>>12208138The question really is what is the worth ZFC? If, as some do, that mathematical statements have a truth value, then so does CH. ZFC does not decide many interesting problems. Do you we need new axioms? I recommend the article "Does set theory need new axioms" by John Steel. Some axioms that could potentially decide CH. There are two main candidates, and this is where it gets technical. The first is Martin's maximum, it is the strongest forcing axiom in a precise way, see the Jech "Multiple Forcing" section. MM implies (relative to a supercompact cardinal) the continuum has size . This contradicts CH but MM has many other nice combinatorial results. See Proper and Imporper Forcing by Shelah. The last is the one I know least about but it the most interesting. The V=Ultimate L conjecture by Woodin. The idea is that Godel's constructible univesre L, cannot accommodate even 'small' large cardinals. So the question is can we construct L like inner models of set theory that accommodate large cardinals. The big open question is at the level of supercompact, if V has supercompact, is there inner mdoel M with that supercompact in V. There are many other technical aspects I am glosssing over.
