Re: Paper and slides on indefiniteness of CH

Dear Peter and Sy,

I would like to add a short comment about the move to \textsf{IMH}^\#. This concerns to what extent it can be formulated without consulting the hyper-universe is an essential way (which is the case for \textsf{IMH} since \textsf{IMH} can be so formulated). This issue has been raised several times in this thread.

Here is the relevant theorem which I think sharpens the issues.

Theorem. Suppose \textsf{PD} holds, Let X be the set of all ctm M such that M satisfies \textsf{IMH}^\#. Then X is not \Sigma_2-definable over the hyperuniverse. (lightface).

Aside: X is always \Pi_2 definable modulo being #-generated and being #-generated is \Sigma_2-definable. So X is always \Sigma_2\wedge \Pi_2-definable. If one restricts to M of the form L_{\alpha}[t] for some real t, then X is \Pi_2-definable but still not \Sigma_2-definable.

So it would seem that internalization \textsf{IMH}^\# to M via some kind of vertical extension etc., might be problematic or might lead to a refined version of \textsf{IMH}^\# which like IMH has strong anti-large cardinal consequences.

I am not sure what if anything to make of this, but I thought I should point it out.

Regards,
Hugh

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

Dear Sy,

I owe you a response to your other letters (things have been busy) but your letter below presents an opportunity to make some points now.

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

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 \textsf{IMH}^\# is compatible with inaccessibles and more.

I don’t buy this. Let’s go back to IMH. It violates inaccessibles (in a dramatic fashion). One way to repair it would have been to simply restrict to models that have inaccessibles. That would have been pretty ad hoc. It is not what you did. What you did is even more ad hoc. You restricted to models that are #-generated. So let’s look at that.

We take the presentation of #’s in terms of \omega_1-iterable countable models of the form (M,U). We iterate the measure out to the height of the universe. Then we throw away the # (“kicking away the ladder once we have climbed it”) and imagine we are locked in the universe it generated. We restrict IMH to such universes. This gives \textsf{IMH}^\#.

It is hardly surprising that the universes contain everything below the # (e.g. below 0^\# in the case of a countable transitive model of V=L) used to generate it and, given the trivial consistency proof of \textsf{IMH}^\# it is hardly surprising that it is compatible with all large cardinal axioms (even choicless large cardinal axioms). My point is that the maneuver is even more ad hoc than the maneuver of simply restricting to models with inaccessibles. [I realized that you try to give an "internal" account of all of this, motivating what one gets from the # without grabbing on to it. We could get into it. I will say now: I don't buy it.]

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.)

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. 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.

We do have “maximality” arguments that give supercompacts and extendibles, namely, the arguments put forth by Magidor and Bagaria. To be clear: I don’t think that such arguments provide us with much in the way of justification. On that we agree. But in my case the reason is that is that I don’t think that any arguments based on the vague notion of “maximality” provide us with much in the way of justification. With such a vague notion “anything goes”. The point here, however, is that you would have to argue that the “maximality” arguments you give concerning HOD (or whatever) and which may violate large cardinal axioms are more compelling than these other “maximality” arguments for large cardinals. I am dubious of the whole enterprise — either for or against — of basing a case on “maximality”. It is a pitting of one set of vague intuitions against another. The real case, in my view, comes from another direction entirely.

An entirely different issue is why supercompacts are necessary for “good set theory”. I think you addressed that in the second of your recent e-mails, but I haven’t had time to study that yet.

The notion of “good set theory” is too vague to do much work here. Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise. The key notion is evidence, evidence of a form that people can agree on. That is the virtue of actually making a prediction for which there is agreement (not necessarily universal — there are few things beyond the law of identity that everyone agrees on — but which is widespread) that if it is proved it will strengthen the case and if it is refuted it will weaken the case.

Best,
Peter

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

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). 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 \textsf{IMH}^\# was a better criterion than the IMH and the \textsf{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.)

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. 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.

An entirely different issue is why supercompacts are necessary for “good set theory”. I think you addressed that in the second of your recent e-mails, but I haven’t had time to study that yet.

To repeat: I am not out to kill any particular axiom of set theory! I just want to take an unbiased look at what comes out of Maximality Criteria. It is far too early to conclude from the HP that extendibles don’t exist.

Thanks,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I guess I should respond to your question as well.

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

My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why?

Let me clarify my position, or at least that part of it which concerns my (frankly extreme) skepticism about your anti-large cardinal principles.

(I am assuming LC axioms persists under small forcing and that is all in the discussion below)

Suppose there is a proper class of Woodin cardinals. Suppose M is a ctm and M has an iteration strategy \mathcal I at its least Woodin cardinal such that \mathcal I is in L(A,\mathbb R) for some univ. Baire set A.

Suppose some LC axiom holds in M above the least Woodin cardinal of M.Then in V, every V_{\alpha} has a vertical extension in which the LC axiom holds above \alpha.

The existence of such an M for the LC axiom is a natural form of consistency of the LC axiom (closely related to the consistency in \Omega-logic).

Thus for any LC axiom (such as extendible etc.), it is compelling (modulo consistency) that every V_{\alpha} has a vertical extension in which LC axiom holds above \alpha.

But then any claim that the LC axiom does not hold in V, is in general an extraordinary claim in need of extraordinary evidence.

The maximality principles you have proposed do not (for me anyway) meet this standard.

Just to be clear. I am not saying that any LC axiom which is consistent in the sense described above, must be true. I do not believe this (there are adhoc LC axioms for which it is false).

I am just saying that the declaration, the LC axiom does not hold in V, in general requires extraordinary evidence, particularly in the case of LC axioms such as the LC axiom: there is an extendible cardinal.

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

Mr. Energy writes (two excerpts):

With co-authors I established the consistency of the following

Maximality Criterion. For each infinite cardinal \alpha, \alpha^+ of \text{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. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.

My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why? I have yet to see an argument that large cardinal existence is needed for “good set theory”, so it does not follow from Type 1 evidence. That is why I think that large cardinal existence is part of Hugh’s personal theory of truth.

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.

There is some ready to hand: At present, Type 2 evidence points towards Forcing Axioms, and these contradict CH and therefore contradict Ultimate L

I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.

This illustrates the pitfalls involved in trying to use an idiosyncratic propogandistic slogan like “HP” to refer to an unanalyzed philosophical conception with language like “intrinsic maximality of the set theoretic universe”. Just look at how treacherous this whole area of “philosophically motivated higher set theory” can be.

E.g., MA (Martin’s axiom) already under appropriate formulations look like some sort of “intrinsic maximality”, at least as clear as many things purported on this thread to exhibit some sort of “intrinsic maximality”, and already implies that CH is false. So have we now completely solved the CH negatively? If so, why? If not, why not? See what happens with an unanalyzed notion of “intrinsic maximality of the set theoretic universe”. Also MM (Martin’s maximum) is even stronger, and implies that 2^\omega = \omega_2. Also looks like “intrinsic maximality of the set theoretic universe”, at least before any convincing analysis of it, and so do we now know that 2^\omega = \omega_2 follows from the “intrinsic maximality of the set theoretic universe”?

I will now take an obvious step toward turning at least some of this very unsatisfying stuff into something completely unproblematic – without the idiosyncratic propogandistic slogans – AND something (hopefully) not needing countable transitive models for straightforward formulations.

Ready? Here is the narraitive.

1. We want to explore the idea that

*L is a tiny part of V* *L is very different from V*

We also want to explore the idea that

**HOD is a tiny part of V. **HOD is very different from V**

Here HOD = hereditarily ordinal definable sets. Myhill/Scott proved that HOD satisfies ZFC, following semiformal remarks of Gödel.

2. There are some interesting arguments that one can give for L being a tiny part of V. These arguments themselves can be subjected to various kinds of scrutiny, and that is an interesting topic in and of its own. But we shall, for the time being, take it for granted that we are starting off with “L is a tiny part of V”.

3. On the other hand, the arguments that HOD is a tiny part of V are, at least at the moment, fewer and much weaker. This reflects some important technical differences between L and HOD. E.g., L is very stable in the sense that L within L is L. However, HOD within HOD may not be HOD (that’s independent of ZFC).

4. Another related big difference between L and HOD is the following. You can prove that any formal extension of the set theoretic universe compatible with the set theoretic universe in a nice sense, must violate V = L if the original set theoretic universe violates V = L. This is the kind of thing that adds to an arsenal of possible arguments that L is only a part or tiny part of V. However, the set theoretic universe demonstrably has a formal extension satisfying V = HOD even if the set theoretic universe does not satisfy V = HOD. This makes the idea that HOD is a tiny part of V a much more problematic “consequence” of “intrinsic maximality of the set theoretic universe”.

5. Yet another difference. Vopenka proved in ZFC that every set can be obtained by set forcing over HOD. That every set can be obtained by set forcing over L is known to be independent of ZFC, and in fact violates medium large cardinals (such as measurable cardinals and even 0^\#). The same is true for set forcing replaced by class forcing.

6. Incidentally, I think there is an open question that goes something like this. Let M be the minimum ctm of ZFC. There exists a ctm extension of M with the same ordinals that is not obtainable by class forcing over M – I think even under a very wide notion of class forcing. Still open?

7. Another way of talking about the problematic nature of V not equal HOD as following from “intrinsic maximality” is that, well, maybe if there were more sets, we would be able to make more powerful definitions, making certain certain sets in HOD that weren’t “before”, and then close this off, making V = HOD. Thus this is an attempt to actually turn V = HOD itself into some sort of “intrinsic maximality”!!

8. So the proper move, until there is more creative analysis of “intrinsic maximality of the set theoretic universe” is to simply say, flat out:

*we are going to explore the idea that HOD is a tiny part of V* *we are going to explore the idea that HOD is very different from V*

and avoid any idiosnyncratic propogandistic slogans like “HP”.

9. So now let’s fast forward to the excerpt from Mr. Energy:

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. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.

Here is a reasonable restatement without the idiosyncratic propoganda – propoganda that papers over all of the issues about HOD raised above.

NEW STATEMENT. With co-authors I (Mr. Energy) established the consistency of the following relative to the consistency of ???

(HOD very different from V). Every infinite set in HOD is the domain of a bijection onto another set in HOD without there being a bijection in HOD.

Furthermore, Hugh and I (Mr. Energy) feel that the above statement refutes the existence of certain kinds of large cardinal hypotheses. If this is confirmed, then it follows that “HOD is very different from V” is incompatible with certain kinds of large cardinal hypotheses.

10. Who can complain about that? Perhaps somebody on the list can clarify just which large cardinal hypotheses might be incompatible with the above statement?

11. Let’s now step back and reflect on this a bit in general terms to make more of it. What can be say about “HOD very different from V” in general terms?

HOD is an elementary substructure of V

is of course very strong. This is equivalent to saying that V = HOD.

But the above statement is an extremely strong refutation of elementary substructurehood.

THEOREM (?). The most severe/simplest possible violation of L being an elementary substructure of V is that “every infinite set in L is the domain of a bijection onto another set in L without there being a bijection in L”.

THEOREM (?). The most severe/simplest possible violation of HOD being an elementary substructure of V is that “every infinite set in HOD is the domain of a bijection onto another set in HOD without there being a bijection in HOD”.

THEOREM (???). The most severe/simplest possible violation of V not equaled to L is that “every infinite set in L is the domain of a bijection onto another set in L without there being a bijection in L”.

THEOREM (???). The most severe/simplest possible violation of V not equaled to HOD is that “every infinite set in HOD is the domain of a bijection onto another set in HOD without there being a bijection in HOD”.

Since this morning I am doing some real time foundations (of higher set theory), I should be allowed to state Theorems without knowing how to state them.

I also reserve the right to stop here.

I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.

Of course, you have chosen to respond to much but not all of what everybody has written here, except me, invoking the “brother privilege”. Actually, I wonder if the “brother privilege” – that you do not have to respond to your brother in an open intellectual forum – is a consequence of the “intrinsic maximality of the set theoretic universe”?

If you are looking for “something genuinely new to say” then you can start with the dozens of emails I have put on this thread, Actually, you have covered very little by serious foundational standards.

On a mathematical note, you can start by talking about #-generation, what it means in generally understandable terms, why it is natural and/or important, and so forth. Why it is an appropriate vehicle for “fixing” IMH (if it is). It is absurd to think that a two line description weeks (or is it months) ago is even remotely appropriate for a list of about 75 readers. Also, continually referring to type 1, type 2, type 3 set theoretic themes without using real and short names is a totally unnecessary abuse of the readers of this list. People are generally not going to be keeping that in their heads – even if they have not been throwing your messages (and mine) into the trash. Are the numbers 1,2,3 canonically associated with those themes? Furthermore, your brief discussion of them was entirely superficial. There are crucial issues involved in just what the interaction of higher set theory is with mathematics that have not been discussed hardly at all here either by you or by others.

BOTTOM LINE ADVICE.

Change HP to CTMP = countable transitive model program. Cast headlines for statements in terms like “HOD is very different from V” or “HOD is a tiny part of V” or things like that. Avoid “intrinsic maximality of the set theoretic universe” unless you have something new to say that is philosophically compelling.

Harvey

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Thu, 30 Oct 2014, Penelope Maddy wrote:

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).

Let me give you a counterexample.

With co-authors I established the consistency of the following

Maximality Criterion. For each infinite cardinal \alpha, \alpha^+ of \text{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. Hugh will conclude that there is something wrong with the above Maximality Criterion and it therefore should be rejected.

My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why? I have yet to see an argument that large cardinal existence is needed for “good set theory”, so it does not follow from Type 1 evidence. That is why I think that large cardinal existence is part of Hugh’s personal theory of truth.

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.

There is some ready to hand: At present, Type 2 evidence points towards Forcing Axioms, and these contradict CH and therefore contradict Ultimate L.

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 … )

That’s a lot to put in Hugh’s mouth. Probably we should invite Hugh to confirm what you say above.

The HP works quite differently. There the picture leads the way —

As with your description above, the “picture” as you call it keeps changing, even with the HP. Recall that the programme began solely with the IMH. At that time the “picture” of V was very short and fat: No inaccessibles but lots of inner models for measurable cardinals. Then came #-generation and the \textsf{IMH}^\#; a taller, handsomer universe, still with a substantial waistline. As we learn more about maximality, we refine this “picture”.

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.

I have offered to work with the height potentialist/width actualist form, and even drop the reduction to ctm’s, to make people happy (this doesn’t affect the mathematical conclusions of the programme). Regarding Peter: Unless he chooses to be more open-minded, what I hear from him is a premature pessimism about the HP based on a claim that there will be “no convergence regarding what can be inferred from the maximal iterative conception”. To be honest, I find it quite odd that (excluding my coworkers Claudio and Radek) I have received the most encouragement from Hugh, who seems open-minded and interested in seeing what comes out of the HP, just as we all want to see what comes out of Ultimate L (my criticisms long ago had nothing to do with the programme itself, only with the way it had been presented).

Pen, I know that you have said that in any event you will encourage the “good set theory” that comes out of the HP. But the persistent criticism (not just from you) of the conceptual approach, aside from the math, while initially of extraordinary value to help me clarify the approach (I am grateful to you for that), is now becoming somewhat tiresome. I have written dozens of e-mails to explain what I am doing and I take it as a good sign that I am still standing, having responded consistently to each point. If there is something genuinely new to be said, fine, I will respond to it, but as I see it now we have covered everything: The HP is simply a focused investigation of mathematical criteria for the maximality of V in height and width, with the aim of convergence towards an optimal such criterion. The success of the programme will be judged by the extent to which it achieves that goal. Interesting math has already come out of the programme and will continue to come out of it. I am glad that at least Hugh has offered a bit of encouragement to me to get to work on it.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Regarding a few recent quotes from Mr. Energy:

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?

It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway).

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 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?

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 assume that Hugh wants to claim that all natural paths in higher set theory, where “all x^\#, x \subseteq \omega, exist” lies there in an early stage of development, lead to PD. Although it would be very nice to have some formalized version of this, it does seem to make sense, and I don’t recall seeing any convincing counterexample to this.

For instance, one can set up a language for natural statements in the projective hierarchy, and try to prove rigorous theorems backing up this statement.

“It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway).”

Any version of HP that I have seen, even for IMH, including any of my own (the only one that is “new” involves Boolean algebras), is awkward compared to using countable transitive models. So “perfectly well” seems like an exaggeration at best. Also I think (have I got this right?) Hugh pointed out a place in the proliferating “fixes” of IMH for which ctms are needed, or perhaps where the awkwardness of not using them becomes really severe? In addition, I think you never answered some of Hugh’s questions about formulating precise and interesting fixes of IMH.

ASIDE: It now seems that any settling of CH via “HP” or CTMP is extremely remote. You did not start the discussion here with this point of view. Recall the subject line of this email.

I think of IMH, with that triple paper, as something not uninteresting. My impression is that you don’t have a comparable second not uninteresting development. IMH has, under standard views at least, a prima facie fatal flaw that calls into doubt the coherence of the very notion you keep talking about – intrinsic maximality of the set theoretic universe. What seems most dubious about “HP” is that it is not robust, and doesn’t have a second not uninteresting success for a wide range of people to really ponder. My back channels indicate to me that the “fixes” artificially layer the idea of IMH on top of large cardinals, which is not a convincing way to proceed.

You should simply rename it CTMP (countable transitive model program), as Hugh and I have said, and then you have a license to pursue practically any grammatically coherent question whatsoever in the realm of ctms as a not uninteresting corner of higher set theory. If something foundational or philosophically coherent comes out of pursuing CTMP then you can try to make something of it foundationally or philosophically. You just don’t have enough success with “HP” to do this with it now. No, you can’t reasonably just invent a branch of set theory called “intrinsic maximality” without more not uninteresting successes. That’s way premature.

Since you spent the bulk of your career on not uninteresting technical work in set theory, it is heroic to try to “get religion” and do something “truly important”, as you are 61. I can see how you got excited with IMH, and got just the right help with the technical complications (Welch, Woodin). But you are trying to dress this up into a foundational/philosophical program under a hopelessly idiosyncratic propogandistic name (HP) way too early, and should have instead stated CTMP and pondered the difficulties with “intrinsic maximality” in an objective and creative way. Incidentally, the way PD is used to prove the consistency of IMH in that triple paper does lend some credence to Hugh conjecturing that “HP” may well simply be another path leading to PD.

OK, I am skeptical in many dimensions of higher set theory, and probably will be raising issues with Koellner/Woodin. You have done some of that already, sometimes with unexpectedly strong language. But I don’t think that the “HP” is strong enough at this point to be using it to set an example that would undermine Koellner/Woodin. You surely can raise some legitimate issues without holding up “HP” as superior.

Harvey