Hugh Woodin has two survey articles on recent work on the Continuum Hypothesis: I and II. Most mathematicians consider the continuum hypothesis as a settled question: since it is independent of ZFC, its truth is unknowable.
Set theorists, on the other hand, sometimes hold out the hope that new, intuitive axioms will be found that will provide a definite answer. Woodin thinks that we are close to finding such an axiom, and it seems to indicate that the cardinality of the reals is aleph two. (The continuum hypothesis states that it’s aleph one.)
There is a simpler example of an intuitive result that implies that the continuum hypothesis is false. Details can be found here and here.
The first article, at least, is full of references to so-called Woodin cardinals. Is that not bad form? Certainly none of my books contains references to objects named after the book authors unless the author absolutely has to (for exammple Conway is pretty well forced to call .0 by its better known name Co_0, and he certainly doesn’t call it the Conway group).
That did read as pretty tacky, but they really are called Woodin cardinals (presumably named that by someone else, though I don’t know that for a fact).
I only managed to understand why CH isn’t settled this past semester as part of my study for my qual (I read at least one of the Woodin articles you’ve linked to on this). Most set theorists don’t find Freiling’s arguments compelling, I think because it feels like he gets too much too cheap. There isn’t any obvious way to cap the power of his axioms, and if you let them get too strong, then you contradict choice as well as requiring the continuum to be at least aleph_omega.
Anyway, the big problem in the traditional mathematical answer of CH being “unknowable” is that it identifies knowability with provability from ZFC, without giving any clear arguments either for the knowability of ZFC or against the knowability of anything else. It seems that the evidence we have in favor of various large cardinal axioms is nearly as good as that in favor of choice and replacement, but the latter two are much more widely accepted. Of course, as is pointed out in the links, the large cardinals don’t do anything to settle CH, but John Steel and others hope that the way the large cardinals were decided will be relevantly similar to the way that we will one day be able to decide CH.
As for talking about “Woodin cardinals”, I believe in the early ’80s, he just talked about “cardinals of a certain sort larger than measurables and smaller than supercompacts”, but the name “Woodin cardinal” has become so ingrained that it would be difficult for him not to use it. And their defining property doesn’t have any nice name like “measurable” or “compact”, and we certainly don’t need any more “inaccessible”, “huge”, or “strong” cardinals, or anything of the sort.
Choice was once controversial. Replacement might be more controversial if the average mathematician knew how powerful it was — it’s like the first large cardinal axiom (or maybe the second, after infinity).
I’m not clear what you mean by “the evidence in favor of various large cardinal axioms is nearly as good as that in favor of choice and replacement”. I’ve heard similar statements before, and I was never really sure what they meant. Anything more compelling than “we tried to find some contradictions, but we can’t”?