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

Re: Paper and slides on indefiniteness of CH

Dear all,

I believe this concludes the discussion that Sy and I were having on $\textsf{IMH}^\#$.  I would like to note one more thing and here the proof does likely requires the machinery Sy was discussing.

Let $\textsf{IMH}^\#(\text{card})$ be $\textsf{IMH}^\#$ together with if there is a cardinal preserving $\#$-generated extension in which $\varphi$ holds then there is a cardinal preserving inner model in which $\varphi$ holds. Similarly define $\textsf{IMH}^\#(\text{card-arith})$ (see earlier in the thread) by restricting to $\#$-generated extensions and inner models, which preserve all cardinals and cardinal arithmetic.

$\textsf{IMH}^\#(\text{card-arith})$ is consistent relative to a bit more than one Woodin cardinal, in fact it holds if every real has a sharp and there is a countable ordinal which is a Woodin cardinal in an inner model. But the proof requires the adaptation of Jensen coding to $\#$-generated models etc.  There is a natural guess as to the statement of that theorem and I am assuming it holds: If M is an inner model of GCH which is $\#$-generated then there is a real $x$ such that M is a definable inner model of $L[x]$, $x^\#$ exists, and such that $L[x]$ is a cardinal preserving extension of M.

As for $\textsf{IMH}(\text{card-arith})$, a natural conjecture is that $\textsf{IMH}^\#(\text{card-arith})$ implies GCH.

I have no idea about the consistency of $\textsf{IMH}^\#(\text{card})$ which looks like the problem of showing $\textsf{SIMH}$ is consistent.

Now a variant of a problem which I raised much earlier in this email-thread seems quite relevant. The new problem is: Can there exist a $\#$-generated M such that if N is a $\#$-generated extension which is cardinal preserving then every set of N is set-generic over M.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Sol and Sy,

I did say in my previous email to this group that it would likely be my last posting to the group.

However I would like to make two final comments which I hope will be taken into account in the final version of Sol’s paper and in a presumably forthcoming paper of Sy’s on HP.

My point is simply that from a set theoretic perspective both comments seem quite natural and dealing with these would help the set theoretically inclined reader (i.e. me) of these papers better understand the issues.

I only cc the list here (and for the last time) because I think others might be in agreement that seeing these issues addressed would be useful.

I am not asking for an email response, I am happy to wait until the respective papers are ready.

For Sol’s paper. It seems very likely by the sketch of Rathjen’s proof in Sol’s slides that the following statements/problems are both not definite logical problems by the criteria of Sol’s draft.

(a) Extreme-CH which asserts $P(\mathbb R) \subset L$.
(b) Extreme-R-AC which asserts that AC holds in $L(R)$

I add (b) because it is closer in syntactic form to CH (both are $\Sigma^2_1$ sentences). For both these problems, I think there is substantial agreement among set theorists that the problems have answers and that both statements are false (i.e. at least 20 set theorists are in agreement on this).

Even if I am wrong and these are logically definite problems then knowing that would be quite useful in understanding the paper.

For Sy’s paper and in particular SIMH and CH. Consider the following scenario by which HP comes to the conclusion that CH is true. There many variations here and I am just identifying one.We have IMH is consistent. Let IMH(card-arith) be IMH together with the following: Suppose there is a card-arith preserving extension of M in which $\phi$ holds. Then there is an card-arith preserving inner model of M in which $\phi$ holds.(card-arith preserving means preserving $2^{\kappa} = \lambda$ for all $\kappa$. So cardinals must be preserved). The consistency proof of IMH shows that IMH(card-arith) is consistent (same model, for a cone of $x$ the minimum model of ZFC containing $x$ satisfies IMH(card-arith)).

“Conjecture” : IMH(card-arith) implies GCH.

My question for Sy’s paper is simply, why if “conjecture” is true does one reject this in favor of SIMH (assuming SIMH is consistent)?

One might argue that if SIMH is consistent then the same technology will refute “conjecture” and so there is no real conflict here within HP.

However, I note that restricting to ccc extensions, SIMH(ccc) is consistent by the consistency proof of IMH but obtaining that not-CH is consistent with IMH(card-arith:ccc) seems far more subtle.

Here SIMH(ccc) for M is IMH together with if a sentence $\phi$ holds in a ccc extension of M then $\phi$ holds in an inner model N of M such that M is a ccc extension of N.

Similarly for IMH(card-arith:ccc) except of course card-arith must be preserved.

Regards,
Hugh