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

Set Theory — The Search for New Axioms

Re: Paper and slides on indefiniteness of CH

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