[FOM] Question about Woodin Conjecture

[Colin McLarty]

John Steel <steel at math.berkeley.edu> wrote:

>       This is not enough--we want T to be consistent with
>supercompacts, huge cardinals, and all our yet undreamt-of large
>cardinal hypotheses of the future. A key move is to take preservation
>under set forcing as characteristic of all large cardinal hypotheses,
>of today and of tomorrow. The consistency requirement on T then
>becomes: true in some set generic extension of V.

Just to see if I understand what is preserved under set forcing. Is it 
right to say that a single supercompact cardinal is not preserved under set 
forcing, since you can collapse it to smaller cardinals? But if each 
supercompact has a larger supercompact, that is preserved under set 
forcing, since any amount of collapse will still leave ordinal-many 
supercompacts?

Colin