# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Wed, 29 Oct 2014, W Hugh Woodin wrote:

My question to Sy was implicitly: Why does he not, based on maximality, reject HOD Conjecture since disregarding the evidence from the Inner Model Program, the most natural speculation is that the HOD Conjecture is false.

Two points:

1. The HP is concerned with maximality but does not aim to make “conjectures”; its aim is to throw out maximality criteria and analyse them, converging towards an optimal criterion, that is all. A natural maximality criterion is that V is “far from $\text{HOD}$” and indeed my work with Cummings and Golshani shows that this is consistent. In fact, I would guess that an even stronger statement that V is “very far from $\text{HOD}$” is consistent, namely that all regular cardinals are inaccessible in $\text{HOD}$ and more. What you call “the $\text{HOD}$ Conjecture” (why does it get this special name? There are many other conjectures one could make about $\text{HOD}$!) presumes an extendible cardinal; what is that doing there? I have no idea how to get extendible cardinals from maximality.

2. Sometimes I make conjectures, for example the rigidity of the Stable Core. But this has nothing to do with the HP as I don’t see what non-rigidity of inner models has to do with maximality. I don’t have reason to believe in the rigidity of $\text{HOD}$ (with no predicate) and I don’t see what such a statement has to do with maximality.

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Tue, 28 Oct 2014, W Hugh Woodin wrote:

My point is that the non-rigidity of HOD is a natural extrapolation of ZFC large cardinals into a new realm of strength. I only reject it now because of the Ultimate L Conjecture and its implication of the HOD Conjecture. It would be interesting to have an independent line which argues for the non-rigidity of HOD.

This is the only reason I ask.

Please don’t confuse two things: I conjectured the rigidity of the Stable Core for purely mathematical reasons. I don’t see it as part of the HP. Indeed, I don’t see a clear argument that the nonrigidity of inner models follows from some form of maximality.

But I still don’t have an answer to this question:

What theory of truth do you have? I.e. what do you consider evidence for the truth of set-theoretic statements?

But I did answer your question by stating how I see things developing, what my conception of V would be, and the tests that need to be passed. You were not happy with the answer. I guess I have nothing else to add at this point since I am focused on a rather specific scenario.

That doesn’t answer the question: If you assert that we will know the truth value of CH, how do you account for the fact that we have many different forms of set-theoretic practice? Do you really think that one form (Ultimate L perhaps) will “have virtues that will swamp all the others”, as Pen suggested?

Best,
Sy

PS: With regard to your mail starting with “PS:”: I have worked with people in model theory. When we get an idea we sometimes say “but that would give an easy solution to Vaught’s conjecture” so we start to look for (and find) a mistake. That’s all I meant by my comments: What I was doing would have given a “not difficult solution to the HOD conjecture”; so on this basis I should have doubted the argument and indeed I found a bug.

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

The Stability Predicate S is the important thing. V is generic over the Stable Core = (L[S],S). As far as I know, V may not be generic over HOD; but it is generic over (HOD,S).

V is always a symmetric extension of HOD but maybe you have something else in mind.

Let A be a V-generic class of ordinals (so A codes V). Then A is (HOD, P)-generic for a class partial order P which is definable in V. So if T is the \Sigma_2-theory of the ordinals then P is definable in (HOD,T) and A is generic over (HOD,T).

Why are you stating a weaker result than mine? I show that for some A, (V,A) models ZFC and is generic over the Stable Core and hence over (HOD,S) where S is the Stability predicate. The Stability Predicate is $\Delta_2$, not $\Sigma_2$. And a crucial point is that its only reference to truth in V is via the “stability relationships” between $V_\alpha$‘s, a much more absolute property than truth which is much easier to analyse. As I said, the Stability Predicate is the important thing in my conjecture.

But you did not answer my question. Are you just really conjecturing that if V is generic over N then there is no nontrivial $j:N \to N$?

But I did answer your question: The Stability Predicate is the basis for my conjecture, not just some arbitrary predicate that makes V generic over HOD. In fact my conjecture looks stronger than the rigidity of (HOD,S), as the Stable Core (L[S],S) is smaller.

Let me phrase this more precisely.

Suppose A is a V-generic class of ordinals, N is an inner model of V, P is a partial order which is amenable to N and that A is (N,P)-generic.

Are you conjecturing that there is no non-trivial $j:N \to N$? Or that there is no nontrivial $j:(N,P) \to (N,P)$? Or nothing along these general lines?

As I said: Nothing along those general lines.

I show that (in Morse-Kelley), the (enriched) Stable Core is rigid for “V-constructible” embeddings. That makes key use of the (enriched) Stability Predicate. I wouldn’t know how to handle a different predicate.

I would think that based on HP etc., you would actually conjecture that there is a nontrivial $j:\text{HOD} \to \text{HOD}. No. This is the "reality check" that Peter and I discussed. Maximality suggests that V is as far from HOD as possible, but we have to acknowledge what is not possible. So maximality considerations have no predictive content. It is an idea which has to be continually revised in the face of new results. Finally you are catching on! I have been trying to say this from the beginning, and both you and Peter were strangely trying to "pin me down" on what the "definitive consequences" or the "make-or-break predictions" of the HP are. It is a study of maximality criteria, with the aim of converging towards the optimal such criterion. How can you expect such a programme to make "definitive predictions" in short time? In recursion theory language, the process is$latex \Delta_2\$ and not $\Sigma_1$ (changes of direction are permitted when necessary; witness the IMH being replaced by the $\textsf{IMH}^\#$). And set-theoretic practice is the big daddy: If you investigate a maximality criterion which ZFC proves inconsistent then you have to revise what you are doing (is “all regular cardinals inaccessible in HOD” consistent? I think so, but may be wrong.)

Yet you propose to deduce the non existence of large cardinals at some level based on maximality considerations. I would do the reverse, revise maximality.

If the goal is to understand maximality then that would be cheating! You may have extrinsic reasons for wanting LCs as opposed to LCs in inner models (important note: for Reinhardt cardinals that would be the only option anyway!) but those reasons have no role in an analysis of maximality of V in height and width.

I guess this is yet another point we just disagree on.

But I still don’t have an answer to this question: “What theory of truth do you have? I.e. what do you consider evidence for the truth of set-theoretic statements?”

Have you read Pen’s “Defending the Axioms”, and if so, does her Thin Realist describe your views? And if so, do you have an argument that LC existence is necessary for “good set theory”?

PS: With embarrassment and apologies to the group, I have to report that I found a bug in my argument that maximality kills supercompacts. I’ll try to fix it and let you know what happens. I am very sorry for the premature claim.

Suppose that there is an extendible and that the HOD Conjecture fails. Then:

1) Every regular cardinal above the least extendible cardinal is measurable in HOD (so HOD computes no successors correctly above the least extendible cardinal).

2) Suppose $\gamma$ is an inaccessible cardinal which is a limit of extendible cardinals. Then there is a club $C \subset \gamma$ such that every $\kappa \in C$ is a regular cardinal in $\text{HOD}$ (and hence inaccessible in HOD).

So, if you fix the proof, you have proved the HOD Conjecture.

I’ll try not to let that scare me

But I’m also not suprised that there was a bug in my proof!

Thanks,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Bob,

On Sun, 26 Oct 2014, Robert Solovay wrote:

Dear Sy,

In the last conjecture of your letter what does ” rigid” mean?

The only elementary embedding from the Stable Core to itself is the identity. I’m thinking of embeddings $j$ such that $(V,j)$ models ZF.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Peter,

On Sun, 26 Oct 2014, Koellner, Peter wrote:

Dear Sy,

I have one more comment on choiceless large cardinal axioms that concerns $\textsf{IMH}^\#$.

It is worth pointing out that Hugh’s consistency proof of $\textsf{IMH}^\#$ shows a good deal more (as pointed out by Hugh):

Theorem: Assume that every real has a sharp. Then, in the hyperuniverse there exists a real $x_0$ such that every #-generated M to which $x_0$ belongs, satisfies $\textsf{IMH}^\#$ and in the following very strong sense:

(*) Every sentence $\phi$ which holds in a definable inner model of some #-generated model N, holds in a definable inner model of M.

There is no requirement here that N be an outer model of M. In this sense, $\textsf{IMH}^\#$ is not really about outer models. It is much more general.

It follows that not only is $\textsf{IMH}^\#$ consistent with all (choice) large cardinal axioms (assuming, of course, that they are consistent) but also that $\textsf{IMH}^\#$ is consistent with all choiceless large cardinal axioms (assuming, of course, that they are consistent).

The point is that $\textsf{IMH}^\#$ is powerless to provide us with insight into where inconsistency sets in.

Before you protest let me clarify: I know that you have not claimed otherwise! You take the evidence for consistency of large cardinal axioms to be entirely extrinsic.

I protest for a different reason: The above argument is too special to $\textsf{IMH}^\#$. For example, consider Hugh’s variant which he called $\textsf{IMH}^\#(\text{card arith})$. I don’t see how you argue as above with this stronger principle, which is known to be consistent, using my proof with Radek based on #-generated Jensen coding.

My point is simply to observe to everyone that $\textsf{IMH}^\#$ makes no predictions on this matter.

So what? How do you know that $\textsf{IMH}^\#(\text{card arith})$ makes no predictions on this matter?

And, more generally, I doubt that you think that the hyperuniverse program has the resources to make predictions on this question since you take evidence for consistency of large cardinal axioms to be extrinsic.

In contrast “V = Ultimate L” does make predictions on this question, in the following precise sense:

Theorem (Woodin). Assume the Ultimate L Conjecture. Then if there is an extendible cardinal then there are no Reinhardt cardinals.

Theorem (Woodin). Assume the Ultimate L Conjecture. Then there are no Super Reinhardt cardinals and there are no Berkeley cardinals.

Theorem (Woodin). Assume the Ultimate L Conjecture. Then if there is an extendible cardinal then there are no Reinhardt cardinals (or Super Reinhardt cardinals or Berkeley Cardinals, etc.)
(Here the Ultimate-L Conjecture is a conjectured theorem of ZFC.)

Interesting. (Did you intend there to be a difference between the first and third theorems above?)

But probably there’s a proof of no Reinhardt cardinals in ZF, even without Ultimate L:

Conjecture: In ZF, the Stable Core is rigid.

Note that V is generic over the Stable Core.

Best,
Sy