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.

So the best one can do with a problem like CH is to say: “Based on a certain Type of evidence, the truth value of CH is such and such.” As said above, Type 1 evidence (the development of set theory as an area of mathematics) will never yield a fixed truth value, we don’t know yet about Type 2 evidence (ST as a foundation) and I still conjecture that Type 3 evidence (based on the Maximality of the universe of sets in height and width) will imply that CH is false.

There will never be such a resolution of CH (for the reasons I gave above). The best one can do is to give a widely persuasive argument that CH (or not-CH) is needed for the foundations of mathematics or that CH (or not-CH) follows from the Maximality of the set-concept. But I would not expect either achievement to draw great acclaim, as nearly all set-theorists care only about the mathematical development of set theory and CH is not a mathematical problem.

This whole discussion about CH is of interest only to philosophers and a handful of philosophically-minded mathematicians. To find the leading open questions in set theory, one has to instead stay closer to what set-theorists are doing. For example: Provably in ZFC, is V generic over an inner model which satisfies GCH?

Why is this last question a leading question? If there is an inner model with a measurable Woodin cardinal it is true, V is a (class) generic extension of an inner model of GCH.

You must mean something else. Focusing on eliminating the assumption of there is an inner model of a measurable Woodin cardinal seems like a rather technical problem.

Regards,

Hugh