# Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 13, 2014, at 4:22 AM, Sy David Friedman  wrote:

I can only repeat what I said to you on 30.September:

Hugh, the HP is (primarily) a study of maximality criteria of the sort we have been discussing. As I have been trying to explain, it is essential to the programme to formulate, analyse, compare and synthesise different criteria, discovering their mathematical consequences. I referred to my formulation of the $\textsf{SIMH}^\#$ as “crude and uncut” as it may have to be modified later as we learn more. Changes in its formulation do not mean a defeat for the programme, but rather progress in our understanding of maximality.

and what I said to Pen yesterday:

Look, the formulations of these maximality criteria are not set in stone; if someone can suggest improvements I am happy to hear them. Indeed Hugh had some interesting suggestions in this regard. The HP is still in its infancy and there is still a lot to learn about the formulation of and the mathematics behind these criteria.

The key issue with forms of the SIMH is what to do when both of the parameters omega_1 and omega_2 are present, and not just the parameter omega_1, anyway.

Why is that the key issue?  The issue of exactly how to formulate $\textsf{SIMH}^\#(\omega_1)$ seems absolutely critical to me which is why I kept pressing you on this.

If one formulates it in terms of the preservation of $\omega_1$ then one gets a principle which strongly denies large cardinals (by implying that there is a real whose sharp does not exist).

If on the other hand, one formulates it in terms of preserving all cardinals then it does not obviously imply $\textsf{IMH}^\#$, so why the “S”?

Further formulating $\textsf{SIMH}^\#$ in terms of preserving all cardinals instead of simply adapting the formulation of $\textsf{SIMH}$ to the #-generated context, yields a principle which one cannot really do anything with current technology.  Of course one gets not-CH but this is trivial.

Let Strong-$\textsf{SIMH}^\#$ be $\textsf{SIMH}^\#$ formulated as you formulate $\textsf{SIMH}$ in your 2006 BSL paper (page 11 for those who wish to look) but for #-generated outer models. But let’s add as possible absolute parameters, definable proper classes, so that Strong-$\textsf{SIMH}^\#$ implies $\textsf{SIMH}^\#$ as you have defined it.

(Thus I am defining a proper (definable) class p to be an “class absolute parameter” if if there is a formula which defines p in all cardinal preserving outer models which are #-generated.  So the class p of all cardinals is trivially such a class. $\textsf{SIMH}^\#$ is then Strong-$\textsf{SIMH}^\#$ just restricted to class absolute parameters).

Thus Strong-$\textsf{SIMH}^\#$ implies $\textsf{IMH}^\#$ (just as SIMH implies IMH). Also Strong-$\textsf{SIMH}^\#$ implies there is a real $x$ such that $\omega_1 = \omega_1^{L[x]}$. The latter is a very deep theorem which is an immediate corollary of the sophisticated machinery of coding by class forcing that you have developed.

Maximality etc., would seem to favor Strong-$\textsf{SIMH}^\#$ over $\textsf{SIMH}^\# unless you have HP-based reasons to reject it. Do you? If not then the move to$latex \textsf{SIMH}^\# instead of to Strong-$\textsf{SIMH}^\#$ looks rather suspicious.

Regards,
Hugh