[FOM] Height and Width of the universe

joeshipman@aol.com joeshipman at aol.com
Sun Jan 20 12:09:08 EST 2008

Here are two theorems of the form "If the Universe is tall enough, it 
can't be too narrow."

If there is a measurable cardinal, there is a nonconstructible set.
If there is an inaccessible cardinal, there is a countable transitive 
model of ZFC.

What other theorems relate the height and width of the Universe? The 
theorem "If there are infinitely many Woodin cardinals, there is no 
projective set of reals which is not determined" seems to go in the 
other direction than the previous two, but it doesn't really, because 
we know there are non-determined sets, and the theorem just says they 
don't fall in the projective hierarchy of reals not that they don't 

-- JS
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