Asaf Karagila
I don't have much choice...

Critical Cardinals

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Yup. I posted a new paper on arXiv. And if you're one of my three regular readers, you know that I am not going to talk about the paper itself (I leave that to the paper), but rather about the process leading to it. If you don't care, that's fine, the paper is on arXiv and you can check the Papers section of the site to see if it's been published or whatnot.

So, this one has been on the back burner for a while. And it actually started as two separate projects that merged and separated and merged again.

The first one starts somewhere in the evening of some wintery Jerusalem night. Yair and I were working on something, I don't even remember what, and we went to grab some bite or went to his car on the way home. And I suggested that we try to find a Silver-like criterion for lifting embeddings to symmetric extensions. It seemed to be an interesting tool to have in one's toolbox, especially with all the recent work about large cardinals as critical points without the axiom of choice.

So we had some rudimentary ideas, that was nice. And we sort of left it in the back to pursue more improtant goals. Namely, our Ph.D. theses, and so on. Every now and then, however, we would take an afternoon and work on that criterion. It culminated with this terribly complicated structural theorem for symmetric extensions somemwhere over the summer. We felt that it sort of works, and we left it for a while, to cook, barely even writing out the thing into some .tex file somewhere.

Fast forward a few months, I am about to go to the Arctic Set Theory meeting of 2017, and I want to give a talk about our work. So we revisit our definitions. I suggest a simpler alternative definition using a concept from my work about iterated symmetric extension, and we check that it does the job. Wonderful. Summer comes, and I leave for Europe. As I come back for a couple of weeks in September, we decide to put some work into this so we can finish the paper and submit it.

Well, we focus a lot of time into trying to work out two examples we originally wanted to include into the paper, but they seem to resist. I end up leaving when we only have the one example working from our second project, and some parts of that one also seem fishy.

So time passes again, and during my time at the TU Wien, I spent more time reworking the definitions and checking again the failures we had. But nothing seems to give, except me, the failures still fail. So I tell Yair that maybe it's time to move on? Maybe we need to cut these results out, and just have a reasonable paper on our hands. Because we keep spending more and more time trying to finish these last two results, and we use it almost as an excuse to not work on the paper itself. Yair agrees, and we set out to write down what we have. We agree that Yair takes charge of the majority of writing, since he knows way more about supercompact Radin forcing than I do, and I will write some of the minor sections. After a couple of months, our slowly-evolving .tex file contains a proof, and a few missing sections for me to write.

Yet again... time passes. Then I finally find the time to read the current version, including margin notes as to what Yair is uncomfortable with, or where he feels we might need to change. But I realize that the only way for me to understand this paper better, is if I just rewrite everything from scratch (also, I am very strict about \(\rm\LaTeX\) in the papers I write, just ask Yair). So we do that. I restructure some of the proofs, and we find a few mistakes that were hiding there. But ultimately, we finish the paper, and after rounds and rounds of proof reading, we post it online, and I write this post.

Sheesh. That's only the first project... well, the second project sort of merged in there, so I don't have to write about it from scratch. But still... in the fall semester at the Newton Institute at Cambridge, back in October 2015, I meet Arthur Apter, who tells me about the following problem: We know that AD proves that there are two consecutive measurable cardinals; starting from any not-yet-inconsistent assumptions, can we get that as a symmetric extension? He also adds immediately that Bull proved a no-go theorem that implies that taking two measurables and doing a symmetric collapse of the top one to be the successor of the other won't work. For a measurable to have a measurable successor, we need to violate choice below the first measurable. There is no way around it.

So of course I bring this back to Yair, and we start working on this project simultaneously to the first project, with more or less similar timelines. Every now and then we meet and talk about it, but mostly it stays on the back burner. Yair proposes using supercompact Radin forcing to proceed. We even get what we think is a new result: a successor of a measurable cardinal can have countable cofinality. And again this takes the back burner. The time is now February 2017, maybe one of the most stressful periods of my life so far, where I put all my energy towards finishing my thesis.

Time passes by, both Yair and I submit our theses. Finally. I am about to leave to Europe, and my journey begins at the European Set Theory Conference in Budapest, where I want to talk about our work. So I write Arthur once more, telling him that we have this and that so far, and I need some historical background on the problem for my short talk. And what Arthur writes reveals a misunderstanding back in Cambridge: the goal is not to get two successive measurable, as much as it is to get two successive measurable successor cardinals, which are not \(\omega_1\) and \(\omega_2\). It turns out that our work was already done by Hugh Woodin (plus collapsing the whole thing to \(\omega_1\)) in the early 1980s and later appeared in a paper of Arthur with James Henle. So all of our work is not only known, but was also published back in the 1980s. Still, Uri Abraham gives a few words of comfort, telling us that if we managed to come up entirely on our own with a proof of Woodin, then this is something to feel good about, we've done a good job.

But we want more. My talk shifts from successors of measurable cardinals to successors of critical cardinals. The day after my talk, Yair finds me on one of the coffee breaks and says that he thinks that we can apply the lifting criterion to the model with a measurable and a singular successor. He is not sure that it works for a measurable successor, but it might. So this joins into the fray of the first project. And for a while we even think we can get a critical cardinal with a measurable successor, at least until I do the rewriting process, and we notice a few missing points which are blatantly false in the case of a measurable successor... so at least for now, that problem still eludes us.

Yeah, so I wanted to put parts of this in the introduction, because I always feel that some context on the research process is a good thing to have in a paper. But this approach keeps getting bad feedback from people (and/or referees). So it's a good thing that I have a blog.

Well. Next time, on the "research adventures" I will write on the process of another paper in the back burner... this time, with David Asperó, about PFA and the axiom of choice. Now it is time for you to go and read the paper and send us some comments!

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