# Re: Paper and slides on indefiniteness of CH

Dear Sy,

I owe you a response to your other letters (things have been busy) but your letter below presents an opportunity to make some points now.

On Oct 31, 2014, at 12:20 PM, Sy David Friedman wrote:

Why? I think I made my position clear enough: I stated a consistent Maximality Criterion and based on my proof (with co-authors) of its consistency I have the impression that this Criterion contradicts supercompacts (not just extendibles). So that leads to a tentative rejection of supercompacts until the situation changes through further understanding of further Maximality Criteria. It’s analagous to what happened with the IMH: It led to a tentative rejection of inaccessibles, but then when Vertical Maximality was taken into account, it became obvious that the IMH# was a better criterion than the IMH and the $\textsf{IMH}^\#$ is compatible with inaccessibles and more.

I don’t buy this. Let’s go back to IMH. It violates inaccessibles (in a dramatic fashion). One way to repair it would have been to simply restrict to models that have inaccessibles. That would have been pretty ad hoc. It is not what you did. What you did is even more ad hoc. You restricted to models that are #-generated. So let’s look at that.

We take the presentation of #’s in terms of $\omega_1$-iterable countable models of the form (M,U). We iterate the measure out to the height of the universe. Then we throw away the # (“kicking away the ladder once we have climbed it”) and imagine we are locked in the universe it generated. We restrict IMH to such universes. This gives $\textsf{IMH}^\#$.

It is hardly surprising that the universes contain everything below the # (e.g. below $0^\#$ in the case of a countable transitive model of V=L) used to generate it and, given the trivial consistency proof of $\textsf{IMH}^\#$ it is hardly surprising that it is compatible with all large cardinal axioms (even choicless large cardinal axioms). My point is that the maneuver is even more ad hoc than the maneuver of simply restricting to models with inaccessibles. [I realized that you try to give an "internal" account of all of this, motivating what one gets from the # without grabbing on to it. We could get into it. I will say now: I don't buy it.]

I also think that the Maximality Criterion I stated could be made much stronger, which I think is only possible if one denies the existence of supercompacts. (Just a conjecture, no theorem yet.)

First you erroneously thought that I wanted to reject PD and now you think I want to reject large cardinals! Hugh, please give me a chance here and don’t jump to quick conclusions; it will take time to understand Maximality well enough to see what large cardinal axioms it implies or tolerates. There is something robust going on, please give the HP time to do its work. I simply want to take an unbiased look at Maximality Criteria, that’s all. Indeed I would be quite happy to see a convincing Maximality Criterion that implies the existence of supercompacts (or better, extendibles), but I don’t know of one.

We do have “maximality” arguments that give supercompacts and extendibles, namely, the arguments put forth by Magidor and Bagaria. To be clear: I don’t think that such arguments provide us with much in the way of justification. On that we agree. But in my case the reason is that is that I don’t think that any arguments based on the vague notion of “maximality” provide us with much in the way of justification. With such a vague notion “anything goes”. The point here, however, is that you would have to argue that the “maximality” arguments you give concerning HOD (or whatever) and which may violate large cardinal axioms are more compelling than these other “maximality” arguments for large cardinals. I am dubious of the whole enterprise — either for or against — of basing a case on “maximality”. It is a pitting of one set of vague intuitions against another. The real case, in my view, comes from another direction entirely.

An entirely different issue is why supercompacts are necessary for “good set theory”. I think you addressed that in the second of your recent e-mails, but I haven’t had time to study that yet.

The notion of “good set theory” is too vague to do much work here. Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise. The key notion is evidence, evidence of a form that people can agree on. That is the virtue of actually making a prediction for which there is agreement (not necessarily universal — there are few things beyond the law of identity that everyone agrees on — but which is widespread) that if it is proved it will strengthen the case and if it is refuted it will weaken the case.

Best,
Peter