Tag Archives: Large cardinals

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

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 Sy,

Pen wrote:

Hugh has talked about how things might go if various conjectures fall in a particular direction: there’d then be a principle ‘V=Ultimate L’ that would at least deserve serious consideration. That’s far short of ‘endorsement’, of course.  Can you point to an HP-generated principle that has that sort of status?

and you responded:

I can come close. It would be the \textsf{SIMH}^\#. But it’s not really analogous to Ultimate L for several reasons:

  1. I hesitate to “conjecture” that the \textsf{SIMH}^\# is consistent.
  2. The \textsf{SIMH}^\# in its crude, uncut form might not be “right”. Recall that my view is that only after a lengthy exploratory process of analysis and unification of different maximality criteria can one understand the Optimal maximality criterion. I can’t say with confidence that the original uncut form of the \textsf{SIMH}^\# will be part of that Optimal criterion; it may have to first be unified with other criteria.
  3. The \textsf{SIMH}^\#, unlike Ultimate L, is absolutely not a “back to square one” principle, as Hugh put it. Even if it is inconsistent, the HP will continue its exploration of maximality criteria and in fact, understanding the failure of the \textsf{SIMH}^\# will be a huge boost to the programme, as it will provide extremely valuable knowledge about how maximality criteria work mathematically.

This is a technical criticism. In brief I am claiming that based on the methodology of HP you have described (though perhaps now rejected), \textsf{IMH}^\# is not the correct synthesis of IMH and reflection. Moreover the correct synthesis, which is significantly stronger, resurrects all the issues associated with IMH regarding “smallness”.

Consider the following extreme version of \textsf{IMH}^\#:

Suppose M is a ctm and M \vDash \text{ZFC}.  Then M witnesses extreme-\textsf{IMH}^\# if:

  1. There is a thickening of M, satisfying ZFC, in which M is a \#-generated inner model.
  2. M witnesses \textsf{IMH}^\# in all thickenings of M, satisfying ZFC, in which M is a \#-generated inner model.

One advantage to extreme-\textsf{IMH}^\# is that the formulation does not need to refer to sharps in the hyperuniverse (and so there is a natural variation which can be formulated just using the V-logic of M). This also implies that the property that M witnesses extreme-\textsf{IMH}^\# is \Delta^1_2 as opposed to \textsf{IMH}^\# which is not even in general \Sigma^1_3.

Given the motivations you have cited for \textsf{IMH}^\# etc., it seems clear that extreme-\textsf{IMH}^\# is the correct result of synthesizing IMH with reflection unless it is inconsistent.

Thm: Assume every real has a sharp and that some countable ordinal is a Woodin cardinal in a definable inner model. Then there is a ctm which witnesses that extreme-\textsf{IMH}^\# holds.

However unlike \textsf{IMH}^\#, extreme-\textsf{IMH}^\# is not consistent with all large cardinals.

Thm:  If M satisfies extreme-\textsf{IMH}^\# then there is a real x in M such that in M, x^\# does not exist.

This seems to be a bit of an issue for the motivation of \textsf{IMH}^\# and \textsf{SIMH}^\#. How will you deal with this?

Regards.
Hugh