Tag Archives: Forcing axioms under V = Ultimate L

Re: Paper and slides on indefiniteness of CH

Dear Sy,

 Type 2 comes down HARD for Forcing Axioms and V = L, as so far none of the others has done anything important for mathematics outside of set theory.

I was assuming that any theory capable of ‘swamping’ all others would ‘subsume’ the (Type 1 and Type 2) virtues of the others.  It has been argued that a theory with large cardinals can subsume the virtues of V=L by regarding them as virtues of working within L.  I can’t speak to forcing axioms, but I think Hugh said something about this at some point in this long discussion.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 10, 2014, at 2:35 AM, Sy David Friedman wrote:

Hmm. What is the value of having the real Calkin algebra in an inner model if you are missing some of its automorphisms? It seems that this is again just a “simulation” of the real thing, albeit a better “simulation” than in an inner model that doesn’t even have the real Calkin algebra.

Not at all, it analyzes the following. Suppose \prec is a well-ordering of the reals. How easily can one generate an automorphism of the Calkin Algebra from \prec which is not inner etc.
This suggests a potential structure theory of well-orderings of the reals, and arguably CH is the better context for such a theory.

The argument that Pen referred to for the preference of LCs over V = L breaks down if you insist that the inner model contain all of the reals (there is no inner model of V = L containing all of the reals if LCs exist).

It seems that with AC we can enjoy the pleasures of AD by going to inner models that satisfy it but may fail to have all of the reals. No doubt it is better to have all of the reals, but I don’t see why that is required […] for enjoying Ultimate-L in an inner model when MM holds in V.

Best is to to have all the sets.

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,

I agree! But I also think that the theory of V = Ultimate-L could flourish in the context of Forcing Axioms! One need only consider inner models that satisfy it but which fail to contain all of the reals.

Maybe you now see my point about large cardinal existence. We don’t need large cardinals in V, it’s enough to have them in inner models, and whenever we want to enjoy LCs we can just go to those inner models. That’s the picture given by the IMH.

This is the point on which we genuinely disagree. You seem to regard the existence of an inner model of a large cardinal axiom as the ultimate manifestation of its consistency. This is simply not true past the level of one Woodin cardinal. It is the existence of iterable models which do this (or correct models). Inner models past the level of one Woodin are like wellfounded (set) models below one Woodin.  It makes no more sense to draw the line at inner models (which you seem to do) past the level of one Woodin than it would be to draw the line at well-founded models below one Woodin (which you do not).

An inner model with infinitely many Woodin cardinals is useless. An iterable inner model with infinitely many Woodin cardinals gives PD and more.  In fact the existence of a transfinitely iterable inner model with infinitely many Woodin cardinals is equivalent to \text{AD}^{L(\mathbb R)} holds in all set-generic extensions of V.

We simply have to agree to disagree here.

I think the mathematics (or lack thereof) will eventually make our points far more clearly than is possible now when so much mathematics remains to be done. So we have our marching orders.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Thu, 9 Oct 2014, W Hugh Woodin wrote:

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.

Hmm. What is the value of having the real Calkin algebra in an inner model if you are missing some of its automorphisms? It seems that this is again just a “simulation” of the real thing, albeit a better “simulation” than in an inner model that doesn’t even have the real Calkin algebra.

The argument that Pen referred to for the preference of LCs over V = L breaks down if you insist that the inner model contain all of the reals (there is no inner model of V = L containing all of the reals if LCs exist).

It seems that with AC we can enjoy the pleasures of AD by going to inner models that satisfy it but may fail to have all of the reals. No doubt it is better to have all of the reals, but I don’t see why that is required for enjoying AD (or for enjoying Ultimate-L in an inner model when MM holds in V).

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

I agree! But I also think that the theory of V = Ultimate-L could flourish in the context of Forcing Axioms! One need only consider inner models that satisfy it but which fail to contain all of the reals.

Maybe you now see my point about large cardinal existence. We don’t need large cardinals in V, it’s enough to have them in inner models, and whenever we want to enjoy LCs we can just go to those inner models. That’s the picture given by the IMH.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dera Sy,

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 \mathbb P_\text{max}-family of axioms show that there are essentially canonical models for forcing axioms at the level of their consequences for H_{\aleph_2} and even H_{\mathfrak c^+}.  So this speculation may not be so unlikely.

However I think it is premature to make a declaration either way at this stage.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Sy,

You quoted from my message

Assuming V = Ultimate L one can have inner models containing the reals of say MM. But assuming MM one cannot have an inner model containing the reals which satisfies V = Ultimate-L.

and then wrote in your message to Pen:

Perhaps (not to put words in Hugh’s mouth) he is saying that Axiom B is better than Axiom A if models of Axiom B produce inner models of Axiom A but not conversely. Is this a start on how to impose a justifiable preference for one kind of good set theory over another? But maybe I missed the point here, because it seems that one could have “MM together with an inner model of Ultimate-L not containing all of the reals”, in which case that would be an even better synthesis of the 2 axioms! (Here we go again, with maximality and synthesis, this time with first-order axioms, rather than with the set-concept and Hyperuniverse-criteria.)

I was responding to what you wrote in your previous message:

To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate L, shoot down forcing axioms, or argue that one of these is not “good Set Theory”.

Let me try again. We accept the Axiom of Choice. This has not suspended or impeded the study of AD. The reason of course is that the study has become the study of certain sets of reals that exist within V; i.e. the study of certain inner models of V which contain the reals.

You have implied above that if we accept V = Ultimate L then that is conflict with the successes of Forcing Axioms. The point that I was trying to make is: not necessarily. One could view the Forcing Axioms as dealing with a fragment of P(\mathbb R); i.e. those sets of reals which belong to inner models of Forcing Axioms etc. It is important to consider inner models here which contain the reals (versus arbitrary inner models) because many of the structures of interest (Calkin Algebra etc), are correctly computed by any such inner model.

This methodology covers essentially all the applications of Forcing Axioms to $latex H-{\mathfrak c^+}.

One cannot reverse the roles here of V = Ultimate L and Forcing Axioms because if Forcing Axioms hold (i.e. CH fails) then there can be no inner model of V = Ultimate L which contains the reals (there can be no inner model containing the reals in which CH holds).

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 8, 2014, at 6:48 AM, Sy David Friedman wrote:

I am most pessimistic about Type 1 truth (Thin Realism). To get any useful conclusions here one would not only have to talk about “good Set Theory” but about “the Best Set Theory”, or at least show that all forms of “good Set Theory” reach the same conclusion about something like CH. Can we really expect to ever do that? To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate-L, shoot down forcing axioms or argue that one of these is not “good Set Theory”. Pen, how do you propose to do that? Forcing axioms are here to stay as “good Set Theory”, they can’t be “shot down”. And even if Ultimate-L dies, there will very likely be something to replace it. Why should we expect this replacement for Ultimate-L to come to the same conclusion about CH that forcing axioms reach (i.e. that CH is false)?

I do not see this at all, In fact, not surprisingly, I completely disagree.

If V = Ultimate L  (and there are large enough cardinals) then one will have inner models containing the reals in which the forcing axioms hold  (including Martin’s Maximum). Thus the theorems of Martin’s Maximum for say H_{\mathfrak c^+} all apply to the objects in such inner models.

For example, consider Farah’s result that all automorphisms of the Calkin Algebra are inner automorphisms assuming MM.  Any inner model containing the reals correctly computes the Calkin Algebra, so Farah’s result applies equally well to automorphisms which belong to such inner models.

One also has inner models containing the reals of the Pmax-axiom and inner models containing  the reals for all of its variations. These axioms are much powerful at the level of H_{\mathfrak c^+}.

This is completely analogous to the theory of determinacy which flourishes in the Axiom of Choice universe through the study of inner models of AD which contain the reals.

Finally note there there is a fundamental asymmetry here. Assuming V = Ultimate L one can have inner models containing the reals of say MM. But assuming MM one cannot have an inner model containing the reals which satisfies V = Ultimate L.

Regards,
Hugh