OK, let’s go for just one more exchange of comments and then try to bring this to a conclusion by agreeing on a summary of our perspectives. I already started to prepare such a summary but do think that one last exchange of views would be valuable.
You have made an important point for me: a rich structure theory together with Gödelian “success” is insufficient to convince number-theorists that ERH is true, and by analogy these criteria should not suffice to convince set-theorists that PD is true.
Unless there is something fundamentally different about LC which there is.
My point here has nothing to do with large cardinals. I am just saying that the tests analogous to those used to argue in favour of PD (success and structure theory) are inadequate in the number theory context. Doesn’t that cast doubt on the use of those tests to justify PD?
Many (well at least 2) set theorists are convinced that PD is true. The issue is why do you think Con PD is true. You have yet to give any coherent reason for this. You responded:
The only “context” needed for Con PD is the empirical calibration provided by a strict “hierarchy” of consistency strengths. That makes no assumptions about PD.
Such a position is rather dubious to me. The consistency hierarchy is credible evidence for the consistency of LC only in the context of large cardinals as potentially true axioms. Remove that context (as IMH and its variants all do) then why is the hierarchy evidence for anything?
My argument is “proof-theoretic”: the consistency strengths in set theory are organised by the consistency strengths of large cardinal axioms. And we have good evidence for the strictness of this hierarchy. There is nothing semantic here.
Aside: Suppose an Oracle informs us that RH is equivalent to Con PD. Then I would say RH is true (and it seems you would agree here). But suppose that the Oracle informs us that RH is equivalent to Con Reinhardt cardinal. Then I actually would conjecture that RH is false. But by your criteria of evidence you would seem to argue RH is true.
I guess you mean Con(Reinhardt without AC). Why would you conjecture in this setting that RH is false? I thought that you had evidence of statements of consistency strength below a Reinhardt cardinal but above that of large cardinals with AC? With such evidence I would indeed conjecture that RH is true; wouldn’t you?
I am not asking how HP could justify the existence of large cardinals. I am simply asking how HP is ever going to even argue for the consistency of just PD (which you have already declared a “truth”). If HP cannot do this then how is it ever credibly going to make progress on the issue of truth in set theory?
Again, I don’t think we need to justify the consistency of large cardinals, the “empirical proof theory” takes care of that.
Yes, theoretically the whole edifice of large cardinal consistency could collapse, even at a measurable, we simply have to live with that, but I am not really worried. There is just too much evidence for a strict hierarchy of consistency strengths going all the way up to the level of supercompactness, using quasi-lower bounds instead of core model lower bounds. This reminds me of outdated discussions of how to justify the consistency of second-order arithmetic through ordinal analysis. The ordinal analysis is important, but no longer necessary for the justification of consistency.
However one conceives of truth in set theory, one must have answers to:
- Is PD true?
I don’t know.
2. Is PD consistent?
You have examples of how HP could lead to answering the first question. But no examples of how HP could ever answer the second question. Establishing Con LC for levels past PD looks even more problematic.
It is not my intention to try to use the HP to justify the already-justified consistency of large cardinals.
There is strong meta-mathematical evidence that the only way to ultimately answer (2) with “yes” is to answer (1) with “yes”. This takes us back to my basic confusion about the basis for your conviction in Con PD.
Note that the IMH yields inner models with measurables but does not imply Pi-1-1 determinacy. This is a “local” counterexample to your suggestion that to get Con(Definable determinacy) we need to get Definable determinacy.
We have had this exchange several times already. Let’s agree to (strongly) disagree on this point.
The fundamental technology (core-model methods) which is used in establishing the “robustness” of the consistency hierarchy which you cite as evidence, shows that whenever “ZFC + infinitely many Woodin cardinals” is established as a lower bound for some proposition (such as PFA, failure of square at singular strong limits, etc), that proposition implies PD. For these results (PFA, square etc.) there are no other lower bound proofs known. There is a higher level consistency hierarchy (which is completely obscured by your more-is-better approach to the hyper-universe).
You also cite strictness of the hierarchy as an essential component of the evidence, which you must in light of the ERH example, and so the lower bound results are key in your view. Yet as indicated above, for the vast majority (if not all) of these lower bound results, once one is past the level of Con PD, one is actually inferring PD. It seems to me that by your own very criteria, this is a far stronger argument for PD then HP is ever going to produce for the negation of PD.
Again: It is not clear that the HP will give not-PD! It is a question of finding appropriate criteria that will yield PD, perhaps criteria that will yield enough large cardinals.
As far as the strictness of the consistency hierarchy we can use quasi-lower bounds, we don’t need the lower bounds coming from core model theory.
And as I have been trying to say, building core model theory into a programme for the investigation of set-theoretic truth like HP is an inappropriate incursion of set-theoretic practice into an intrinsically-based context.
All those comments aside, we have an essential disagreement at the very outset. I insist that any solution to CH must be in the context of strong rank maximality (and assuming the provability of the Conjecture this becomes a perfectly precise criterion). You insist that this is too limited in scope and that we should search outside this “box”.
No, we may be able to stay “within the box” as you put it:
I said that SIMH(large cardinals + -generation) might be what we are looking for; the problems are to intrinsically justify large cardinals and to prove the consistency of this criterion. Would you be happy with that solution?