Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was the continuum hypothesis born?
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.
- $2^{\omega}=\omega_1$
- 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
Best Answer
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.