[FOM] Height and Width of the universe
colin.mclarty at case.edu
Mon Jan 21 12:59:59 EST 2008
Joe Shipman offered
> Here are two theorems of the form "If the Universe
> is tall enough, it can't be too narrow."
of which one was
> If there is a measurable cardinal,
> there is a nonconstructible set.
But this does not say a tall enough unverse cannot be narrow. Of
course universes of constructible sets can be just as tall as any
universes at all -- including any universe with nonconstructible
sets. I think the point is clear but to be sure I'll say: given any
universe satisfying ZF, the subuniverse of constructibles in it is
just as tall.
What propeties of cardinals are preserved under narrowing -- in the
sense of remaining true in all inner models with the same ordinals?
Certainly cardinals remain cardinals, and inaccessibles remain
inaccessible. And famously no large cardinal property from
measureable up does remain under narrowing down to the constructibles.
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