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 Harvey, There’s quite a bit of terminology being used here that most people, including me, are not familiar with. parasitic (simple generally understandable examples, please) The term “parasitic” is not a precise mathematical term — it is a metaphor. I thought it was a helpful metaphor. Perhaps I should have put it in scare quotes. But if we (including you) put every term that is not a precise mathematical term in scare quotes then it would lead to an unreadable typographic forest. If you feel more comfortable with scare quotes then please imagine invisible scare quotes around most of the terms I use that are not precise mathematical terms. (And please don’t put scare quotes around all such terms when you use them!) To determine the sense of “parasitic” and whether you agree with the statements I have made involving that term you have to look at the examples — the three (precise) examples I gave in the letter in which I first used this term (all of which were examples of “parasitic” statements) and the examples in my last letter (like “all sets of reals are determined” and “all open sets of reals are determined, which were examples of statements that were not “parasitic”). extendible cardinal (as in Kanamori?) Yes, as in Kanamori, in the context of ZFC. In the context of ZF one defines it as Hugh did in his letter of Oct. 18. To repeat: Def. (ZF) $\kappa$ is extendible if for all $\alpha > \kappa$, there exists an elementary embedding $j:V_{\alpha} \to V_{j(\alpha)}$ such that $\text{crt}(j) = \kappa$ and $j(\kappa) > \alpha$ Reinhardt cardinal (I know what Reinhardt’s axiom is, j:V into V) Def. (ZF) $\kappa$ is Reinhardt if there is a non-trivial elementary embedding$latex j:V\to V\$ with $\text{crt}(j)=\kappa$.

(So, a Reinhardt cardinal is simply the critical point of an embedding given by Reinhardt’s axiom.)

super Reinhardt cardinal

Def. (ZF) $\kappa$ is Super Reinhardt if for every $\alpha$ there is a non-trivial elementary embedding $j:V\to V$ with $\text{crt}(j)=\kappa$ such that $j(\kappa)>\alpha$.

Berkeley cardinal

I will give a definition in my letter to Pen on choiceless large cardinals.

Can someone step up here and explain these in generally understandable terms?

Peter’s message indicates how strong a grip Hugh has on $\textsf{IMH}^\#$, which if I recall, was offered as a “fix” for IMH (inner model hypothesis) being incompatible with even an inaccessible cardinal.

Is $\textsf{IMH}^\#$ merely a layering of the IMH idea on top of large cardinal infrastructure?

Sy gave a precise definition of this — see e.g. his letter to Bob for the definition of #-generation.

NOTE: I had concluded that, on the basis of this extensive traffic, this “HP” is not a legitimate foundational program, and should be renamed CTMP = countable transitive model program, a not uninteresting technical program that has been around for quite some time, with no artificial philosophical pretensions. Just the study of ctms. However, if Peter Koellner is going to put such an enormous amount of painstaking and skilled effort in continuing corresponding about it, I am more than happy to suspend disbelief and ask these questions.

The fact that I am trying to understand the program and its claims
should not be understood as an endorsement or a projection of whether the program is tenable. One cannot asses a program until one understands it.  I am still in the understanding phase.

NOTE: My own position is that “intrinsic maximality of the set
theoretic universe” is prima facie a deeply flawed notion, fraught with nonrobustness (inconsistencies, particularly) that may or may not be coherently adjusted in order to lead to anything foundationally interesting.

I have said something similar.

Best,
Peter