On Oct 15, 2014, at 6:56 AM, Sy David Friedman
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 and without collapsing cardinals. If the GCH holds at unboundedly many cardinals in then has a cardinal-preserving extension which is not a set-generic extension.
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 in which the GCH fails at all sufficiently large cardinals.
I agree, these seem like the very difficult cases.