On Oct 15, 2014, at 6:56 AM, Sy David Friedman
Dear Harvey,
Maybe I can be of some help with the set theory question (Hugh, feel free to correct or amplify what I say):
If the GCH holds at some infinite cardinal
in
then one can add a new subset of
without adding a new subset of
in 
then one can add a new subset of 
without adding a new subset of Maybe you need GCH holds on a club class? Otherwise I do not see that the Easton products do not collapse for example the double successors of limits. But maybe you have something else in mind at limit stages?
So the difficulty is with models
I agree, these seem like the very difficult cases.