# Re: Paper and slides on indefiniteness of CH

Regarding a few recent quotes from Mr. Energy:

I want to know what you mean when you say “PD is true”. Is it true because you want it to be true? Is it true because ALL forms of good set theory imply PD? I have already challenged, in my view successfully, the claim that all sufficiently strong natural theories imply it; so what is the basis for saying that PD is true?

It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway).

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

I want to know what you mean when you say “PD is true”. Is it true because you want it to be true? Is it true because ALL forms of good set theory imply PD? I have already challenged, in my view successfully, the claim that all sufficiently strong natural theories imply it; so what is the basis for saying that PD is true?

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

I assume that Hugh wants to claim that all natural paths in higher set theory, where “all $x^\#$, $x \subseteq \omega$, exist” lies there in an early stage of development, lead to PD. Although it would be very nice to have some formalized version of this, it does seem to make sense, and I don’t recall seeing any convincing counterexample to this.

For instance, one can set up a language for natural statements in the projective hierarchy, and try to prove rigorous theorems backing up this statement.

“It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway).”

Any version of HP that I have seen, even for IMH, including any of my own (the only one that is “new” involves Boolean algebras), is awkward compared to using countable transitive models. So “perfectly well” seems like an exaggeration at best. Also I think (have I got this right?) Hugh pointed out a place in the proliferating “fixes” of IMH for which ctms are needed, or perhaps where the awkwardness of not using them becomes really severe? In addition, I think you never answered some of Hugh’s questions about formulating precise and interesting fixes of IMH.

ASIDE: It now seems that any settling of CH via “HP” or CTMP is extremely remote. You did not start the discussion here with this point of view. Recall the subject line of this email.

I think of IMH, with that triple paper, as something not uninteresting. My impression is that you don’t have a comparable second not uninteresting development. IMH has, under standard views at least, a prima facie fatal flaw that calls into doubt the coherence of the very notion you keep talking about – intrinsic maximality of the set theoretic universe. What seems most dubious about “HP” is that it is not robust, and doesn’t have a second not uninteresting success for a wide range of people to really ponder. My back channels indicate to me that the “fixes” artificially layer the idea of IMH on top of large cardinals, which is not a convincing way to proceed.

You should simply rename it CTMP (countable transitive model program), as Hugh and I have said, and then you have a license to pursue practically any grammatically coherent question whatsoever in the realm of ctms as a not uninteresting corner of higher set theory. If something foundational or philosophically coherent comes out of pursuing CTMP then you can try to make something of it foundationally or philosophically. You just don’t have enough success with “HP” to do this with it now. No, you can’t reasonably just invent a branch of set theory called “intrinsic maximality” without more not uninteresting successes. That’s way premature.

Since you spent the bulk of your career on not uninteresting technical work in set theory, it is heroic to try to “get religion” and do something “truly important”, as you are 61. I can see how you got excited with IMH, and got just the right help with the technical complications (Welch, Woodin). But you are trying to dress this up into a foundational/philosophical program under a hopelessly idiosyncratic propogandistic name (HP) way too early, and should have instead stated CTMP and pondered the difficulties with “intrinsic maximality” in an objective and creative way. Incidentally, the way PD is used to prove the consistency of IMH in that triple paper does lend some credence to Hugh conjecturing that “HP” may well simply be another path leading to PD.

OK, I am skeptical in many dimensions of higher set theory, and probably will be raising issues with Koellner/Woodin. You have done some of that already, sometimes with unexpectedly strong language. But I don’t think that the “HP” is strong enough at this point to be using it to set an example that would undermine Koellner/Woodin. You surely can raise some legitimate issues without holding up “HP” as superior.

Harvey