Re: Paper and slides on indefiniteness of CH

Dear Hugh,

In ZF if there is an extendible cardinal (or just proper class of supercompact cardinals formulated properly in ZF) then there is a class forcing extension in which AC holds and one preserves all supercompact cardinals (and much more).

Thanks for pointing this out. It suggests that for the choiceless-HP, AC might be compatible with maximality if the existence of supercompacts is (still unclear), and also that any “good set theory” compatible with supercompacts has a chance of being at least “simulated” in models of useful axioms compatible with AC.

Of course I’d prefer just to always assume choice but do want to stay open-minded about that as we don’t know what the set theory of the 22nd century might look like.


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