Dear Sy,

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):

2) New-

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:

3) Revised-New-

(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.]

Regards,

Hugh