Tag Archives: Resolving CH using maximality

Re: Paper and slides on indefiniteness of CH

Dear Pen and Geoffrey,

On Wed, 24 Sep 2014, Penelope Maddy wrote:

Thanks, Geoffrey. Of course you’re right. To use Peter’s terminology, if you’re a
potentialist about height, but an actualist about width, then CH is determinate in the usual
way. I was speaking of Sy’s potentialism, which I think is intended to be potentialist about
both height and width.

You both say that if one hangs onto width actualism then “CH is determinate in the usual way”. I have no idea what “the usual way” means; can you tell me?

But nothing you have said suggests that there is any problem at all with determining the CH as a radical potentialist. Again:

… solving the CH via the HP would amount to verifying that the pictures of V which optimally exhibit the Maximality feature of the set concept all satisfy CH or all satisfy its negation. I do consider this to be discovering something about V. But I readily agree that it is not the “ordinary way people think of that project”.

And in more detail:

We have many pictures of V. Through a process of comparison we isolate those pictures which best exhibit the feature of Maximality, the “optimal” pictures. Then we have three possibilities:

a. Does CH hold in all of the optimal pictures?
b. Does CH fail in all of the optimal pictures?
c. Otherwise

In Case a, we have inferred CH from Maximality, in Case b we have inferrred -CH from Maximality and in Case c we come to no definitive conclusion about CH on the basis of Maximality.

OK, maybe this is not the “usual way” (whatever that is), but don’t you acknowledge that this is a programme that could resolve CH using Maximality?

I also owe Pen an answer to:

… at some point could you give one explicit HP-generated mathematical principle that you endorse, and explain its significance to us philosophers?

As I tried to explain to Peter, it is too soon to “endorse” any mathematical principle that comes out of the HP! The programme generates different possible mathematical consequences of Maximality, mirroring different aspects of Maximality. For example, the IMH is a way of handling width maximality and \#-generation a way of handling height maximality. Each has its interesting mathematical consequences. But they contradict each other! The aim of the programme is to generate the entire spectrum of possible ways of formulating the different aspects of Maximality, analysing them, comparing them, unifying them, … until the picture converges on an optimal Maximality criterion. Then we can talk about what to “endorse”. I conjecture that the negation of CH will be a consequence, but it is too soon to make that claim.

The IMH is significant for many reasons. First, it refutes the claim that “Maximality in width” implies the existence of large cardinals; indeed the IMH is the most natural formulation of “Maximality in width” and it refutes the existence of large cardinals! Second, it illustrates how one can legitimately talk about “arbitrary thickenings” of V in discussions of Maximality, without tying one hands to the restrictive notion of forcing extension. Third, as discussed at length in my papers with Tatiana, it inspires a re-think of the role of large cardinals in set theory, explaining this in terms of their existence in inner models as opposed to their existence in V.

But the HP has moved beyond the IMH to other criteria like \#-generation, unreachability and (possibly) omniscience, together with different ways of unifying these criteria into new “synthesised” criteria. It is an ongoing study with a lot of math behind it (yes Pen, “good set theory” that people can care about!) but this study is still in its infancy. I can’t come to any definitive conclusions yet, sorry to disappoint. But I’ll keep you posted.