@Prof. Hamkins: Sorry to have misquoted you. Still and all, it would be nice to have a synopsis of set theorist's experience in CH and GCH worlds. Is there a survey article you know of that provides such a synopsis? For my part, I side with the not-CH advocates because I myself believe that there is no reason to believe that Cohen (or for that matter Random) reals are not 'real' and because I believe that the universal quantifier in the Power Set Axiom should be interpreted as 'all possible subsets'. I hope to find the philosophical flaws in my own position. Hope you can help.

For Andrej Bauer: When you state "Questions about Cantor's Hypothesis were historically important for the development of set theory, but nowadays we understand the situation very well.", what is this "situation" set theorists understand very well. What are the intuitions set theorists have developed regarding the models of ZFC where CH and GCH hold? I accept your answer as being the orthodox(?) view of set theorists, who of course have every right to study any and all models of ZFC.

Also, what happens to set theory as the foundation of mathematics under the multiverse view? Is essentially the 'foundationalist' program dead and buried?

what is the larger context that makes the set-theoretic multiverse a coherent whole? Has one been discovered yet?

I have questions regarding the analogy between set theory and geometry, primarily the following: We know that in some cases we can represent alternatives to Euclid's fifth postulate as holding in spaces having curvature. The notion of curvature in spaces provides a larger context in which it makes sense to speak of the equality of alternative geometries. What is the larger context, if any, that allows one to speak of the equality of models of set theory coherently? This isn't to say that set theorists can't explore every possible model of set theory--

But if one can coherently interpret the universal quantifier as 'all possible', even as a type of generalized quantifier, couldn't one have a set theory that satisfies the universalist demands, even though it would be different (slightly?) from ZFC (in the manner in which you specify)?

Thanks, Prof. Hamkins, that is helpful. I have another question that you in particular might be able to answer, since you co-wrote the the paper relating modal logic to forcing. The question is this: can the phrase 'all possible' be a coherent quantifier? As I mentioned in my question, I am interested in interpreting the universal quantifier in the power set axiom as 'all possible' subsets of a given set A. Of course the usual interpretation of the universal quantifier limits its interpretation to all sets in a given model(and such limitations give rise to the multiverse view)

By the way, Chang in his paper had the closed interval [0,1] as the truth-values for the infinite-valued Lukasiewicz logic. It seems rather telling that the elements of [0,1] can be put in a 1-1 correspondence to a complete Boolean algebra.

Emil: I'm looking at the article "Boolean-valued Model" from Springer's Online Encyclopedia of Mathematics (www.encyclopediaofmath.org/index.php/Boolean-valued_model) and found an interesting statement..."If a Boolean algebra B is a two-element algebra (i.e. B={0,1} then the B-model M is the classical two-valued model." This seems to suggest that if B had an infinite number of elements, then the B-model M would essentially have an infinite-valued logic, which under certain conditions could be the infinite-valued logic of Lukasiewicz?

Emil: Thanks for the comment. Very helpful. What I would like to know is how Chow could come to be so misguided. What was the source of his confusion, in your opinion? Also, when you say, "the number of truth values as such is mostly irrelevant, what matters most is what equations hold in the algebra of truth values," are you referring to taking the quotient M^B/U where U is an ultrafilter to form an actual model(in Chow's case, a model of ZFC)? Also, can one actually do forcing in the infinite-valued predicate logic of Lukasiewicz? Thanks for your help.

To Emil Jerabek: Also, I understand one can do Forcing in the infinite-valued predicate logic of Lukasiewicz. Is that sort of forcing essentially similar to Forcing with Boolean-valued models?

To Emil Jerabek: I guess what I expect to get is a return to Naive Set Theory, since in the infinite-valued logic of Lukasiewicz, at least Chang may have correctly proven that the Axiom of Comprehension without parameters is consistent. My understanding of boolean-valued models comes from Timothy Chow's "A Beginner's Guide To Forcing" in which he claims that Boolean-valued Models are a type of 'fuzzy set theory'. Can't a Boolean-valued model based on a Boolean algebra with an infinite number of elements be construed as an infinite-valued logic?