Maybe I can be of some help with the set theory questions (Hugh, feel free to correct or amplify what I say):
On Wed, 15 Oct 2014, Harvey Friedman wrote:
Suppose is a ctm and . Must have an outer model of which is cardinal preserving and not a set forcing extension? Number Theory is full of problems which seem completely out of reach. And one can easily generate such problems in a foundational guise.
I have a couple of questions about Hugh’s question.
1. The question as formulated involves both forcing extensions and general (outer) extensions. Are there appropriate formulations that
a. Do not refer to forcing at all.
Here is one (but it is cheating, and I’ll explain why below): Does every ctm of ZFC have an outer model of ZFC with the following covering property? For some cardinal of , every function (with ordinal domain) is covered by a multi-valued function with the same domain and at most values? (I.e., for each in the domain of , has size at most and is an element of ).
This is “cheating” because I have just used a theorem of Bukovsky to reformulate “set-generic extension” in terms of a covering property!
b. Given any ctm , refer only to forcing extensions.
Here is one: Does every have a cardinal-preserving forcing extension which adds a new set but no new real? This was first asked by Matt Foreman. I think it is still open.
2. What are the known natural conditions on that are known to be sufficient? Also for variants in 1 above.
If the GCH holds at some infinite cardinal in M then one can add a new subset of without adding a new subset of kappa 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.
So the difficulty is with models in which the GCH fails at all sufficiently large cardinals.