Tag Archives: HOD

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,

This is an addendum to what I wrote to you about large cardinal existence on 25.October.

It just occurred to me that we already have a consistent maximality principle which provably contradicts large cardinal existence.

Consider:

(*) \alpha^+ of HOD is less than \alpha^+ for all infinite
cardinals \alpha, and

({*}{*}) Every infinite cardinal is regular in HOD.

Cummings, Golshani and I proved the consistency of the conjunction of (*) and (**).

Using work of Cummings-Schimmerling, Gitik and Dzamonja-Shelah we get:

Fact. The conjunction of (*) and ({*}{*}) implies that there are no supercompacts.

[One can weaken the hypothesis to just: For cofinally many singular strong limit cardinals \alpha of cofinality \omega, \alpha is regular in HOD and \alpha^+ of HOD is less than \alpha^+.]

This is not enough evidence to infer the nonexistence of supercompacts from maximality but this is definitely pointing in that direction!

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Fri, 24 Oct 2014, Penelope Maddy wrote:

Dear Sy,

We already know why CH doesn’t have a determinate truth value, it is because there are and always will be axioms which generate good set theory  which imply CH and others which imply not-CH. Isn’t this clear when one looks at what’s been going on in set theory?

Well, I’m not sure it is clear that there will never be a theory whose virtues swamp the rest.

What evidence do you see for the existence of such a theory? All the evidence points to the contrary: The number of different valuable directions in set theory just keeps multiplying.

What I could imagine is that a particular truth value of CH will be required for an optimal foundation for mathematics (Type 2 evidence), but that is just wild speculation at this point. If that occurred, then maybe it would tip the balance between axioms which are valuable for the mathematical development of set theory (Type 1 evidence) yet give different verdicts on CH.

I am however inclined to think that Type 3 evidence (the HP) will not have as much influence as Type 2 evidence, simply because people regard the foundations of mathematics as more important than what can be derived from the maximality of the set concept.

 

Is CH one of the leading open questions of set theory?

No! The main reason is that, as Sol has pointed out, it is not a mathematical problem but a logical one. The leading open questions of set theory are mathematical.

I didn’t realize that you’d been convinced by Sol’s arguments here.  My impression was that you thought it  might be possible to resolve CH mathematically:

I started by telling Sol that the HP might give a definitive refutation of
CH! You told me that it’s OK to change my mind as long as I admit it, and I admit it now!

That’s why I posed the question to you as I did.

You misunderstood me. I didn’t need Sol to convince me that CH is a logical but not mathematical problem. The HP is a programme based on logic, so any conclusion about CH via the HP would be a logical solution, not a mathematical one.

With apologies to all, I want to say that I find this focus on CH to be exaggerated. I think it is hopeless to come to any kind of resolution of this problem, whereas I think there may be a much better chance with other axioms of set theory such as PD and large cardinals.

PD has a decent chance of winning the blessing of all 3 forms of evidence: It may be that you need it for the best set theory (as mathematics), if mathematicians ever start worrying about the higher projective levels they may appreciate having the Lebesgue measurability of the projective sets, and as Hugh and I mentioned, PD may be a consequence of maximality according to the HP. (Hugh if you thought that I was claiming otherwise then you got confused; where did I say that?) Of course it is too soon to come to any definitive conclusion about PD, but there is a fighting chance for its truth.

I am less optimistic about large cardinals. This past week I was at AIM (American Institute of Mathematics) and based on the work we did I am willing to conjecture that the following is consistent:

(*) Every uncountable cardinal is inaccessible in HOD (the hereditarily ordinal definable sets).

(Cummings, Golshani and I already got the consistency of a weaker version of this: alpha^+ of HOD is less than alpha^+ for all infinite cardinals alpha.)

Now (*) is clearly a maximality principle, but the mathematical evidence is that it contradicts the existence of large cardinals. Indeed, the consistency proofs of these “V is fatter than HOD” principles break when you try to accomodate large cardinals, and indeed Hugh has plausible conjectures which imply that such obstacles are unsurmountable.

So where this is pointing is that maximality denies large cardinal existence. This happened before with the IMH but that got fixed by marrying the IMH to a vertical maximality principle; I don’t see how large cardinals are going to escape from this latest dilemma.

All the best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Pen, Geoffrey, Hugh and others,

As discussed, the aim of the HP is to derive consequences of the maximality feature of the set concept via mathematical criteria of maximality which can be formulated in a way which is consistent with width actualism and which can be analysed through a mathematical analysis carried out within the Hyperuniverse.

With this background, the real work of the programme consists of the formulation, analysis and synthesis of such criteria for the selection of those countable transitive models of ZFC which are optimal in terms of their maximality properties.

As a guide to the choice of criteria I’d like to tentatively suggest (yes, this is subject to change as we learn more about Maximality) the following three-step

Maximality Protocol

  1. Impose Height-Maximality via the criterion of #-generation.
  2. Impose Cardinal-Maximality (the class of cardinals should be as thin as possible): A tentative criterion for this is that for any infinite ordinal alpha and subset x of \alpha, (\alpha^+)^{\text{HOD}_x} < \alpha^+. [\text{OD}_x consists of those sets which are definable with x and ordinals as parameters; a set is in \text{HOD}_x if it and all elements of its transitive closure belong to \text{OD}_x.]
  3. Impose Width-Maximality via the criterion \textsf{SIMH}^\# (Levy absoluteness with absolute parameters for cardinal-prserving, #-generated outer models).

I don’t know if Cardinal-Maximaity as formulated above is consistent. Cummings, Golshani and I showed that one can consistently get (\alpha^+)^{\text{HOD}} < \alpha^+ for all infinite cardinals \alpha, and this is good evidence for the consistency of Cardinal-Maximality, but it is significantly weaker. And as already said, I don’t know if the \textsf{SIMH}^\# is consistent.

As I see it, there are two options. Either the Maximality Protocol can be successfully implemented (i.e. relative to large cardinals there are #-generated, cardinal-maximal models obeying the \textsf{SIMH}^\#) and this will be strong evidence that the negation of CH is derivable from Maximality. Or there will be new inconsistency arguments which will tell us a lot about the nature of Maximality in set theory and lead to new, compelling and consistent criteria.

As always I welcome your thoughts.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I’d like to take a crack at

Can you or John tell me what evidence you have that the iterability problem will be solved positively to enable the construction of an inner model for a supercompact in the foreseeable future?

There’s a lot of evidence that there are canonical inner models with supercompacts, and that their canonicity derives from a form of iterability. The evidence consists mostly of approximations to such a result that have already been proved. The large cardinals/determinacy/inner models theory has a great deal of internal unity, and people keep extending it to reach higher consistency strengths. It’s not going to break down between many Woodin cardinals and supercompacts. “Subtle is the lord, but he is not malicious.”

Whether and when humanity will prove that there are canonical, iterable inner models with supercompacts is another question. I would guess that there will always be at least a few people interested in the higher reaches of the consistency strength hierarchy, and so eventually some (perhaps AI-enhanced) human will prove it. I would also guess that it won’t happen in the next few years. I think we see a road, one that involves developing the theory of HOD in models of AD.

Best,
John