# 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

# Re: Paper and slides on indefiniteness of CH

Well, it is hard not to respond. So I guess I will violate my “last message” prediction. Hopefully my other predictions are not so easily refuted. My apologies to the list. Never say never I suppose.

Hugh:

Regarding your two points directed at me and the HP:

1) Having established Con LC, one has established that every set X belongs to an inner model in which LC is witnessed above X.

I don’t necessarily agree. There are only countably many LC axioms and a perfectly coherent scenario is that each holds in an inner model which is coded by some real. That demands only countably many reals and in no way suggests that there should be large LCs in inner models containing all of the reals.

But please don’t misunderstand me: The HP is a programme for discovering new first-order properties via intrinsically-based criteria for the choice of preferred universes. It is open-ended, meaning that one cannot exclude the possibility of arriving at the statement you express above or even at the existence of LCs in V at some point in the future. But so far the evidence is just not there.

2) I challenge HP to establish that there is an inner model of “ZFC + there are infinitely many Woodin cardinals” without establishing PD. For this HP can use Con LC for any LC up to Axiom I0.

I think I understand the point you want to make here, which is that the HP so far offers no new techniques for producing consistency lower bounds beyond core model theory. I agree. But the intuitive (not mathematical) step from Con LC to inner models for LC is straightforward: Extrinsically we understand that LC is perfectly compatible with both the well-foundedness of the membership relation and with ordinal-maximality. From this we can conclude that LCs exist in countable transitive models of ZFC which are ordinal-maximal (i.e. #-generated). From this it provably follows that they exist in inner models.

We completely disagree on the Con LC issue as our email thread to this point clearly shows, no need to make a further comment on that.

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 $V = \text{HOD}$ 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 $V = L$.

The issue I seek clarified is exactly how HP will lead to a new axiom. At some point HP must declare some new sentence as “true” . What are the HP protocols? You seem to suggest that SIMH if consistent is such a “truth” but I am not even sure you make that declaration.

You said:

“Let IMH(card-arith) be IMH together with the following:

Suppose there is a card-arith preserving extension of M in which \phi holds. Then there is an card-arith preserving inner model of M in which \phi holds.

“Conjecture” : IMH(card-arith) implies GCH.

My question for Sy’s paper is simply, why if “conjecture” is true does one reject this in favor of SIMH (assuming SIMH is consistent)?”

It is good that you posed this question because it illustrates very well how the HP is meant to work. If the “conjecture” is true then the SIMH is almost surely inconsistent and this would be exciting progress in the HP. Indeed I welcome the exploration of a wide range of such criteria in order to gain a better understanding of absoluteness, constantly refining our picture of the universe based on these forms of maximality. Of course some criteria, like the SIMH, are very natural and well-motivated, whereas others, such as the IMH for ccc extensions (roughly speaking: Levy absoluteness with cardinal-absolute parameters for ccc extensions) are not. Note that the latter is consistent and solves the continuum problem! But in my view its downfall is simply that the notion of “ccc extension” is unmotivated.

Continuing the point I make above. I agree with you that if “conjecture” is true then SIMH is probably inconsistent. But it is also possible that both “conjecture” is true and SIMH is consistent. What then?

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.

I am really trying to be helpful here. These are natural issues that I think need to be addressed in making the case that HP has the potential to discover and validate new axioms.

Regards,
Hugh