# Re: Paper and slides on indefiniteness of CH: My final mail to the Thread

Dear Sy,

Before we close this thread, it would be nice if you could state what the current version of $\textsf{IMH}^\#$ is. This would at least leave me with something specific to think about.

Is it:

1) (SDF: Nov 5) M is weakly #-generated and for each phi, if
for each countable alpha, phi holds in an outer model of M which
is generated by an alpha-iterable presharp then phi holds in an inner model of M.

2) (SDF: Nov 8) M is weakly #-generated and for all $\phi$: Suppose that whenever $g$ is a generator for M (iterable at least to the height of M), $\phi$ holds in an outer model M with a generator which is at least as iterable as $g$. Then $\phi$ holds in an inner model of M.

or something else? Or perhaps it is now a work in progress?

Regards,
Hugh

# 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

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Nov 5, 2014, at 7:40 AM, Sy David Friedman wrote:

Hugh:

1. Your formulation of $\textsf{IMH}^\#$ is almost correct:

M witnesses $\textsf{IMH}^\#$ if

1) M is weakly #-generated.

2) If $\phi$ holds in an outer model of M which is weakly
#-generated then $\phi$ holds in an inner model of M.

But as we have to work with theories, 2) has to be: If for each countable $\alpha$, $\phi$ holds in an outer model of M which is generated by an $\alpha$-iterable presharp then $\phi$ holds in an inner model of M.

Let’s call this New-$\textsf{IMH}^\#$.

Are you sure this is consistent?

Assume coding works in the weakly #-generated context:

Coding Assumption: if M is weakly #-generated then M can be coded by a real in an outer model which is weakly #-generated.

Then:

Theorem. Assume PD. Then there is a real $x$ such that for all ctm M, if x is in M then M does not satisfy New-$\textsf{IMH}^\#$.

(So in any case, one cannot get consistency by the determinacy proof).

2. Could you explain a bit more why V = Ultimate-L is attractive?

Shelah has the informal notion of a semi-complete axiom.

V = L is a semi-complete axiom as is $AD^{L(\mathbb R)}$ in the context of $L(\mathbb R)$ etc.

A natural question is whether there is a semi-complete axiom which is consistent with all large cardinals. No example is known.

If the Ultimate L Conjecture is true (provable) then V = Ultimate L is arguably such an axiom and further it is such an axiom which implies V = HOD (being “semi-complete” seems much stronger in the context of V = HOD).

Of course this is not a basis in any way for arguing V = Ultimate L. But is certainly makes it an interesting axiom whose rejection must be based on something equally interesting.

You said: “For me, the “validation” of V = Ultimate L will have to come from the insights V = Ultimate L gives for the hierarchy of large cardinals beyond supercompact.”
But why would those insights disappear if V is, for example, some rich generic extension of Ultimate L? If Jack had proved that $0^\#$ does not exist I would not favour V = L but rather V = some rich outer model of L.

I think if our evolving understanding of the large cardinal hierarchy rests primarily on the context of V = Ultimate L then very likely the rich generic extensions are not playing much of a role in understanding the large cardinal hierarchy.

This for me would build the case for V = Ultimate L and against these rich extensions. It would then take something quite significant in the theory of the rich extensions to undermine that.

But such speculations seem very premature. We do not even know if the HOD Conjecture is true. If the HOD Conjecture is not true then the entire Ultimate L scenario fails.

3. I told Pen that finding a GCH inner model over which V is generic is a leading open question in set theory. But you gave an argument suggesting that this has to be strengthened. Recently I gave a talk about HOD where I discussed the following four properties of an inner model M:

Genericity: V is a generic extension of M.

Weak Covering: For a proper class of cardinals $\alpha$, $\alpha^+ = \alpha^{+M}$.

Rigidity: There is no nontrivial elementary embedding from M to M.

Large Cardinal Witnessing: Any large cardinal property witnessed in V is witnessed in M.

(When 0# does not exist, all of these hold for M = L except for Genericity: V need not be class-generic over L. As you know, there has been a lot of work on the case M = HOD.)

Now I’d like to offer Pen a new “leading open question”. (Of course I could offer the PCF Conjecture, but I would prefer to offer something closer to the discussion we have been having.) It would be great if you and I could agree on one. How about this: Is there an inner model M satisfying GCH together with the above four properties?

Why not just go with the HOD Conjecture? Or the Ultimate L Conjecture?

There is is another intriguing problem which has been suggested by this thread.

Suppose $\gamma^+$ is not correctly computed by HOD for any infinite cardinal $\gamma$.Must weak square hold at some singular strong limit cardinal?

This looks like a great problem to me and it seems clearly to be a new problem.

Regards,
Hugh