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