Tag Archives: Zero sharp

Re: Paper and slides on indefiniteness of CH

Dear Bob,

What is the precise definition of “maximality”. Is it evident from that definition, that maximality implies there are reals not in HOD? If not, can you give a cite as to where this is proved?

I do not know of a precise definition of “maximality”. Rather I regard “maximality” as an “intrinsic feature” of the universe of sets via what Pen has referred to as “the usual kind of conceptualism: there is a shared concept of the set-theoretic universe (something like the iterative conception); it’s standardly characterized as including ‘maximality’, both in ‘width’ (Sol’s ‘arbitrary subset’) and in the ‘height’ (at least small LCs).” [Pen: I dropped the bit about "reflection" as I wasn't sure what you meant; but I don't think that its omission will affect this discussion.]

“Maximality” is indeed formulated mathematically in a number of different ways in the HP, but I don’t know if there is an ultimate mathematical formulation which fully captures it and therefore cannot claim that there will be a precise definition of this intrinsic feature.

Nevertheless I do regard the existence of reals not in HOD to be derivable from “maximality” for the following reason, which I expect to be shared by others who share the maximal iterative conception: Part of this conception is that the powerset of omega consists of arbitrary subsets of omega. This is violated by V = L, which insists that the only subsets of omega are those which are predicatively definable relative to the ordinals, and also by V = HOD, which insists that the only subsets of omega are those which are definable relative to ordinals.

Returning to V = Ultimate L: I would expect that anyone claiming this to be true would also claim V = L to be true had Jack succeeded in showing that 0^\# cannot exist. But V = L is in my view clearly wrong (irregardless of whether 0^\# can exist), as well-expressed by Gödel:

“From an axiom in some sense opposite to [V=L], the negation of Cantor’s conjecture could perhaps be derived. I am thinking of an axiom which … would state some maximum property of the system of all sets, whereas [V=L] states a minimum property. Note that only a maximum property would seem to harmonize with the concept of set …”

This argument seems to refute V = Ultimate L.