Re: Paper and slides on indefiniteness of CH

Dear Sy,

Dear Hugh,

I agree that HP is part of the very interesting study of models of ZFC.  There are many open and studied questions here. For example suppose \phi is a sentence such that there is an uncountable wellfounded model of ZFC+ \phi but only at most one model of any given ordinal height. Must all the uncountable wellfounded models of ZFC + \phi satisfy V= L? (The wellfounded models must all satisfy V = HOD and that there are no measurable cardinals). The answer could well be yes and the proof extremely difficult etc., but to me this would be no evidence that V=L.

See my 2012 MALOA lectures:

A positive answer to your question follows from item 13 there; the proof is not difficult.

Thanks. I was aware of content of item 13 but had not realized it completely solved the problem I stated. So you have given me a homework problem.

Nevertheless, there are many similar questions. For example if $\phi$ is a sentence and ZFC + $\phi$ has only one transitive model M, must $M \vDash \textsf{CH}$?

Of course you will argue this is not relevant to HP. OK, here is another question which I would think HP must deal with.

Question: Suppose M is a countable transitive model of ZFC. Must there exist a cardinal preserving extension of M which is not a set-generic extension?

No such M can satisfy IMH since within any such M, all sets have sharps and much more.

Conjecture:  If M is such a model then $M \vDash \textsf{PD}$.

I guess you could predict based on your conviction in HP that this latter case will not happen just as I predict PD is consistent. For me an inconsistency in PD is an extreme back-to-square-one event. I would like to see (at some point) HP make an analogous declaration.

Huh? The HP is a programme for truth in general, it is not aimed at a particular statement like CH. Even if the SIMH is inconsistent there is still plenty for the programme to explore. I don’t yet know if CH will have a constant truth value across the “preferred universes” (Sol is right, a better term would be something more flattering, like “optimal universes”), especially as a thorough investigation of the different intrinsically-based criteria has only just begun and it will take time to develop the optimal criteria together with their first-order consequences. I guess it is possible that an unavoidable “bifurcation” occurs, i.e. there are two conflicting optimal criteria, one implying CH and the other its negation. It is much too early to know that. Perhaps this would be what you call a “back-to-square-one event” regarding CH.7.

From your message to Pen on 19 Aug:

Goal: The goal is to arrive at a single optimal criterion which best synthesises the different intrinsically-based criteria, or less ambitiously a small set of such optimal criteria. Elements of the Hyperuniverse which obey one of these optimal criteria are called “preferred universes” and first-order properties shared by all preferred universes are regarded as intrinsically-based set-theoretic truths. Although the process is dynamic and therefore the set of such truths can change the expectation is that intrinsically-based truth will stabilise over time. (I expect that more than a handful will consider this to be a legitimate notion of intrinsically-based truth.)

I will try one more time. At some point HP must identify and validate a new axiom. Otherwise HP is not a program to find new “axioms”. It is simply part of the study of the structure of countable wellfounded models no matter what the motivation of HP is.

It seems that to date HP has not done this. Suppose though that HP does eventually isolate and declare as true some new axiom. I would like to see clarified how one envisions this happens and what the force of that declaration is. For example, is the declaration simply conditioned on a better axiom not subsequently being identified which refutes it? This seems to me what you indicate in your message to Pen.

Out of LC comes the declaration “PD is true”. The force of this declaration is extreme, within LC only the inconsistency of PD can reverse it.

Regards,
Hugh