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I don’t know what to make of this, added to regular cardinal by David Roberts:
Assuming the consistency of ’there is a proper class of strongly compact cardinals’, it is consistent that every uncountable cardinal is singular, a result due to Moti Gitik.
How does this square with the easy proof that is uncountable and regular? (And of course is uncountable and singular, so it’s not a matter of getting it exactly backwards.) So in what context is this occurring, or what is it supposed to say?
I’m guessing that Gitik is working in the absence of AC?
That’s certainly one possibility, but if so, then the phrasing is quite odd. In any case, we must ask, consistent with what?
My guess would be, consistent with ZF. But David should tell us, since he presumably looked at the paper. (-:
He also might have read Gitik’s bio on Wikipedia, which is equally vague. At least that makes it unlikely to be a mistake, so it probably is just all over , as you suggested.
Sorry for the vagueness. The correct statement is:
Con(ZFC + there is an unbounded class of strongly compact cardinals) Con(ZF + for all ordinals )
and hence the only regular cardinal in this model is . Got to run, I can edit the page later if needed.
Thanks, David; I’ve edited the page.
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