Tag Archives: CH as an open problem

Re: Paper and slides on indefiniteness of CH

Dear Sy,

This doesn’t really bear on any of the debates we’ve been having, but …

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously. … Resolving CH was certainly never my goal; I got into the HP to better understand large cardinals and internal consistency, with no particular focus on CH. … It would be interesting to ask other set-theorists (not Hugh or I) what the goals of set theory are; I think you might be very surprised by what you hear, and also surprised by your failure to hear “solve CH”.

The goal I mentioned was resolving CH as part of a full theory of sets of reals more generally. I said ‘resolving’ to leave open the possibility that the ‘resolution’ will be a understanding of why CH doesn’t have a determinate truth value, after all (e.g., a multiverse resolution).

It’s not a matter of how many people are actively engaged in the project: there might be lots of perfectly good reasons why most set theorists aren’t (because there are other exciting new projects and goals, because CH has been around for a long time and looks extremely hard to crack, etc.). I would ask you this: is CH one of the leading open questions of set theory? Is it the sort of thing that would draw great acclaim if someone were to come up with a widely persuasive ‘resolution’?

All best,