[FOM] Height and Width of the universe

[Colin McLarty]

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.

best, Colin