# Re: Paper and slides on indefiniteness of CH

From Mr. Energy:

Why? I think I made my position clear enough: I stated a consistent Maximality Criterion and based on my proof (with co-authors) of its consistency I have the impression that this Criterion contradicts supercompacts (not just extendibles). So that leads to a tentative rejection of supercompacts until the situation changes through further understanding of further Maximality Criteria. It’s analagous to what happened with the IMH: It led to a tentative rejection of inaccessibles, but then when Vertical Maximality was taken into account, it became obvious that the IMH# was a better criterion than the IMH and the IMH# is compatible with inaccessibles and more.

I also think that the Maximality Criterion I stated could be made much stronger, which I think is only possible if one denies the existence of supercompacts. (Just a conjecture, no theorem yet.)

Looks like I have been nominated long ago (smile) to try to turn this controversy into something readily digestible – and interesting – for everybody.

A main motivator for me in this arguably unproductive traffic is to underscore the great value of real time interaction. Bad ideas can be outed in real time! Bad ideas can be reformulated as reasonable ideas in real time!! Good new ideas can emerged in real time!!! What more can you want? Back to this situation.

This thread is now showing even more clearly the pitfalls of using unanalyzed flowery language like “Maximality Criterion” to try to draw striking conclusions (technical advances not yet achieved, but perhaps expected). Nobody would bother to complain if the striking conclusions were compatible with existing well accepted orthodoxy.

So what is really being said here is something like this:

“My (Mr. Energy) fundamental thinking about the set theoretic universe is so wise that under anticipated technical advances, it is sufficient to overthrow long established and generally accepted orthodoxy”.

What is so unusual here is that this unwarranted arrogance is so prominently displayed in a highly public environment with several of the most well known scholars in relevant areas actively engaged!

What was life like before email? We see highly problematic ideas being unravelled in real time.

What would a rational person be putting forward? Instead of the arrogant

*Maximality Criteria tells us that HOD is much smaller than V and this (is probably going to be shown in the realistic future to) refutes certain large cardinal hypotheses*

the entirely reasonable

**Certain large cardinal hypotheses (are probably going to be shown in the realistic future to) imply that HOD has similarities to V. Such similarities cannot be proved or refuted in ZFC. This refutes certain kinds of formulations of “Maximality in higher set theory, under relevant large cardinal hypotheses.**

and then remark something like this:

***The notion “intrinsic maximality of the set theoretic universe” is in great need of clear elucidation. Many formulations lead to inconsistencies or refutations of certain large cardinal hypotheses. We hope to find a philosophically coherent analysis of it from first principles that may serve as a guide to the appropriateness of many set theoretic hypotheses. In particular, the use of HOD in formulations can be criticized, and raises a number of unresolved issues.***

Again, what was life like before email? We might have been seeing students and postdocs running around Europe opening claiming to refute various large cardinal hypotheses!

Harvey

# Re: Paper and slides on indefiniteness of CH

On Oct 31, 2014, at 12:20 PM, Sy David Friedman wrote:

Dear Hugh,

On Fri, 31 Oct 2014, W Hugh Woodin wrote:

Ok we keep going.

Why? I think I made my position clear enough: I stated a consistent Maximality Criterion and based on my proof (with co-authors) of its consistency I have the impression that this Criterion contradicts supercompacts (not just extendibles).

But why do you have that impression? That is what I am interested in. You have given no reason and at the same time there seem to be many reasons for you not to have that impression. Why not reveal what you know?

I also think that the Maximality Criterion I stated could be made much stronger, which I think is only possible if one denies the existence of supercompacts. (Just a conjecture, no theorem yet.)

Let the Strong HOD Hypothesis be: No successor of a singular strong limit of uncountable cofinality is $\omega$-strongly measurable in HOD

(Recall: this is not known to consistently fail with appealing to something like Reinhardt Cardinals. The restriction to uncountable cofinality is necessary because of the Axiom I0: Con (ZFC + I0) gives the consistency with ZFC that there is a singular strong limit cardinal  whose successor is $\omega$-strongly measurable in HOD.)

If the Strong HOD Hypothesis holds in V and if the Maximality Criterion holds in V, then there are no supercompact cardinals, in fact there are no cardinals $\kappa$ which are $\omega_1+\omega$-extendible; i.e. no $\kappa$ for which there is $j:V_{\kappa+\omega_1+\omega} \to V_{j(\kappa +\omega_1+\omega)}$.

If ZFC proves the HOD Hypothesis, it surely proves the Strong HOD Hypothesis.

First you erroneously thought that I wanted to reject PD and now you think I want to reject large cardinals! Hugh, please give me a chance here and don’t jump to quick conclusions; it will take time to understand Maximality well enough to see what large cardinal axioms it implies or tolerates.

I see you making speculations for which I do not yet see another explanation of. But fine, take all the time you want. I have no problem with agreeing that HP is in a (mathematically) embryonic phase and we have to wait before being able to have a substantive (mathematical) discussion about it.

There is something robust going on, please give the HP time to do its work. I simply want to take an unbiased look at Maximality Criteria, that’s all. Indeed I would be quite happy to see a convincing Maximality Criterion that implies the existence of supercompacts (or better, extendibles), but I don’t know of one.

But if the synthesis of maximality, in the sense of failure of the HOD Hypothesis, together with large cardinals, in the sense of there is an extendible cardinal, yields a greatly enhanced version of maximality, why is this not enough?

That is what I am trying to understand.

Regards.
Hugh

# Re: Paper and slides on indefiniteness of CH

Ok we keep going.

On Oct 31, 2014, at 3:30 AM, Sy David Friedman wrote:

Dear Pen,

With co-authors I established the consistency of the following Maximality Criterion. For each infinite cardinal $\alpha$, $\alpha^+$ of HOD is less than $\alpha^+$.

Both Hugh and I feel that this Criterion violates the existence of certain large cardinals. If that is confirmed, then I will (tentatively) conclude that Maximality contradicts the existence of large cardinals.

It seems that you believe the HOD Conjecture (i.e. that the HOD Hypothesis is a theorem of ZFC). But then HOD is close to V in a rather strong sense (just not in the sense of computing many successor cardinals correctly). This arguably undermines the whole foundation for your maximality principle (Maximality Criterion stated above). I guess you could respond that you only think that the HOD Hypothesis is a theorem of ZFC + extendible and not necessarily from just ZFC.

If the HOD Hypothesis is false in V and there is an extendible cardinal, then in some sense, V is as far as possible (modulo trivialities) from HOD. So in this situation the maximality principle you propose holds in the strongest possible form. This would actually seem to confirm extendible cardinals for you. Their presence transforms the failure of the HOD Hypothesis into an extreme failure of the closeness of V to HOD, optimizing your maximality principle. So in the synthesis of maximality, in the sense of the failure of the HOD Hypothesis, with large cardinals, in the sense of the existence of extendible cardinals, one gets the optimal version of your maximality principle.

The only obstruction is the HOD Conjecture. The only evidence I have for the HOD Conjecture is the Ultimate L scenario. What evidence do you have that compels you not to make what would seem to be strongly motivated conjecture for you (that ZFC + extendible does not prove the HOD Hypothesis)?

I find your position rather mysterious. It is starting to look like your main motivation is simply to deny large cardinals.

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

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