# Re: Paper and slides on indefiniteness of CH

Dear Peter,

My apologies for the actualism/potentialism confusion! The situation is this: We have been throwing around 3 views:

1. Actualism in height and width (Neil Barton?)
2. Actualism only in width (Pen and Geoffrey?)
3. Actualism in neither.

Now the problem with me is that I have endorsed both 2 and 3 at different times! What I have been trying to say is that the choice between 2 and 3 does not matter for the HP, the programme can be presented from either point of view without any change in the mathematics. In 3 the universes to which V is compared actually are there, as part of the background multiverse (an extreme multiverse view) and in 2 you can only talk about them with “quotes”, yet the question of what is true in them is internal to (a mild lengthening of) V.

I have been a chameleon on this: My personal view is 3, but since no one shares that view I have offered to adopt view 2, to avoid a philosophical debate which has no pracatical relevance for the HP.

It is similar with the use of countable models! Starting with view 2 I argue that the comparisons that are made of V with other “universes” (in quotes) could equally well be done by replacing V by a ctm and removing the quotes. But again, this is not necessary for the programme, as one could simply refuse to do that and awkardly work with quoted “universes” all of the time. I don’t understand why anyone would want to do such an awkward thing, but I am willing to play along and sadly retitle the programme the MP (Maximality Programme) instead of the Hyperuniverse Programme, as now the countable models play no role anymore. In this way the MP is separated from the study of countable transitive models altogether.

In summary: There is some math going on in the HP which is robust under changes of interpretation of the programme. My favourite interpretation would be View 3 above, but I have settled on View 2 to make people happy, and am even willing to drop the reduction to countable models to make even more people happy.

I am an extreme potentialist who is willing to behave like a width actualist.

The mathematical dust has largely settled — as far as the program as it currently stands is concerned –, thanks to Hugh’s contributions.

What? There is plenty of unsettled mathematical dust out there, not just with the future development of the HP but also with the current discussion of it. See my mail of 25.October to Pen, for example. What do we say about the likelihood that maximality of V with respect to HOD likely contradicts large cardinal existence? Even if the HP leads to the failure of supercompacts to exist, can one at least get PD out of the HP and if so, how?

More broadly, a lot remains unanswered in this discussion regarding Type 1 evidence (for “good set theory”): If $\text{AD}^{L(\mathbb R)}$ is parasitic on $\text{AD}$ how does one argue that it is a good choice of theory? When we climb the interpretability hierarchy, should we drop AC in our choice of theories and instead talk about what happens in inner models, as in the case of AD? Similarly, why is large cardinal existence in V preferred over LC existence in inner models? Are Reinhardt cardinals relevant to these questions? And with regard to Ultimate L: What theory of truth is to be used when assessing its merits? Is it just Thin Realism, and if so, what is the argument that it yields “the best set theory” (“whose virtues swamp all the others” as Pen would say) and if not, is there something analagous to the HP analysis of maximality from which Ultimate L could be derived?

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Fri, 24 Oct 2014, W Hugh Woodin wrote:

Dear Sy,

You wrote to Pen:

But to turn to your second comment above: We already know why CH doesn’t have a determinate truth value, it is because there are and always will be axioms which generate good set theory which imply CH and others which imply not-CH. Isn’t this clear when one looks at what’s been going on in set theory? (Confession: I have to credit this e-mail discussion for helping me reach that conclusion; recall that I started by telling Sol that the HP might give a definitive refutation of CH! You told me that it’s OK to change my mind as long as I admit it, and I admit it now!)

ZF + AD will always generate “good set theory”…   Probably also V=L…

This seems like a rather dubious basis for the indeterminateness of a problem.

I guess we have something else to put on our list of items we simply have to agree we disagree about.

What theory of truth do you have? I.e. what do you consider evidence for the truth of set-theoretic statements? I read “Defending the Axioms” and am convinced by Pen’s Thin Realism when it comes to such evidence coming either from set theory as a branch of mathematics or as a foundation of mathematics. On this basis, CH cannot be established unless a definitive case is made that it is necessary for a “good set theory” or for a “good foundation for mathematics”. It is quite clear that there never will be a case that we need CH (or not-CH) for “good set theory”. I’m less sure about its necessity for a “good foundation”; we haven’t looked at that yet.

We need ZF for good set theory and we need AC for a good foundation. That’s why we can say that the axioms of ZFC are true.

On the other hand if you only regard evidence derived from the maximality of V as worthy of consideration then you should get the negation of CH. But so what? Why should that be the only legitimate relevant evidence regarding the truth value of CH? That’s why I no longer claim that the HP will solve the continuum problem (something I claimed at the start of this thread, my apologies). But nor will anything like Ultimate L, for the reasons above.

I can agree to disagree provided you tell me on what basis you conclude that statements of set theory are true.

Best,
Sy