## Solutions to the Continuum Hypothesis

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(A related MO question: http://mathoverflow.net/questions/14338/what-is-the-general-opinion-on-the-generalized-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.

## 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 paradize" (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 asnwer is by Lucca Belloti (also suggested by Haim).

The proposed asnwer $2^{\aleph_0}=\aleph_2$ 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 Continuuum 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 Anders Caicedo and several other useful answers. (Unfortunately, I can accept only one answer.)

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

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Another noteworthy reference is Gödel's unpublished note 1970a where he collects evidence in favor of $2^{\aleph_0} = \aleph_2$ - books.google.com/… – François G. Dorais♦ May 7 2010 at 8:00

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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. – ogerard May 8 2010 at 12:40

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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. – Mark Meckes May 10 2010 at 13:36

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Gil, I agree with your characterization of what the question is. But to my ears, a "hypothesis" is a statement, not a question. You're asking about answers to the question (or solutions to the problem) of whether the Continuum Hypothesis should be accepted; what I object to is using "Continuum Hypothesis" as a shorthand name for that question. – Mark Meckes May 13 2010 at 23:52

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The presentation was very delightful. – To be cont'd May 18 2010 at 11:30
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## 9 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.

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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. – Colin Tan May 19 2010 at 8:32

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I agree, Colin; the analogy with geometry is very strong and extends to many facets of how we think about the various geometries. – Joel David Hamkins May 19 2010 at 16:18

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Is there any name for something like "formalism with a preferred model", where one acknowledges that there is no reason apart from personal preference to study a given system of axioms, but where one also acknowledges that one has a preferred system? It seems somewhere in between the formalism and pragmatism and seems to be a view partially justified by advances in topos theory. – Harry Gindi May 20 2010 at 7:21

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Because it admits a preferred model, that view seems much closer to Platonism or realism than formalism. It is similar to what occurs in (first order) arithmetic, where we have the standard model of the natural numbers. In set theory, however, we seem to lack such a standard model. Topos theory does not provide a standard model, since all the diverse models of set theory that have been constructed by forcing and so on each have their own incompatible version of Set. The nature and truths of any absolute set-theoretic background is what is at stake and it remains disapointingly murky. – Joel David Hamkins May 20 2010 at 12:23

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"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." – gowers Feb 6 2011 at 17:28
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(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) that is required to established 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. This 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 where 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 bi-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.

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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). – John Stillwell May 18 2010 at 23:47
Could you give some more details about the connection with the recent work of Sargsyan? – Simon Thomas Nov 23 2010 at 13:08
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) – Andres Caicedo 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. – Andres Caicedo 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. – Andres Caicedo Nov 23 2010 at 15:45
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Regarding Shelah's approach, I believe that the following paper should be quite accessible to non-professionals: http://arxiv.org/pdf/math.LO/0102056

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 page might be useful: http://www2.units.it/episteme/L&PS_Vol3No1/bellotti/bellotti-html/bellotti_L&PS_Vol3No1.html

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Thanks Haim, I think I vaguely had this paper of Shelah also in mind but did not find it when I wrote the question. – Gil Kalai May 8 2010 at 8:54

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 here). 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).

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 paper.

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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 $\kappa$.

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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.)

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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.

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

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I believe Gödel and Cohen's arguments are not acceptable. My reasoning is a common sense one:

A mathematical proof is a text whose spelling correctness can be checked by a machine. The text in itself has no meaning. Only an interpretation of it can have any meaning. So, at best, Gödel and Cohen proved some mathematical statements which can be interpreted as implying that the continuum hypothesis is undecidable (in a given axiom system).

It's surprising that Bourbaki, in his Set Theory book, takes Gödel and Cohen's arguments for granted. I think that Bourbaki's approach, coherently followed, leads to a rejection of these arguments.

http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Continuum_Hypothesis/

I think many people share this opinion. (For some reason they don't express it.)

In another answer I gave three quotations from Cohen's book "Set theory and the Continuum Hypothesis". For the reader's convenience I'll paste them below. I think we'll all find them interesting.

Quotation 1 (pages 26-27)

It should be emphasized that these functions are "real" mathematical objects and not objects of any formal system ...

Question. What is a "real mathematical object"? (And what is an "unreal" or "irreal" mathematical object?)

Quotation 2 (page 39)

The theorems of the previous section are not results about what can be proved in particular axiom systems; they are absolute statements about functions.

Questions. What is an "absolute statement"? Is there an "axiom free" mathematics?

Quotation 3 (page 41)

We have now arrived at a rather peculiar situation. On the one hand $\sim A$ is not provable in $Z_1$ and yet we have just given an informal proof that $\sim A$ is true. (There is no contradiction here since we have merely shown that the proofs in $Z_1$ do not exhaust the set of all acceptable arguments.)

Question. What is an "acceptable argument"? It seems that "true" is defined as "following from an acceptable argument".

Here is another quotation (page 13):

If a set of statements S has a model then it is consistent.

If the underlying set theory is inconsistent, then any set of statements has a model.

I suggest the following three books to the interested reader.

(1) P. Cohen, Set theory and the Continuum Hypothesis.

(2) S.C. Kleene, Introduction to Metamathematics.

(3) N. Bourbaki, Theory of Sets. French original: Théorie des ensembles.

Let me ask explicitly the question which is implicit in this post:

Is the Continuum Hypothesis true, false or undecidable in Bourbaki's set theory?

There is no mathematical questions whose answer I more ardently wish I knew.

Thank you to all the people in charge of MathOverflow for everything you're doing!

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If one is a pragmatic formalist (believing that formal systems are no more nor less than working models of other systems), one may still ask about formal systems modeling the reachability of propositions in other formal system, such as ZFC. My impression is that these are precisely the sorts of results that Gödel and Cohen produced with respect to CH; that is, they indeed prove that a mathematician who works correctly cannot prove CH in ZFC. This is no more objectionable than proving that an infinite graph is disconnected, even though one can never prove this by exhausting all possible paths. – Niel de Beaudrap May 20 2010 at 8:00
Dear Niel - Thanks for your comment. This is just to see if I understand you correctly. Your opinion is (as I understand it) that Gödel and Cohen proved some mathematical statements which can be interpreted as implying that the continuum hypothesis is undecidable (in ZFC to quote your comment). Please correct me if I'm wrong. – Pierre-Yves Gaillard May 20 2010 at 9:50

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It is true that Goedel and Cohen proved that if ZF is consistent, then so are ZFC+CH and ZFC+$\neg$CH. But their proofs provided more than just this relative consistency result, as a formal syntactic result. Rather, what they provided were construction methods for building new models of set theory from old models---from any universe of sets in which ZFC is true, we can build closely related models where ZFC+CH and ZFC+$\neg$CH hold. So the methods are more semantic than syntactic, and perhaps the situation is like interpreting the various Euclidean and hyperbolic geometries within one another. – Joel David Hamkins May 20 2010 at 12:31
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