Re: Paper and slides on indefiniteness of CH

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-\textsf{IMH}^\#

This is the #-generated version. In an attempt to provide a V-logic formulation
you proposed a principle which I called (in my message of Nov 5):

2) New-\textsf{IMH}^\#

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 phi: Suppose that whenever g is a generator for V (iterable at least to the height of V), \phi holds in an outer model M of V with a generator which is at least as iterable as g. Then \phi holds in an inner model of V.

Let’s call this:

3) Revised-New-\textsf{IMH}^\#

(There are too many (*) principles)

But: Revised-New-\textsf{IMH}^\# is just the disjunct of Original-\textsf{IMH}^\# and New-\textsf{IMH}^\#

So Revised-New-\textsf{IMH}^\# is consistent. But is Revised-New-\textsf{IMH}^\# really what you had in mind?

(The move from New-\textsf{IMH}^\# to the disjunct of Original-\textsf{IMH}^\# and New-\textsf{IMH}^\# seems a bit problematic to me.)

Assuming Revised-New-\textsf{IMH}^\# is what you have in mind, I will continue.

Thus, if New-\textsf{IMH}^\# is inconsistent then Revised-New-\textsf{IMH}^\# is just Original-\textsf{IMH}^\#.

So we are back to the consistency of New-\textsf{IMH}^\#.

The theorem (of my message of Nov 8 but slightly reformulated here)

Theorem. Assume PD. Then there is a countable ordinal \eta and a real x such that if M is a ctm such that
1) x is in M and M \vDash ``V = L[t]\text{ for real }t"
2) M satisfies Revised-New-\textsf{IMH}^\# with parameter \eta
then M is #-generated (and so M satisfies Original-\textsf{IMH}^\#)

strongly suggests (but does not prove) that New-\textsf{IMH}^\# is
inconsistent if one also requires M be a model of “V = L[Y] for some set Y”.

Thus if New-\textsf{IMH}^\# is consistent it likely must involve weakly #-generated models M 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-\textsf{IMH}^\#:

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

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