioms, look at the metamathematical properties of axiom candidates. There is an asymmetry between axioms that imply CH and those that imply $$\sim CH.$$ Woodin’s ­$$\Omega$$-conjecture.

Edit As it is stated in some other answers, Woodin has changed his mind and he believes in the continuum hypothesis. This is part of his "Ultimaate L" project. See the following slide for his very recent expository talk on the $$CH$$:

The Continuum hypothesis

• Mohammad Golshani kindly mentioned in a FB discussion two recent approaches: "Matteo Viale has arguments and claims the continuum is $\aleph_2$. Sakae Fuchino claims the continuum is either $\aleph_1$ or $\aleph_2$ or is very large. Commented Sep 11, 2020 at 6:34
• (cont.) For Viale see his papers Tameness for set theory I and Tameness for set theory II or the slide logicatorino.altervista.org/.../TAMSTslides.pdf Commented Sep 11, 2020 at 6:35
• (cont.) For Fuchino's work see fuchino.ddo.jp/slides/kobe2020-07-28-pf.pdf Commented Sep 11, 2020 at 6:36

In Aug. 2020, I gave a talk at Wuhan with the title "How many real numbers are there?", taking into account my result with D. Asperó on MM++ => (*). There is a recording: https://m.bilibili.com/video/BV1TV411h714 , and there is also a set of notes: https://ivv5hpp.uni-muenster.de/u/rds/Notes_Wuhan.pdf . Comments are very welcome.

• To answer the question in the title of your talk: 4, obviously. Commented May 3, 2021 at 10:29
• The paper in question was just published in the Annals: doi.org/10.4007/annals.2021.193.3.3 Commented May 3, 2021 at 15:47
• A quanta magazine article: quantamagazine.org/… Commented Jul 16, 2021 at 11:32
• Hi Ralf, I think the problem is your bias, no? When one works under one theory the other looks unnatural, when one switches back and fourth between both MM and Ultimate-L then both seem natural. Its a choice people make, they choose not to get into the other theory, but surely we cannot be biased this way towards the cardinality of R. More objective arguments are needed. Commented Jan 12, 2022 at 23:36
• For example, one can now ask what is $2^{\omega_2}$? If $2^\omega=\omega_2$ and if we know this then surely we must also have a definite answer for $2^{\omega_2}$ and $2^{2^{\omega_2}}$ and etc . MM is silent on those. Commented Jan 12, 2022 at 23:41

This is a nice survey:

Koellner, Peter (2011). "The Continuum Hypothesis". Exploring the Frontiers of Independence (Harvard lecture series).

(Pasted from Wikipedia article on CH.)

Since this old question has been bumped to the front page, I'm surprised nobody so far mentioned the following opinion expressed by Paul Cohen at the end of his 1966 book Set Theory and the Continuum Hypothesis:

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it is absurd to think that the process of adding one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now $\aleph_1$ is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set $\mathfrak{c}$ is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach $\mathfrak{c}$. Thus $\mathfrak{c}$ is greater than $\aleph_n$, $\aleph_\omega$, $\aleph_\alpha$ where $\alpha = \aleph_\omega$ etc. This point of view regards $\mathfrak{c}$ as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.

(The emphasis is Cohen's. The $\mathfrak{c}$ is actually a plain C in the typewritten text.)

• A similar view is embraced by Dana Scott in the foreword to J. L. Bell's Boolean-valued models and independence proofs in set theory, Clarendon Press, Oxford, 1977. He writes  Perhaps we would be pushed in the end to say that all sets are countable (and that the continuum is not even a set) when at last all cardinals are absolutely destroyed.'' Commented May 14, 2017 at 13:23
• FWIW, I'm not sure that Cohen held fast to that opinion throughout his life. Elsewhere he said, "Well, philosophically, if you really believe sets exist -- I mean, if you adopt the extreme platonic position -- you can ask, what is the answer? Certainly Goedel himself had the platonic view that the question demanded an absolute answer and that, therefore, neither his proof of the consistency of the continuum hypothesis nor mine of its independence from them was a final answer. My personal view is that I regard the present solution of the problem as very satisfactory. (cont.) Commented May 14, 2017 at 14:47
• "I think it is the only possible solution. It gives one a feeling for what's possible and what's impossible, and in that sense I feel one should be very satisfied... If I were a betting man, I'd bet no one is going to come up with any other kind of solution. There will be philosophical papers, but I don't think any mathematical paper will say that there is any answer other than the answer that it's undecidable." (Interview in July 1985 for More Mathematical People.) Commented May 14, 2017 at 14:51

my answer was too fast, and so i deleted it, but it seems that Gil refers to it, so i will again very quickly explain the matter.

in general, i agree with the view that some mathematicians express, namely that there is no fact of the matter here. the issue has two aspects.

Aspect 1. Cantor originally did not ask an AC cardinality question, namely if $$2^\omega=\omega_1$$. more or less at the beginning he was asking if the perfect set property is true. of course, then this becomes (under AC) a cardinality question.

the trouble with the AC question is that it seems as if one is asking how many real numbers there are, and this question, to me, makes no sense. i can only speculate, judging from Cantor's work elsewhere (Cantor-Bendixon analysis, closed sets have the perfect set property), that Cantor was originally concerned with properties that sets of reals have rather than "how many real numbers there are", where "how many" is asked the same way when one asks how many people there are in the room. to begin with, where exactly these real numbers are supposed to be? i simply cannot make sense of these sort of mysterious questions.

more down to earth question would be whether every set of reals has a perfect set property. here, the dividing line is perfect. Large Cardinals imply that every definable set has the perfect set property and the Axiom of Choice does what it always does, namely it produces a set, using a magical enumeration of the real numbers, without a perfect set property.

the question whether every "definable" set has the perfect set property is a mathematical question provided one fixes a notion of definability. the question is resolved under Large Cardinals for almost all reasonable notions of definability, projective, in $$L(\mathbb{R})$$ and etc.

in fact, the story is even better. Large Cardinals give exact answers to these questions. meaning, if you want all Projective Sets to behave the way Borel and Analytic sets behave (Lebesgue measurability, perfect set property, Baire property, determinacy and etc) then you must have these or that large cardinal in your universe hiding somewhere (usually in the form of a countable model that "captures" the level of definability in question).

In the context of definability, CH (even the cardinality version) is a mathematical rather than a philosophical question.

Aspect 2. if i may say so, usually two notions get mixed up. one is "the platonistic universe of sets" and the other is "foundations of mathematics".

the universe of sets is the way it is, it doesn't depend on what i or anybody else likes or doesn't like. the actual universe is also the way it is, it also doesn't depend on what i or anybody else likes, it is just the way it is.

it must be abundantly clear to anyone following physics that there are several ways of talking about the universe, the exact same universe that doesn't depend on any one person.

the parallel is exactly that. "foundations of mathematics" is the language we pick, according to our licking, to discuss the properties of the universe that we see or intuit.

my goal is to avoid debate here, so i will not mention specific views (except one mentioned just as a clarification), but most views expressed here are proposals for a choice of language, a way of speaking about sets. one view, which i will mention but not for the purpose of debating, the Multiverse View, is actually a mixture of language and the set theoretic reality which in my personal view is rather confused (i am not saying this to start a debate, forgive me for the word).

when interpreting MV as a language, namely as a type of foundations of math, one faces challenges that are outlined in Steel's "Goedel's Program". what is the actual logical language, what is the satisfaction relation and etc and etc, it is easy to declare success under MV but a lot is left unexplained.

when interpreting MV as a view of the universe, namely that the universe has that form, i am perplexed because no reason is given for knowing the existence of it. we only truly know the existence of one universe, namely Goedel's L. the rest, the forcing extensions, are nothing but generalized compactness constructions, the way we get non-standard numbers from the actual numbers. they exist in the language, we can talk about them, but asserting that they exist is going a long way. it is like saying that number theorists study all models of PA rather than the properties of (N, 0, 1, +, ., Suc, exp, <). i am not so sure if number theorists would agree with this.

at any rate, what exactly one is asking when one is asking whether CH is true or not?

one can ask if CH is true in the actual set theoretic universe independently of the language we use to talk about it, in some absolute sense. i believe this interpretation is not a meaningful interpretation unless CH or not CH is somehow implied (in the logical sense) by some basic principles of sets that one can discover, these principles must be independent of the language. i doubt this can happen. for all i know, ZFC may not even be true in the actual universe, it seems more plausible that the universe is expanding and every set is actually countable from some point of view, and so in the actual universe the set of reals may not exist.

so for me, CH, asked as a cardinality question, only makes sense when one is discussing language. does PFA imply it? does this or that foundational framework imply it? and etc.

the following point is perhaps overlooked by many set theorists and mathematicians.

the trouble for set theory isn't that there are undecidable questions. this just says something about our language being incomplete, which is not so difficult to accept. the trouble is that there could be two ways of speaking about reality with no way of interpreting one into another.

the parallel would be this. there could be creatures out there somewhere living in the same universe as we do, yet their laws of nature that make the exact same predictions as ours do, are simply not translatable into ours. meaning there is no way for us to understand what they are saying, yet we can see them do their physics which from our perspective is total gibberish that somehow makes the very same predictions as ours. this seems impossible, how would our laws explain their existence? it would seem that our laws are incomplete and we need to do more research to explain what they are saying.

it could be that PFA and Ultimate L line are examples of two ways of discussing the universe that are simply disconnected from one another. they do not discuss the same reality. this must be false, they both stem from large cardinals, they both are large cardinal theories in disguise. but how can we prove this or at least feel confident that they are the same.

i believe that Steel's Program is the most important program that reasonably deals with this issue. Steel's Program is an interpretation of Goedel's Program, and the idea is to develop tools to translate between mathematical frameworks. Steel views large cardinals and inner model theory in this sense, as a way of translating PFA into the realm of determinacy axioms and etc. the two big open problems in this line of research are the following.

1. is there a natural determinacy theory T such that PFA can be forced over any "nice" model of T.
2. does PFA imply that HOD of $$L(\Gamma_{uB}, \mathbb{R})$$ has superstrong cardinals.

i will not go any further than this.

There is a recent (and unfinished) attempt to view the Shelah's approach to Continuum Hypothesis in terms of homotopy theory.

• Please change a link, It does not work anymore. Commented Mar 20, 2017 at 6:19
• I've redirected the link to Gabrilovich's website, which has a number of relevant texts. Commented Jun 16, 2017 at 2:23

Shelah's approach from his paper in his "The Generalized Continuum Hypothesis revisited", concerns mainly the generalized continuum hypothesis. His main theorem addresses an appealing variation of the GCH which is based on a revised notion of "power." Let me explain what is this notion of $\lambda^{[\kappa]}$ (in words: $\lambda$ to the revised power of $\kappa$,) which is central to his approach and is also of independent interest. $\lambda^{[\kappa]}$ is the minimum size of a family of subsets of size $\kappa$ of a set $X$ of cardinality $\lambda$ such that every subset of cardinality $\kappa$ of $X$ is covered by less than $\kappa$ members of the family. (Of course we need that $\kappa < \lambda$ and also we need that $\kappa$ is a regular cardinal.)

The introduction to the paper is readable and motivated.