Re: Paper and slides on indefiniteness of CH

Dear Sy,

You proposed “strong unreachability” as intrinsically justified on the basis of the maximal iterative conception of set, writing: “It is compelling that unreachability (and strong unreachability) with reflection is faithful to maximality but these criteria have not yet been systematically investigated”. Now, we know a bit more, in light of Hugh’s result: If you accept strong unreachability then you have to accept either V=HOD or PD.

But you have rejected V = HOD on grounds of maximality, writing (in your 21.8.14 to Hugh):

[It] cannot be “true” because it violates the maximality of the universe of sets. Recall Sol’s comment about “sharpenings” of the set concept that violate what the set concept is supposed to be about. Maximality implies that there are sets (even reals) which are not ordinal-definable.

So what now? Do you accept PD? Do you claim that we now know that PD is intrinsically justified on the basis of the maximal iterative conception of set?

Or do you retract one of the above claims about what is intrinsically justified on the basis of the maximal iterative conception of set? And if “maximality” keeps suggesting principles that conflict and must be either revised or rejected, does that not indicate that we are not here dealing with a robust notion? Or do you see enough convergence and underlying unity to allay this worry? And, if so, can you, in hindsight, explain what went wrong in this case?

Best,
Peter

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