TITLE: Very Large Cardinal Axioms and Continuum Hypothesis
QUESTION [10 upvotes]: Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?

REPLY [2 votes]: I just want to make one simple observation, which I think got lost in the history of the subject. Unfortunately, I do not speak German and so it will be nice if someone who knows German actually reads the sources (and corrects me if I am wrong).
I believe Cantor's original conjecture was that any infinite subset of reals either has the same size as the natural numbers or the same size as the real numbers.
Under AC this becomes the most cited version of CH, namely that $2^\omega=\omega_1$.
However, I believe it is a mistake to interpret Cantor's conjecture under ZFC. To me, one gets the feeling that Cantor's original CH is true by simply playing with actual sets, he himself showed that every closed set satisfies his original CH. And so the proper interpretation of Cantor's original CH under ZFC should be that every definable subset of reals is either countable or has the same size as the reals.
Of course, "definable" is open to interpretation as well, but large cardinals did settle a lot of these. For example, under large cardinals, there is no counterexample to Cantor's original CH among projective sets, sets in $L(\mathbb R)$,  etc. 
In this sense, Goedel was not wrong at all when he suggested that large cardinals may help settle the continuum hypothesis. In fact, far from it. The large cardinals freeze so much of the definable part of the universe that it is astonishing how right Goedel was.  For example, Woodin showed that under large cardinals the theory of Chang model cannot be changed, he also showed that after collapsing the power set of the supercompact, the theory of $L(\Gamma_{uB},\mathbb R)$ cannot be changed, all of these is what Goedel anticipated and it is, at its roots, motivated by his beliefs.