I respond below to that part of your message directed to me.
On Oct 9, 2014, at 1:06 PM, Sy David Friedman wrote:
Dear Pen and Hugh,
Yes, there is a definite appeal to this style of argument. But I see a problem.
Consider a special case of your third “old” example: MC (measurable cardinal) is better than V = L because models with MC have inner models with V = L, but not conversely. The problem here is with the theory T = “V = L[x] for some real x and there is an inner model satisfying MC”. This theory gives all of the mathematical benefits of V = L but also allows you to simulate MC in an inner model. So it looks better than just MC. But conversely, MC gives you all of the benefits of MC (obviously) and simulates V = L in an inner model. So which theory is better? This is not an artificial example, as the theory T is precisely what you get from the (original, unsynthesised form of the) IMH!
I’m worried that the same problem arises with Ultimate L vs. MM. As Hugh points out, Ultimate L gives an inner model of MM.
Just to be clear: V = Ultimate L only gives inner models of MM containing the reals if one has large cardinals. Further we do not yet know if V = Ultimate L is consistent will “all” large cardinals.
Very good. But what do we do with the theory T = “MM and there is an inner model of Ultimate L”? Hugh, let me reveal my ignorance by asking if this theory could be consistent?
Of course MM is consistent with an inner model of V = Ultimate L, in fact MM surely implies there is an inner model of V = Ultimate L.
If not then we are in business! But if so, it looks like we are losing the advantage of Ultimate-L over MM. I do understand that you can’t have MM and an inner model of Ultimate-L containing all reals, but I don’t see the relevance of this as we don’t need the inner model to contain all reals to simulate Ultimate L, do we?
The relevance for requiring the inner models contain the reals is so that the inner models correctly compute the structures of interest. This is not simulation. One has the structures of interest in these inner models and by varying the inner models these structures exhibit different behaviors.
Look, all I am suggesting is that maybe the theory of Forcing Axioms can flourish in the context of say V = Ultimate L just as the theory of AD has flourished in the context of the Axiom of Choice, the latter is not through simulation. The -family of axioms show that there are essentially canonical models for forcing axioms at the level of their consequences for and even . So this speculation may not be so unlikely.
However I think it is premature to make a declaration either way at this stage.