Below is an addendum to the end of my last message.
(Sorry, this addendum was intended to be short, but it grew quite long. I hope and expect that this won’t happen again, as I think am finally running out of things to say in this discussion!)
I should also have mentioned that even among valuable axioms of set theory that imply not-CH there are strong disagreements about the size of the continuum.
There is a vibrant area of set theory called “cardinal characteristics of the continuum”. This is the study of cardinal numbers defined in the following way: We take some natural property that a set of reals could have and associate to it the least size of a set of reals that could have that property. For example a (= the almost disjoint number) is the smallest size of an infinite maximal almost disjoint set of subsets of omega (we cheat here slightly and identify the reals with subsets of omega). b = the bounding number, c = the size of the continuum, d = the dominating number, …, there is a cardinal characteristic for almost every letter of the alphabet. There are also characteristics of a less combinatorial flavour, like the least size of a set of reals that is not Lebesgue measurable.
Each cardinal characteristic is at least and at most . So under CH all cardinal characteristics collapse together into one size, namely . Under forcing axioms, they all have size $mathfrak c$ and $mathfrak c$ is at least ; if the forcing axiom is strong enough (like PFA) then so all cardinal characteristics again collapse to the same size, this time they all equal $latex \aleph_2$,
Now the point is that the approximately 30 different cardinal characteristics have distinct intuitions behind them and there is recent work (Bartoszynski, Brendle, A.Fischer, Goldstern, Kellner, Shelah, …) using refinements of Shelah’s methods that shows that it is indeed possible to find models of ZFC where many of these characteristics (well, at least 5 of them) take different values. Of course to separate even 3 of these characteristics in the same model one needs that the continuum is at least (to get at least 3 different uncountable cardinals up to the size of the continuum), so this contradicts forcing axioms.
So to respect the rich theory of cardinal characteristics of the continuum we’ll need a big continuum (as I think that Bob suggested?), much bigger than the answers given by CH or forcing axioms, and more in line with the answer suggested by the HP.
(Type 2) We don’t know yet what the best axioms for the foundation of math outside of set theory will say about CH. No prediction here, and the answer will take a lot of research that has not yet even begun (I hope my grant proposal gets approved!).
(Type 3) Not-CH is derivable from the maximal iterative conception. This hasn’t been shown yet (the set theory is very hard) but has a good chance of being shown in my lifetime (I am 61).
So it doesn’t look good for resolving the continuum problem (long face). But I see no argument that CH is in principle undecidable, it is just a case of bad luck.
Second PS (where Sy shoots himself in the foot):
I am worried now about AC. I don’t see an argument that it follows from the maximal iterative conception, unlike the other axioms of ZFC. Does everyone agree with me? I hope not, because I would love for it to follow from the MIC, i.e. to be “intrinsic” in the sense that I am using the word. For otherwise I see the need to deal with the folliowing “argument” that not-AC follows from Maximality:
A result of Morris (I don’t know him) from the 1970s is the following: There is a first-order sentence S (a strong denial of AC) such that
- Any (countable, transitive) model of ZFC has an outer model (thickening) in which S holds.
- No (countable, transitive) model of ZF + S has an outer model in which AC holds.
In other words if you consider the larger Hyperuniverse of countable transitive models M of ZF which may or may not satisfy AC, the ones which are powerset maximal (maximal in width) satisfy not-AC! The reason is that if M satisfies ZFC then M must have an outer model satisfying S by 1 above but cannot have an inner model satisfying S by 2 above.
This worries me, because one may ask: Why do we assume AC at the start of an investigation of “intrinsic” sources of truth? Maybe the HP should be reformulated without any assumption about AC?
I see 2 ways out of this jam: It may be that a more thorough examination of the choiceless-HP will kill off the Morris example, i.e. models of a more sophisticated form of the HP will satisfy AC. For example I don’t know yet if Morris’ argument can tolerate cardinal-preservation when GCH fails in the ground model (probably not). So just like with the original form of the IMH in the standard HP programme, the situation changes when other maximality considerations are taken into account. The secod escape would be either to argue that AC follows from the MIC (my appeal above) or to simply declare that the HP is a study of what is derivable from the MIC + AC, in other words from what is “intrinsic” to a slightly enhanced set-concept, that in which AC is assumed. That would be disappointing but not really damaging to the programme. Of course it would in any case good to also keep the choiceless-HP in mind, to see what first-order consequences it has.
Now before you start making fun of me, I’d like to point out that AC may not be just a worry for the HP but also with Type 1 considerations: Even if AC is totally convincing for its value in the foundations of mathematics outside set theory, it is quite possible that there are forms of “good set theory” which deny AC. The trick with AD is to regard the failure of AC as just a feature of inner models, but such a trick may not be available for every form of “good mathematics” that demands the failure of AC (consider the Morris example or Gitik’s result that every uncountable cardinal could be singular if AC fails). But maybe Jose will argue that AC is just as essential to set theory as it is to math outseide of set theory? I’d like to understand that argument.
Best regards to all,