Re: Paper and slides on indefiniteness of CH

Dear Sy,

Pen and Peter, can you please help here? Pen hit me very hard for developing what could be regarded as “Sy’s personal theory of truth” and it seems to me that we now have “Hugh’s personal theory of truth”, i.e., when Hugh develops a powerful piece of set theory he wants to declare it as “true” and wants us all to believe that. This goes far beyond Thin Realism, it goes to what Hugh calls a “conception of V” which far exceeds what you can read off from set-theoretic practice in its many different forms. Another example of this is Hugh’s claim that large cardinal existence is justified by large cardinal consistency; what notion of “truth” is this, if not “Hugh’s personal theory of truth”?

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

I’m pretty sure Hugh would disagree with what I’m about to say, which naturally gives me pause. With that understood, I confess that from where I sit as a relatively untutored observer, it looks as if the evidence Hugh is offering is overwhelming of your Type 1 (involving the mathematical virtues of the attendant set theory). My guess is he’d also consider type 2 evidence (involving the relations of set theory to the rest of mathematics) if there were some ready to hand. He has a ‘picture’ of what the set theoretic universe is like, a picture that guides his thinking, but he doesn’t expect the rest of us to share that picture and doesn’t appeal to it as a way of supporting his claims. If the mathematics goes this way rather than that, he’s quite ready to jettison a given picture and look for another. In fact, at times it seems he has several such pictures in play, interrelated by a complex system of implications (if this conjecture goes this way, the universe like this; if it goes that way, it looks like that…) But all this picturing is only heuristic, only an aide to thought — the evidence he cites is mathematical. And, yes, this is more or less how one would expect a good Thin Realist to behave (one more time: the Thin Realist also recognizes Type 2 evidence). (My apologies, Hugh. You must be thinking, with friends like these…)

The HP works quite differently. There the picture leads the way — the only legitimate evidence is Type 3. As we’ve determined over the months, in this case the picture involved has to be shared, so that it won’t degenerate into ‘Sy’s truth’. So far, to be honest, I’m still not clear on the HP picture, either in its height potentialist/width actualist form or its full multiverse form. Maybe Peter is doing better than I am on that.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I see no virtue in “back to square one” conjectures. In the HP the whole point is to put out maximality criteria and test them; it is foolish to make conjectures without doing the mathematics.

Of course you do not see any virtue in “back to square one” conjectures. Fine, we have different views (yet again),

Why should your programme be required to make “make or break” conjectures, and what is so attractive about that?

I find it quite interesting if philosophical considerations lead to specific “make or break” conjectures. Especially if there is no obvious purely mathematical basis on which to make the conjecture. The HOD Conjecture is a good example. There is no purely mathematical reason (that I know of) to make that conjecture (that the HOD Hypothesis is provable from say ZFC + extendible).  It is a prediction from the Ultimate L scenario (just as is the (provability) of the \Omega Conjecture).

Of course there is another reason for identifying such conjectures. They provide test questions for future progress. If one can refute from large cardinals that the \Omega Conjecture holds then one refutes the Ultimate L Conjecture and moreover shows that there is a failure of inner model theory based on sequences of extenders.

One more question at this point: Suppose that Jack had succeeded in proving in ZFC that 0^\# does not exist. Would you infer from this that V = L is true? On what grounds?

Not necessarily. But I certainly would no longer declare as evident that V is not L. The question of V versus L would for me,  reopen.

Your V = Ultimate-L programme (apologies if I misunderstand it) sounds very much like saying that Ultimate L is provably close to V so we might as well just take V = Ultimate-L to be true. If I haven’t misunderstood then I find this very dubious indeed. As Pen would say, axioms which restrict set-existence are never a good idea.

If the Ultimate L Conjecture is true (provable in ZFC + extendible) then V = Ultimate L becomes a serious possibility which (to me anyway) cannot just be dismissed as is now the possibility that V = L.

For me, the “validation” of V = Ultimate L will have to come from the insights V = Ultimate L gives for the hierarchy of large cardinals beyond supercompact. (These are the “other tests which will have to be passed”).

If that does not happen or if the genuine insights come from outer models of V = Ultimate L, or even from something entirely unrelated to Ultimate L, then for me the case for V = Ultimate L will weaken, possibly significantly.

On the other hand, if in the setting of V = Ultimate L, a whole new hierarchy of large cardinals is revealed, otherwise invisible, then things get interesting. Here it might be the Axiom \textsf{I}0, in but the context of V = Ultimate L, which could be key.

You will respond that is sheer speculation without foundation or solid evidence. It is sheer speculation. We shall see about the evidence.

Maybe it is time to try once again to simply agree that we disagree and wait for future mathematical developments before continuing this debate.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

In light of your HOD Dichotomy I interpreted the HOD conjecture to say that if there is an extendible delta then HOD correctly computes successors of singulars above \delta correctly. All I meant was that if you drop the extendible then this conclusion need not hold. I am guessing (I really don’t know) that if there is an extendible then this conclusion does hold (and hence the HOD Conjecture is true).

Unless you can derive extendibles from some form of maximality the consequence I would draw from the HOD conjecture would be that maximality violates the existence of extendible cardinals.

Best, Sy

Re: Paper and slides on indefiniteness of CH

Dear Bob,

I guess I have used it both ways. But also I am most interested in (A) but in the form ZFC + extendible proves the formal statement of the HOD Conjecture i.e. that there is a proper class of regular cardinals which are not \omega-strongly measurable in HOD.

I suppose I should have called the formal statement of the HOD Conjecture, the HOD Hypothesis; and then defined the HOD Conjecture as the conjecture that ZFC (or ZFC + extendible) proves the HOD Hypothesis. Probably it is too late to make that change.

Here is a simple version the HOD Dichotomy theorem:

Theorem. Suppose \delta is extendible. Then the following are equivalent.

  1. HOD Hypothesis.
  2. There is a regular cardinal above \delta which is not \omega-strongly measurable in HOD.
  3. There is a regular cardinal above \delta which is not measurable in HOD.
  4. For every singular cardinal \gamma > \delta, \gamma is singular in HOD and \gamma^+ is the \gamma^+ of HOD.
  5. \delta is supercompact in HOD witnessed by the restriction to HOD of supercompactness measures in V.

Regards, Hugh

Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 30, 2014, at 6:28 AM, Sy David Friedman wrote:

Dear Hugh,

Regarding your “HOD Conjecture”: I look at it very differently. My guess is that it is true, but this only means that maximality (V far from HOD) implies that extendible cardinals don’t exist! Probably this can be improved to “supercompacts don’t exist”!

So one could reasonably take the view that the HOD Conjecture is as misguided now as would have been the conjecture that L is close to V given the Jensen Covering Theorem. (Let’s revise history and pretend that Jensen’s Covering Theorem was proved before measurable cardinals etc. had been defined and analyzed).

Unless you can somehow get extendible cardinals into the picture, what you call the HOD conjecture is indeed misguided, as Cummings, Golshani and I have shown.

The \text{HOD} Conjecture asserts there is a proper class of regular cardinals which are not \omega-strongly measurable in \text{HOD}. Your results here in no way show this is misguided and moreover, while interesting, these results are completely irrelevant to the \text{HOD} Conjecture. I have already pointed this out to you several times.

Why?

1) It is not known (without appealing to Reinhardt cardinals) if there can exist even 4 regular cardinals which are \omega-strongly measurable in \text{HOD}, even getting 3 requires \textsf{I}0 and the \Omega-Conjecture.

2) It is not known if the successor of a singular strong limit of uncountable cofinality can be \omega-strongly measurable in \text{HOD}.

To refute the \text{HOD} Conjecture one must produce a model in which all sufficiently large regular cardinals are \omega-strongly measurable in \text{HOD}.

Regards,
Hugh

PS: \kappa is \omega-strongly measurable in \text{HOD} if there exists \lambda < \kappa such that (2^{\lambda})^{\text{HOD}} < \kappa and such that
there is no partition of S = {\alpha < \kappa: \text{cf}(\alpha) = \omega} into \lambda many sets \langle S_{\alpha}: \alpha < \lambda\rangle \in \text{HOD} such
each set S_{\alpha} is stationary in V.

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Wed, 29 Oct 2014, W Hugh Woodin wrote:

My question to Sy was implicitly: Why does he not, based on maximality, reject HOD Conjecture since disregarding the evidence from the Inner Model Program, the most natural speculation is that the HOD Conjecture is false.

Two points:

1. The HP is concerned with maximality but does not aim to make “conjectures”; its aim is to throw out maximality criteria and analyse them, converging towards an optimal criterion, that is all. A natural maximality criterion is that V is “far from \text{HOD}” and indeed my work with Cummings and Golshani shows that this is consistent. In fact, I would guess that an even stronger statement that V is “very far from \text{HOD}” is consistent, namely that all regular cardinals are inaccessible in \text{HOD} and more. What you call “the \text{HOD} Conjecture” (why does it get this special name? There are many other conjectures one could make about \text{HOD}!) presumes an extendible cardinal; what is that doing there? I have no idea how to get extendible cardinals from maximality.

2. Sometimes I make conjectures, for example the rigidity of the Stable Core. But this has nothing to do with the HP as I don’t see what non-rigidity of inner models has to do with maximality. I don’t have reason to believe in the rigidity of \text{HOD} (with no predicate) and I don’t see what such a statement has to do with maximality.

Re: Paper and slides on indefiniteness of CH

That doesn’t answer the question: If you assert that we will know the truth value of CH, how do you account for the fact that we have many different forms of set-theoretic practice? Do you really think that one form (Ultimate L perhaps) will “have virtues that will swamp all the others”, as Pen suggested?

Look, as I have stated repeatedly I see the subject of the model theory of ctm’s as separate from the study of V (but this is not to say that theorems in the mathematical study of ctm’s cannot have significant consequences for the study of V). I see nothing wrong with this view or the view that the practice you cite is really in the subject of ctm’s, however it is presented.

??? My question has nothing to do with ctm’s! It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway). I was referring to the many different forms of set-theoretic practice which disagree with each other on basic questions like CH. How do you assign a truth value to CH in light of this fact?

For your second question, If the tests are passed, then yes I do think that V = Ulitmate-L will “swamp all the others” but only in regard to a conception of V, not with regard to the mathematics of ctm’s. There are a number of conjectures already which I think would argue for this. But we shall see (hopefully sooner rather than later).

Here come the irrelevant ctm’s again. But you do say that V = Ultimate L will “swamp all the others”, so perhaps that is your answer to my question. Now do you really believe that? You suggested that Forcing Axioms can somehow be “part of the picture” even under V = Ultimate L, but that surely doesn’t mean that Forcing Axioms are false and Ultimate L is true.

Pen and Peter, can you please help here? Pen hit me very hard for developing what could be regarded as “Sy’s personal theory of truth” and it seems to me that we now have “Hugh’s personal theory of truth”, i.e., when Hugh develops a powerful piece of set theory he wants to declare it as “true” and wants us all to believe that. This goes far beyond Thin Realism, it goes to what Hugh calls a “conception of V” which far exceeds what you can read off from set-theoretic practice in its many different forms. Another example of this is Hugh’s claim that large cardinal existence is justified by large cardinal consistency; what notion of “truth” is this, if not “Hugh’s personal theory of truth”?

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

Look: There is a rich theory about the projective sets in the context of not-PD (you yourself have proved difficult theorems in this area). There are a number of questions which remain open about the projective sets in the context of not-PD which seem very interesting and extremely difficult. But this does not argue against PD. PD is true.

I want to know what you mean when you say “PD is true”. Is it true because you want it to be true? Is it true because ALL forms of good set theory imply PD? I have already challenged, in my view successfully, the claim that all sufficiently strong natural theories imply it; so what is the basis for saying that PD is true?

If the Ultimate-L Conjecture is false then for me it is “back to square one” and I have no idea about an resolution to CH.

I see no virtue in “back to square one” conjectures. In the HP the whole point is to put out maximality criteria and test them; it is foolish to make conjectures without doing the mathematics. Why should your programme be required to make “make or break” conjectures, and what is so attractive about that? As I understand the way Pen would put it, it all comes down to “good set theory” for your programme, and for that we need only see what comes out of your programme and not subject it to “death-defying” tests.

One more question at this point: Suppose that Jack had succeeded in proving in ZFC that 0^\# does not exist. Would you infer from this that V = L is true? On what grounds? Your V = Ultimate L programme (apologies if I misunderstand it) sounds very much like saying that Ultimate L is provably close to V so we might as well just take V = Ultimate L to be true. If I haven’t misunderstood then I find this very dubious indeed. As Pen would say, axioms which restrict set-existence are never a good idea.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

I realize that I am not sure how you use the word “conjecture”. Here are two sample readings of this word in your “Hod Conjecture”. (I realize this does not exhaust the possibilities.)

A) It is provable in ZFC that if $kappa$ is a huge cardinal, then the HOD conjecture holds in V_\kappa.

B) It is simply true that the HOD conjecture holds. No implication concerning provability is intended.

(I realize position B is incomprehensible (and/or absurd) to the Friedman brothers.) My position that CH is false (and that \mathfrak c is weakly inaccessible) is much like this suggested alternative B.

— Bob

Re: Paper and slides on indefiniteness of CH

Dear all,

Here is some background for those who are interested. My apologies to those who are not, but delete is one key stroke away.

Jensen’s Covering Theorem states that V is either very close to L or very far from L. This opens the door for consideration of 0^\# and current generation of large cardinal axioms.

Details: “close to L” means that L computes the successors of all singular (in V) cardinals correctly. “far from L” means every uncountable cardinal is inaccessible in L.

The HOD Dichotomy Theorem [proved here] is in some sense arguably an abstract generalization of Jensen’s Covering Theorem. This theorem states that if there is an extendible cardinal then V is either very close to \text{HOD} or very far from \text{HOD}.

Details: Suppose \delta is an extendible cardinal. “very close to \text{HOD}” means the successor of every singular cardinal above \delta is correctly computed by \text{HOD}. “very far from \text{HOD}” means that every regular cardinal above \delta is a measurable cardinal in \text{HOD} and so \text{HOD} computes no successor cardinals correctly above \delta.

Aside: The restriction to cardinals above \delta is necessary by forcing considerations and the close versus far dichotomy is much more extreme than just what is indicated above about successor cardinals.

The pressing question then is: Is the \text{HOD} Dichotomy Theorem really a “dichotomy” theorem?

The \text{HOD} Conjecture is the conjecture that it is not; i.e. if there is an extendible cardinal then \text{HOD} is necessarily close to V.

Given set theoretic history, arguably the more plausible conjecture is that \text{HOD} Dichotomy Theorem is a genuine dichotomy theorem and so just as 0^\# initiates a new generation of large cardinal axioms (that imply V is not L) there is yet another generation of large cardinal axioms which corresponds to the failure of the \text{HOD} Conjecture.

But now there is tension with the Inner Model Program which predicts that \text{HOD} Conjecture is true (for completely unexpected reasons).

My question to Sy was implicitly: Why does he not, based on maximality, reject \text{HOD} Conjecture since disregarding the evidence from the Inner Model Program, the most natural speculation is that the \text{HOD} Conjecture is false.

The point here is that the analogous conjecture for L is false (since 0^\# exists).

So one could reasonably take the view that the \text{HOD} Conjecture is as misguided now as would have been the conjecture that L is close to V given the Jensen Covering Theorem. (Let’s revise history and pretend that Jensen’s Covering Theorem was proved before measurable cardinals etc. had been defined and analyzed).

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Sy,

My point is that the non-rigidity of HOD is a natural extrapolation of ZFC large cardinals into a new realm of strength.  I only reject it now because of the Ultimate-L Conjecture and its implication of the HOD Conjecture. It would be interesting to have an independent line which argues for the non-rigidity of HOD. This is the only reason I ask.

Please don’t confuse two things: I conjectured the rigidity of the Stable Core for purely mathematical reasons. I don’t see it as part of the HP. Indeed, I don’t see a clear argument that the nonrigidity of inner models follows from some form of maximality.

It would be nice to see one such reason (other than then non V-constructible one).

You seem to feel strongly that maximality entails some form of V is far from HOD. It would seem a natural corollary of this to conjecture that the HOD Conjecture is false, unless there is a compelling reason otherwise. If the HOD Conjecture is false then the most natural explanation would be the non-rigidity of HOD but of course there could be any number of other reasons.

In brief: HP considerations would seem to predict/suggest the failure of the HOD Conjecture. But you do not take this step. This is mysterious to me.

I am eager to see a well grounded argument for the HOD Conjecture which is independent of the Ultimate-L scenario.

Why am I so eager?  It would “break the symmetry” and for me anyway argue more strongly for the HOD Conjecture.

But I did answer your question by stating how I see things developing, what my conception of V would be, and the tests that need to be passed. You were not happy with the answer. I guess I have nothing else to add at this point since I am focused on a rather specific scenario.

That doesn’t answer the question: If you assert that we will know the truth value of CH, how do you account for the fact that we have many different forms of set-theoretic practice? Do you really think that one form (Ultimate-L perhaps) will “have virtues that will swamp all the others”, as Pen suggested?

Look, as I have stated repeatedly I see the subject of the model theory of ctm’s as separate from the study of V (but this is not to say that theorems in the mathematical study of ctm’s cannot have significant consequences for the study of V). I see nothing wrong with this view or the view that the practice you cite is really in the subject of ctm’s, however it is presented.

For your second question, If the tests are passed, then yes I do think that V = Ulitmate L will “swamp all the others” but only in regard to a conception of V, not with regard to the mathematics of ctm’s. There are a number of conjectures already which I think would argue for this. But we shall see (hopefully sooner rather than later).

Look: There is a rich theory about the projective sets in the context of not-PD (you yourself have proved difficult theorems in this area). There are a number of questions which remain open about the projective sets in the context of not-PD which seem very interesting and extremely difficult. But this does not argue against PD. PD is true.

Sample current open question: Suppose every projective set is Lebesgue measurable and has the property of Baire. Suppose every light-face projective set has a light-face projective uniformization. Does this imply PD? (Drop light-face and the implication is false by theorems of mine and Steel, replace projective by hyper projective and the implication holds even without the light-face restriction,  by a theorem of mine).

If the Ultimate L Conjecture is false then for me it is “back to square one” and I have no idea about an resolution to CH.

Regards,
Hugh