In an attempt to move things along, I would like to both summarize where we are
and sharpen what I was saying in my (first) message of Nov 8. My points were
possibly swamped by the technical questions I raised.
1) We began with Original-
This is the #-generated version. In an attempt to provide a -logic formulation
you proposed a principle which I called (in my message of Nov 5):
I raised the issue of consistency and you then came back on Nov 8 with the principle :
What this translates to for a countable model V is then this:
V is weakly #-generated and for all : Suppose that whenever is a generator for V (iterable at least to the height of V), holds in an outer model M of V with a generator which is at least as iterable as . Then holds in an inner model of V.
Let’s call this:
(There are too many principles)
But: Revised-New- is just the disjunct of Original- and New-
So Revised-New- is consistent. But is Revised-New- really what you had in mind?
(The move from New- to the disjunct of Original- and New- seems a bit problematic to me.)
Assuming Revised-New- is what you have in mind, I will continue.
Thus, if New- is inconsistent then Revised-New- is just Original-.
So we are back to the consistency of New-.
The theorem (of my message of Nov 8 but slightly reformulated here)
Theorem. Assume PD. Then there is a countable ordinal and a real such that if is a ctm such that
1) is in and
2) satisfies Revised-New- with parameter
then is #-generated (and so satisfies Original-)
strongly suggests (but does not prove) that New- is
inconsistent if one also requires be a model of “ for some set ”.
Thus if New- is consistent it likely must involve weakly #-generated models which cannot be coded by a real in an outer model which is #-generated.
So just as happened with SIMH, one again comes to an interesting CTM question whose resolution seem essential for further progress.
Here is an extreme version of the question for New-:
Question: Suppose M is weakly #-generated. Must there exist a weakly #-generated outer model of M which contains a set which is not set-generic over M?
[This question seems to have a positive solution. But, building weakly #-generated models which cannot be coded by a real in an outer model which is weakly #-generated still seems quite difficult to me. Perhaps Sy has some insight here.]