# Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I think it would be nice to revisit all of these topics. Let me say two things about the axiom “V = Ultimate L” and your request that it be presented in “generally understandable terms”.

(1) The development of inner model theory has involved a long march up the large cardinal hierarchy and has generally had the feature that when you build an inner model for one key level of the large cardinal hierarchy — say measurable, strong, or Woodin — you have to start over when you target the next level, building on the old inner model theory while adding a new layer of complexity (from measures to extenders, from linear iterations to non-linear iterations) — because the inner models for one level are not able to accommodate the large cardinals at the next (much as L cannot accommodate a measurable).

Moreover, the definitions of the inner models — especially in their fine-structural variety — are very involved. One essentially has to develop the theory in tandem with the definition. It looked like it would be a long march up the large cardinal hierarchy, with inner models and associated axioms of the form “V = M” of increasing complexity.

One of the main recent surprises is that things change at the level of a supercompact cardinal: If you can develop the inner model theory for a superpact cardinal then there is a kind of “overflow” — it “goes all the way” — and the model can accommodate much stronger large cardinals. Another surprise is that one can actually write down the axiom — “V = Ultimate L” — for the conjectured inner model in a very crisp and concise fashion.

(2) You will, however, find that the axiom “V = Ultimate L” may not meet your requirement of being explainable in “generally understandable terms”. It is certainly easy to write down. It is just three short lines. But it involves some notions from modern set theory — like the notion of a Universally Baire set of reals and the notion of $\Theta$. These notions are not very advanced but may not meet your demand or being “generally understandable”. Moreover, to appreciate the motivation for the axiom one must have some further background knowledge — for example, one has to have some knowledge of the presentation of HOD, in restricted contexts like $L(\mathbb R)$, as a fine-structural inner model (a “strategic inner model”). Again, I think that one can give a high-level description of this background but to really appreciate the axiom and its motivation one has to have some knowledge of these parts of inner model theory.

I don’t see any of this as a shortcoming. I see it as the likely (and perhaps inevitable) outcome of what happens when a subject advances. For comparison: Newton could write down his gravitational equation in “generally understandable terms” but Einstein could not meet this demand for his equations. To understand the Einstein Field Equation one must understand the notions a curvature tensor, a metric tensor, and stress-energy tensor. There’s no way around that. And I don’t see it as a drawback. It is always good to revisit a subject, to clean it up, to make it more accessible, to strive to present it in as generally understandable terms as possible. But there are limits to how much that can be done, as I think the case of the Einstein Field Equations (now with us for almost 100 years) illustrates.

Best, Peter

# Re: Paper and slides on indefiniteness of CH

Dear Peter,

On Mon, 27 Oct 2014, Koellner, Peter wrote:

Dear Sy,

The reason I didn’t quote that paragraph is that I had no comment on it. But now, upon re-reading it, I do have a comment. Here’s the paragraph:

Well, since this thread is no stranger to huge extrapolations beyond current knowledge, I’ll throw out the following scenario: By the mid-22nd cenrury we’ll have canonical inner models for all large cardinals right up to a Reinhardt cardinal. What will simply happen is that when the LCs start approaching Reinhardt the associated canonical inner model won’t satisfy AC. The natural chain of theories leading up the interpretability hierarchy will only include theories that have AC: they will assert the existence of a canonical inner model of some large cardinal. These theories are better than theories which assert LC existence, which give little information.

Here’s the comment: This is a splendid endorsement of Hugh’s work on Ultimate L.

??? It is a scenario (not endorsement) of an inner model theory of some kind; why Hugh’s version of it?

Let us hope that we don’t have to wait until the middle of the 22nd century.

We appear to disagree on whether AD^L(R) is “parasitic” on AD in the way that “I am this [insert Woodin's forcing] class-sized forcing extension of an inner model of L”, where L is a choiceless large cardinal axiom. At least, I think we disagree. It is hard to tell, since you did not engage with those comments (which addressed the whole point at issue).

I have given up on trying to understand the word “parasitic”.

Let us push the analogy [between AD and choiceless large cardinals].

Shortly after AD was introduced $L(\mathbb R)$ was seen as the natural inner model. And Solovay conjectured that $\text{AD}^{L(\mathbb R)}$ follows from large cardinal axioms, in particular from the existence of a supercompact.

This leads to a fascinating challenge, given the analogy: Fix a choiceless large cardinal axiom C (Reinhardt, Super Reinhardt, Berkeley, etc.) Can you think of a large cardinal axiom L (in the context of ZFC) and an inner model M such that you would conjecture (in parallel with Solovay) that L implies that C holds in M?

You have overstretched the analogy to the point where it doesn’t work any more. $\text{AD}^{L(\mathbb R)}$ is not about large cardinals and we had little reason to believe that it would outstrip LC axioms consistent with AC. Reinhardt cardinals are likely stronger (in consistency strength) than any LC axiom consistent with AC (I think they are just plain inconsistent). So we cannot expect an inner model for Reinhardt’s axiom just from a LC axiom consistent with AC! We need some other way of extending ZFC for that. Maybe the latter is the “fascinating challenge” that you want to formulate? I.e. how can we extend ZFC + LCs to yield an inner model for a Reinhardt cardinal?

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

There is a great deal of disconnect between your reaction to Hugh’s letter and the impression I got from his letter, so much so that it feels like we read different letters.

Right at the start, Hugh’s letter has the line “I want to emphasize that what I describe below is just my admittedly very optimistic view” and later it has the line “Now comes (really extreme) sheer speculation.” It is thus clear that the letter is presenting an optimistic view.

Now, on the one hand, you seem to realize it is the presentation of an optimistic scenario — e.g. when you speak of “fantasy” and the difficulty of solving some of the underlying conjectures on which it rests, like, the iterability problem — but then later you switch and write:

You give the feeling that you are appearing at the finish line without running the race. … It gives the false impression that you have figured everything out, while in fact there is a lot not yet understood even near the beginning of your story.

I didn’t get that impression at all and I don’t know how you got it. I got the impression of someone presenting an “very optimistic view”, one that is mathematically precise and has the virtue of being sensitive to mathematical conjectures. ["There are rather specific conjectures which if proved would, I think argue strongly for this view. And if these conjectures are false then I would have to alter my view".] Far from getting the impression of someone who made it look like he was “at the finish line without running the race” I got the impression of someone who had a clear account of a finish line, was working hard to get there, realized there was a lot to do, and even thought that the finish line could disappear if certain conjectures turned out to be false.

(The mathematics behind this is considerable. In addition to the massive amount of work in inner model theory over the last forty years the new work is quite involved. E.g. even the monographs “Suitable Extender Models” 1 and 2 and the monograph on fine structure, alone amount to more than 1000 pages of straight mathematics, which, given my experience with the “expansion factor” in this work, is a misleadingly small number.)

It is remarkable to me that we now have such a scenario in inner model theory, one that is mathematically precise and has mathematical traction in that if certain conjectures turn out to be true one would have a strong case for it. The point is that given the incremental nature of inner model theory, a decade ago no one would have advocated such a view since, e.g., once one reached one supercompact the task of reaching a huge cardinal would not thereby be solved (any more than solving the inner model problem for strong cardinals also solved the inner model problem for Woodin cardinals). But now there has been a shift in landscape — a shift due to mathematical discoveries, showing that in a precise sense one just has to reach one supercompact and that at that point there is `overflow’ — and one can articulate such a scenario in a mathematically precise manner.

It is a virtue of a foundational program if it can articulate such a scenario. A foundational program should be able to list a sequence of conjectures which if true would make a case for the program and which if false would be a mark against the program, and even, in an extreme case set one back to square one. To do this is not to indulge in sheer fantasy. It is to give a program mathematical traction.

I would like to see you do the same for your program. You really should, at some stage, be able to do this. There must be a line of conjectures that you can point to which if true would make a strong case for the program and if false would be a setback; otherwise, it is not open to certification or refutation and one starts to wonder whether it is infinitely revisable and so “not even wrong”. I’m sure you agree. So please tell us whether you are at the stage where you can do that and if you are then I for one would like to hear some of the details (or be pointed to a place where I can find them).

Best,
Peter

P.S. I owe you a response to your request for feedback on one of the points at issue between you and Pen. I’m sorry for not doing that yet. The semester started. I’ll send something soon.

Also, a high bar must be met to send an email to so many people and I doubt I will meet that high bar. I’ll send it here since it was requested here. But eventually I think this should all be moved to a blog or FOM, something where people can subscribe.

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

I don’t think that we have a big disagreement here. The Friedman-Holy models are certainly not canonical and I agree that a key question is whether there are canonical, fine-structural inner models for large cardinals.

But no unverified hypotheses are needed to create the Friedman-Holy models and they witness the compatibility of arbitrary large cardinals with a key component of fine-structure theory: acceptability. We also force Global Square and I suspect that these models can be built as inner models using “internal consistency” arguments for reverse Easton forcings. (Of course the real motivation for the models was to use ideas of Neeman to show that a degree of supercompactness is a “quasi” lower bound on the consistency of the Proper Forcing Axiom).

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

For the uninitiated I think we should be clear about the difference between what has been proved and what has only been conjectured. As I understand it:

1. The Friedman-Holy models actually exist (they can be forced) and fulfill John’s 3 conditions.

I do not agree here. I do not think that the Friedman-Holy models are a starting point for fine-structure. That was the point I was trying to make.

For example, (global) strong condensation does not even imply $\square_{\omega_1}$ and $\square$ is a central feature of fine-structure.

For me anyway, fine-structure is a feature of canonical models and it is the canonical models which are important here. Identified instances of fine-structure can be forced but this in general is a difficult problem.  Could one create a list of such features, force them all while retaining all large cardinals, and finally argue that the result is a canonical model?

This seems extremely unlikely to me.

2. The models you are discussing are only conjectured to exist. In the final paragraph above you hint at a way of actually producing them.

3. If your models do exist then they also fulfill John’s 3 conditions but have condensation properties which are rather different from those of the Friedman-Holy models.

You are giving “my” models far more relevance here than they deserve.

Very technical point here: The models only reach the finite levels of supercompact and so do not fulfill John’s conditions. In fact, fine-structural extender models can never work since any such model is always a generic extension once one is past the level of one Woodin cardinal. One needs the hierarchy of fine-structural strategic-extender models. Though the latter occur naturally they have been much more difficult to explicitly construct. For example it is not yet known (as far as I know) if there can exist such an inner model at the level of a Woodin limit of Woodin cardinals no matter what iteration hypothesis and large cardinals one assumes in V.

If the Ultimate L Conjecture is true then V = Ultimate L meets John’s conditions. If this conjecture is false or more generally if there is an anti-inner model theorem (say at supercompact) then the Friedman-Holy models and their generalizations may be the best one can do (and to me this is the essence of the the inner model versus outer model debate).

But as you know the history of inner model theory has been full of surprises, and in particular we can’t just assume that the iterability hypotheses will be verified. For this reason, I do think it important to be clear about what has been proved and what has only been conjectured.

Of course I agree with this last point. The entire of theory of iterable models at the level of measurable Woodin cardinals and beyond could be vacuous and not because of an inconsistency. But it has not yet happened that a developed theory of canonical inner models as turned out to be vacuous. Will it happen? That is an absolutely key question right now.

Regards,
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Sun, 31 Aug 2014, W Hugh Woodin wrote:

The condensation principle that you force (local club condensation) actually does not hold in fine-structural models for the finite levels of supercompact which have been constructed (assuming the relevant iteration hypothesis). There are new fine-structural phenomena which happen in the long-extender fine structure models and which do not have precursors in the theory of short-extender models. (These models are generalizations of the short-extender models with Jensen indexing, the standard parameters are solid etc.)

When you say “Jensen indexing” do you mean the one that I proposed: index at the successor of the image of the critical point?

At the same time these models do satisfy other key condensation principles such as strong condensation at all small cardinals (and well past the least weakly compact). I believe that it is still open whether strong condensation can be forced even at all the $\aleph_n$’s by set forcing. V = Ultimate L implies strong condensation holds at small cardinals and well past the least inaccessible.

Very interesting! I guess we provably lose strong condensation at the level of $\omega$-Erdős, but it would of course be very nice to have it below that level of strength.

Finally the fine structure models also satisfy condensation principles at the least limit of Woodin cardinals which imply that the Unique Branch Hypothesis holds (for strongly closed iteration trees) below the least limit of Woodin cardinals. If this could be provably set forced (without appealing to the $\Omega$ Conjecture) then that would be extremely interesting since it would probably yield a proof of a version of the Unique Branch Hypothesis which is sufficient for all of these inner model constructions.

For the uninitiated I think we should be clear about the difference between what has been proved and what has only been conjectured. As I understand it:

1. The Friedman-Holy models actually exist (they can be forced) and fulfill John’s 3 conditions.
2. The models you are discussing are only conjectured to exist. In the final paragraph above you hint at a way of actually producing them.
3. If your models do exist then they also fulfill John’s 3 conditions but have condensation properties which are rather different from those of the Friedman-Holy models.

But don’t your models, if they exist, have some strong absoluteness properties that the Friedman-Holy models are not known to have? That’s why I suggested that John’s list of 3 conditions may have been incomplete.

Hugh, it is wonderful that you have the vision to see how inner model theory might go, and the picture you paint is fascinating. But as you know the history of inner model theory has been full of surprises, and in particular we can’t just assume that the iterability hypotheses will be verified. For this reason, I do think it important to be clear about what has been proved and what has only been conjectured.

Thanks,
Sy