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?
This concerns the (clarified) notion of being strongly unreachable from the outline on HP you circulated yesterday. (M is strongly unreachable if for all proper inner models N of M, for all sufficiently large M-cardinals as computed in N is strictly less than as computed in M).
Suppose V is strongly unreachable (and just relative to -definable classes from parameters to make this explicitly first order). Then there are no measurable cardinals and either
- V = HOD and in fact V = K, (so GCH holds, and much more); or
- global-PD holds.
(K refers to a natural generalization of the usual core model—the union of “lower-parts of structures”—and this could be L of course. This K must be very L-like because of having no measurable cardinals. Global-PD is the assertion that PD holds in all set-generic extensions).
These are not mutually exclusive possibilities. But I actually do not know if (2) is possible. This leads to some rather subtle questions about correctness, for example suppose that M is countable and M is the minimum correct model of ZFC+global-PD. Must M be strongly unreachable? It seems likely that the answer should be yes, but this looks quite difficult (to me anyway).
(“Correct” here means that the set-generic extensions of M are projectively correct)
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 cannot exist. But V = L is in my view clearly wrong (irregardless of whether 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.
The axiom V = Ultimate L implies V = HOD
So V = Ultimate L 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.
PS: This is of course not to say that V = Ultimate L is mathematically uninteresting or cannot play a role in the formulation of some future “true” axiom of set theory.
PPS: Since the Reinhardt fiasco I think it would be best to refer to statements that are not known to be consistent (relative to LCs) as “hypotheses” and not as “axioms”, especially in the context of a discussion over truth in set theory.
Quick answer to the question directed to me below. —Hugh
On Aug 20, 2014, at 2:49 AM, Sy David Friedman wrote:
As I understand it (please correct me) your [Solomon Feferman's] valid point is that by for example taking “set” to mean “constructible set” we have violated the intrinsic feature of “maximality”, a feature which the concept of set is meant to exhibit. (Aside: I can then well imagine that on similar grounds you would hesitate to accept an axiom called V = Ultimate-L! But perhaps Hugh will clarify that this need not even imply V = HOD, so it should not be regarded as an anti-maximality statement.)
The axiom V = Ultimate L implies V = HOD and that V is not a set-generic extension of any inner model.