(Wherein I quote from Twitter.)
This is my excuse to put this page to use. It all started, more or less, around here:
Villaveces, Andrés (gavbenyos). “@bluelephant lo siento bastante crudo – me parece que Joel simplemente pone en versión impresa cierto consenso, pero falta argumentar” 2 April 2012, 4:32 p.m. Tweet.
(Loosely translated: “This is a very good article by Hamkins on something akin to sociology of set theory” “I find it weak — I think that Joel is simply putting in writing a known consensus, but needs to argue for it more”) Long story short, I feel Andrés is right, but I thought I should elaborate my view, at least somewhat. Originally I considered writing a blog entry, but it quickly became apparent it would grow longer than I can afford time-wise. So, I used twitter instead.
(But there is a serious caveat, namely, it seems that the paper is intended for a general mathematical and philosophical audience, so the omission of technical issues is most likely intentional. Javier even remarked as much.)
What follows is the series of Tweets I posted. It is not a transcription; to ease reading, I have added a couple of links, reformatted the posts rather than continuing with the MLA suggested approach, very lightly edited the most obvious typos, and added a couple of phrases where I felt more clarity was needed. [Edit, April 22: I also removed a line on vs Woodin’s , as the proof of the underlying claim has been withdrawn.]
I started at 10:44 p.m., with a warning: “(Technical pseudo-philosophical thoughts for a few posts.)”
What Joel calls “the dream solution” is a two step procedure:
- Find an obviously true new set-theoretic principle.
- Show that the principle decides the Continuum Hypothesis.
He then proceeds to argue that such a solution is unattainable.
“Our extensive experience […] prevents us from looking upon any statement settling CH as being obviously true.”
I could only find one convincing argument in the paper, the most familiar one: By very mild forcing, we can change the truth value of CH. The point here is that, even if we admit large cardinals, we expect them to satisfy the Levy-Solovay theorem. (“Even if”, because Joel doesn’t seem to admit large cardinals as true.)
Is Levy-Solovay, plus the belief that we understand the large cardinal hierarchy well enough, all there is to the Continuum problem?
I think there is strong mathematical evidence for admitting large cardinals as part of the set theoretic universe. See
The main reason why we favor large cardinals is interpretative power. A more sophisticated presentation of Joel’s argument would have to show that large cardinals should force us to have bi-interpretability between (suitable) theories where CH holds and where it fails. That is what Woodin‘s “Ultimate L” theory would really accomplish: If
- We can isolate true K without anti-large cardinal restrictions, and
- Prove the Omega-conjecture,
then, and only then, would we be able to use Levy-Solovay as a convincing argument. Prior to these, or analogous developments, its use would be premature. (So, if one were to advocate an unattainability situation, a more serious mathematical discussion than in the paper seems needed.)
Woodin had previously advanced a theory that would suggest a way of settling CH, by refuting it. This was based on Omega-completeness. But unfortunately the key point of the argument was a “simple definability” result, based on what we now know was a mistake. (Roughly, Woodin thought that what we now call hod-mice could not accommodate too high large cardinals. This was refuted by Sargsyan.)
The bi-interpretability position is actually rather radical. We abandon the pursuit of “the right theory” beyond its large cardinal core. If we didn’t, then one could make a strong argument for not-CH based on
- The naturalness of reflection principles. (Which I would go so far as to call the “true” part of forcing axioms.)
- Mathematical experience. (Which goes against some of the other arguments in Joel’s paper.)
In effect, there is not perfect symmetry between CH universes and not-CH ones. CH universes are good for anti-classification results. Not-CH universes (by which I really mean universes with rich reflection principles) are good for structure theorems.This has been argued well by Todorcevic and others, and weakens a naive defense of CH that an analyst could make; I briefly argued this here. And, if there is a way of rescuing some of Woodin’s “simple definability” approach, then I think we will be making progress towards strong set theoretic evidence against CH.
(The point of Twitting was so this wouldn’t grow out of control and never-ending…)
As I was posting, Andrés Villaveces made comments along the way, which prompted some responses:
- Yes, I agree Joel’s position with respect to large cardinals seems very strange to me. It goes beyond not advocating for them, even. I find set theory to be different from “model theory of ZFC”, which is the closest I can come to a coherent description of part of his position. I do not think that all models share the same status, even if they are ill-founded or not even omega-models. And definitely large cardinals are part of the set-theoretic landscape. (So I disagree with Shelah, it seems. So I may very well be wrong, what do I know?)
- Of course, the assertion that reflection principles are the true part of forcing axioms is meant to be provocative. The way I see it, the justified extensions of ZFC come in two stages. In the first one, we get large cardinals. In the second, we definitely need to go beyond first-order, and the closest we currently get to true maximality principles is forcing axioms. But I do not see how to justify that forcing axioms are true, in whatever sense one uses that word. On the other hand, they provide us with a rich and coherent theory, that usually traces back to reflection principles. And reflection can itself be naturally justified by very similar arguments to those we first advance for large cardinals. (The argument of course needs elaboration, but this is how I usually begin. I know I’ve written on this before, but couldn’t quite locate where at the moment — will add a link if I find something.)
There is one last point I feel I need to make: Levy-Solovay depends rather explicitly in certain large cardinal behavior. Woodin has suggested some large cardinal schema (at the level of rank-to-rank embeddings and beyond) that are fragile under forcing. The schema are far from reasonably understood at the moment, and they may end up being inconsistent. Rather, the point is that they suggest that Levy-Solovay may not be the end-all, and large cardinals, properly understood, may end up after all settling CH.
So: Does CH have a truth value, or can we expect a “dream solution”? I do not know. But a more serious argument is needed in any case.
And with that, I stopped: “(Apologies for the length.)” at 11:46 p.m.