Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCHHow 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
