This negation of CH is based on Woodin's work: https://en.wikipedia.org/wiki/Ω-logic Of course, you can only believe his result because you need to believe his axioms first. But for me it is really convincing for multiple reasons: 1. While it is, of course, a negation of CH, it does not really break everything because the sequence of Aleph numbers is preserved, just the names assigned to different alephs change 2. His conclusion is based on the quantification over possible forcings, and it looks really powerful - as forcing is used to prove independence of so many large cardinal axioms, so quantification over forcings must be extremely powerful. So it is like (in physics) expanding universe of sets into the multiverse! 3. [itex]\omega_1[/itex] is now less than continuum. And [itex]\omega_1[/itex], at least for me, looks intuitively "almost" countable, as the sequence of ordinals is explicitly well ordered. Of course, any set can be well ordered if we assume AC, but often no constructive example of such ordering can be provided. 4. Interestingly enough, Goedel himself had suspected that continuum = [itex]\aleph_2[/itex] I am Platonist, so for me it sounds more like a discovery. Not like a formal game (with this axiom we can do this, and with another we can do that). Do you feel the same?