I saw a few papers on arXiv recently that were very clearly prepared by asking ChatGPT or some other GenAI to analyse an idea and write a paper. Of course, the papers contained obvious mistakes, but obvious to whom? It is very well-known to experts on any topic who interacted with GenAI that one of the reasons they are so terrible is their need to appease the user. This is such a wonderful…
I was recently reminded of an old answer I posted on Math.SE with advice on how to give a talk . I wrote that answer in 2012 when I was only finishing my masters, and while I had given a couple of seminar talks by that points, it's 13 years later and I look at this post and it looks like the advice a toddler gives an infant. And so, now, with much more experience, I figured that it's a good time…
I spent the past few months training a few neural nets on large cardinal axioms and related results. I've put the notes in the Papers page. More seriously, of course, I gave a course on large cardinal axioms during the spring semester. It wasn't meant to be particularly deep, and I am sure that other resources will do the material justice, but I figured it's worth putting this up there. Continue…
This is a piece of advice that I found myself giving to many early career researchers, students, and colleagues supervising and advising those as well. For years, actually. A mathematician, the joke says, is a blind man, in a dark room, searching for a black cat that isn't there. I don't know about that, but I think we can still agree that a researcher, in most fields, is sailing a boat, at night,…
Well, that was quite the week! As the last few posts had possibly clued you in, we hosted a conference in Leeds this past week. We had around 50 participants, including the speakers and poster presentations. I am very grateful for everyone who came here, those that spoke and did such a wonderful job, the posters that were so fascinating, and the audience that asked questions. I'm really grateful…
Continuing from last two posts, this is a final reminder to register to 120 Years of Choice . Registration ends next week on the 20th of June . We are very excited about the speakers and the poster presentations! The titles and abstracts are now available through our website (a few are TBD, but should be filled in soon enough). Continue reading...
Continuing from the previous post, we have a website for the conference now , and you can now register for the conference. We are planning to have two poster sessions, and we might be able to offer some financial support for people who are presenting posters. We will prioritise early career researchers, and given the budget limitations we might not accept all poster submissions. Continue…
We are happy to make the initial announcement for the "120 Years of Choice" conference that will take place in Leeds between the 8th and 12th of July, 2024 ! We will have a proper website, poster, and the whole shebang soon enough. But in the meantime, here is the list of confirmed speakers, in no particular order, for you to enjoy! Continue reading...
It was foretold of a legendary scientist, one who would master all of mathematics, all of physics, all of chemstiry, all of biology, some of engineering, bits of psychology, and none of economics. Truly, they were a Master of Science, complete with an M.Sc. and all that. One night, drinking around a campfire with their students, one of them asked: "Master, what is best in science?" Continue…
So, I spent the last year training a bunch of neural nets how to do forcing, with and without choice, how to work with symmetric extensions, and how to force over symmetric extensions. It was pretty damn good. If it wasn't clear already, I gave a year-long course here in Leeds on these topics. You can find the notes on the Papers page . Continue reading...
Oddly enough, one of the questions I hear from starting Ph.D. students is "how do you prove theorems ?", so let's talk about that. I'm saying "oddly enough", by the way, because first of all, I am someone they come to with this question, and in my mind I had just finished my Ph.D. (no, do not tell me it's been six years since I submitted my thesis), and secondly I remember having similar thoughts…
Sometimes, as a man in mathematics, the topic of diversity comes up. Either discussing with colleagues, or discussing with friends, or whatever. How can we bring more women to mathematics, how can we bring more people of colour to mathematics, how can we bring more people to mathematics? One answer is kind of obvious. We need more role models. Especially in a culture where for a very long time…
Nothing. Nothing is wrong with that movie. If you haven't watched it in the last couple of weeks, go and watch it tonight. Continue reading...
Which answer is the most correct? All of the above. All of the above. All of the above. All of the above. None of the above. Continue reading...
I got an email a few days ago from Lucas Polymeris, a Chilean student, who asked me a very nice question. I want to go through the journey I took from that question to the answer. Before we get to the question, let's review some definitions. Continue reading...
It's been a great ride at the University of East Anglia. I've come there as a postdoc and left with a permanent job. In today's climate, that's nothing to scoff at. But this is the end of this part of the journey. I'm leaving UEA and soon I will be leaving Norwich. Come the morning, I will officially be a University Academic Fellow at the University of Leeds. In Leeds, as you might have guessed at…
No, this isn't some personal blog post. Derek Muller of the Veritasium fame had posted a new video this afternoon. I'm going to spoil the crap out of the video, so you might want to watch it before reading on. Continue reading...
Well, it seems that there's a new paper about Flow and the Partition Principle on arXiv . This time claiming to prove that in ZF+Atoms the Partition Principle does not imply the Axiom of Choice. I'll start reading the paper and post live commentary in a couple of days, but at the moment, this is to let you know that I took notice. There seem to be some overlap with the previos paper and I hope…
Recursion n. : (see recursion). As the joke goes. But that's actually a misnomer, since that would be an ill-founded definition, which is exactly the point where you can't do a recursive definition. I'm not here to analyse that joke, though. I'm here to talk about something else. Some time ago, I was looking for something, and I couldn't even tell you what, and I came across an algorithm for…
I know. I already did a looking back post last year when the decade was coming to its inevitable end. But I couldn't pass on a post titled "Hindsight: 2020". And this year was a good year for reflection, not just compactness. So, with the winter solstice just behind us, we can say that this is the dawn of 2021. Let's look back. I know, yes, it's been a crap year with the pandemic, and everything…
My new postdoc, Jiachen Yuan, suggested it will be a good idea to have a one-day workshop on Prikry forcing. And I agreed, so we're doing this. You can find more details on the website right here . Here are the highlights: We'll do this online on December 14th; there are four invited speakers: Alejandro Poveda (HUJI), Tom Benhamou (TAU), Sittinon Jirattikansakul (CMU), and Chris Lambie-Hanson…
So. Just over two weeks ago a paper on arXiv claimed the proof that the Axiom of Choice does not follow from the Partition Principe in \(\ZF\). This is quite a claim, coming out of left field and laying the ground for a new theory called \(\Flow\). I spent two weeks reading the paper carefully, documenting my efforts in the previous post and on Twitter (where it is now a whole mess that is…
Some of you already saw this new preprint on arXiv , and some of you even emailed me about it. I'm reading the paper, but I decided to do something drastic and join Twitter , temporarily , so I can more easily have discussions about this paper. I will update this post on occasion while I read it, to reflect new understanding, kind of like a live journal, if you will. Continue reading...
Back in autumn 2015 I took a functional analysis course with Prof. Matania Ben-Artzi, and he let me write a term paper about uses of the axiom of choice in functional analysis for my final grade. One year later, in October 2016, I finally posted the note here . It then received some feedback from some people, and about a year after that I posted a small revision. Earlier this week I suggested my…
As some of you may already know, I was recently awarded a UKRI Future Leaders Fellowship. This is a well-funded project, which is why I can afford hiring postdocs and (once the pandemic is over) organise events and travel. When I was writing the proposal I was talking to one of the good people at the university about set theory. She had a Ph.D. in biology, so she had some vague idea what is a set,…
I recently posted an answer on MathOverflow where I explained a bit about the approaches to forcing in the literature, at least as I experienced them. Hanul Jeon took some of these words, and made them into a real nice comic. Originally appearing on his Twitter account . He was kind enough to let me post it here with his permission. Continue reading...
Three years ago I was looking for jobs, I was at the last stretch of my PhD and without clear prospects on what's next. Now I am offering jobs. Unfortunately, this is not as simple as me just looking at some people's emails and choosing from them. The university has a rigorous and exhausting hiring process involving applying online, shortlisting, interviews, whatnot. Continue reading...
I am not here to solve racism. I am not here to solve discrimination. I can't do that. I'm just an early career mathematician, working on very impractical ideas whose influence on society is immeasurably small and far away. (That is not to say that these things are not important. They are). Oddly enough, I am moved to write this by a Ben & Jerry's Silence is NOT an Option campaign. Continue…
Those who know me in real life will know that I enjoy cooking. I particularly enjoy cooking pasta. Most authentic Italian recipes are so simple (algio e olio anyone?) that it's just wonderful. I was cooking dinner today, bucatini alla matriciana if you must know, and I realised that cooking pasta and writing papers have nothing in common. Exactly one of those things is enjoyable, and it is not the…
Last night someone asked a question on Math.SE regarding a lemma used in proving certain chain conditions hold when iterating forcing with finite support. The exact details are not important. The point is that the authors, almost everywhere, regarded this as a trivial case. Indeed, in my answer I also viewed this as trivial. It was tantamount to the claim: If \(\cf(\alpha)\neq\cf(\kappa)\), then…
Oh, have I been waiting to tell you something... Yes, I am a Future Leaders Fellow. But as my three regular readers know, this blog is not about announcements, it's about my experience. In early January 2019, I was told that I can try and apply for a new scheme in the United Kingdom called "Future Leaders Fellowship". At the time not a lot was known about it, the first round winners were due to be…
I've decided to design a small "choose your own adventure". For fatalists. You can also play the interactive version here! . This was originally "in-blog", but I decided that the interactive version is a bit more interesting. Enjoy it while it lasts! Continue reading...
One of the classic results of Sierpinski is that if there are as many countable sets of reals as there are reals, then there is a set which is not Lebesgue measurable. (You can find a wonderful discussion on MathOverflow .) This is fact is used in the paradoxical decomposition theorems (which I often enjoy bringing up as a counter-argument to bad arguments that the Banach–Tarski paradox implies we…
With most people under quarantine, I spent some time going over older files in my computer. Exercise sheets, notes, whatever. Several years ago, when I was a teaching assistant for Itay Kaplan on Mathematical Logic 2 (incompleteness and basic model theory), I had put "bonus questions" in most homework sheets. Here are a few, translated from Hebrew. (Week 2, arithmetic hierarchy,…
The decade is drawing to a close, and while it is entirely arbitrary, it's a great excuse to look back at this decade. At the end of 2009 I started my senior year as an undergraduate. I both read the first part of "Introduction ot Cardinal Arithmetic" to get a hold on the basics of set theory, and also took my first course on set theory (I'm omitting the introductory course from my freshman year…
The meeting in Norwich is over. Here are my thoughts. It felt haphazard, without a concrete plan. And that was great. In the first day, Tadatoshi Miyamoto and David (Asperó) presented two problems in the morning and in the early afternoon. We then had a discussion about them, and it was just a general discussion, that went very well. Continue reading...
Methods in Higher Forcing Axioms (or MEHIFOX, for short) is an experimental workshop hosted in Norwich by David Asperó and myself. You can find the website, right here . This workshop is sponsored by the London Mathematical Society, and the School of Mathematics in UEA. The idea is to have a workshop, where we actually work. This is contrary to the normal use of "workshop" (in set theory, at…
Everybody wants to improve their mathematical skills! And quickly, too! Since it's so hard to do just that, I've written down some quick and dirty hacks for quickly improving your mathematical skills! 1. Get a graduate-level degree in mathematics! Getting a PhD in mathematics is not really about getting the PhD itself. It's more about getting much better at learning mathematics. So if you get a…
I have posted two new lecture notes online. The one is from a course in functional analysis I took in the autumn of 2015/16 with Prof. Matania Ben-Artzi, and the second is from the course I taught in axiomatic set theory in the autumn of 2016/17. Just as a general caveat for the set theory notes, since all the students in the course were also my students in the basic set theory course that I…
I have often seen people complain about Replacement axioms. For example, this MathOverflow question , or this one , or that one , and also this one . This technical-looking schema of axioms state that if \(\varphi\) defines a function on a set \(x\), then the image of \(x\) under that function is a set. And this axiom schema is a powerhouse! It is one of the three component that give \(\ZF\) its…
I just posted another problem in the problems page . The prize, by the way, is a bottle of port wine, or equivalent. And I truly hope to make good on that prize. In another problem there, coming from a work with David Asperó, we asked if an \(\omega_2\)-closed forcing must preserve the property of being proper. Yasou Yoshinobu provided us with a negative answer based on Shelah's "Proper and…
Those who know me, also know my strong liking of the amazing Spreewaldhof Get One! pickles (go, get one!). I got one as a present from a friend who visited Germany, and after that, I started obsessing over them. I found them in Vienna on my first visit (with the help of Jakob Kellner), and they became the standard Viennese gift from visitors to Jerusalem for me. Mozart chocolate balls for the…
We all know and love Cohen's first model where the axiom of choice fails. It is the O.G. symmetric extension. But Cohen didn't invent the idea on his own, he used Fraenkel's ideas from his work on set theory with atoms and permutation models. The two results, however, are significantly different. Fraenkel's construction does not affect sets of ordinals, in particular the real numbers can still be…
Talks. Giving talks. We usually don't give talks about past research. Talks are meant to present recent research, things you've just finished, that you're finishing right now, that you've found out! So often times, it seems, it is very tempting to talk about theorems that you haven't finished writing their proofs in full. Usaully, we put "work in progress" to indicate that this is something not…
Suppose that a parameter formula defines an inner model. Does that inner model satisfy choice? Well, obviously, if choice failed then the answer is no, just by taking \(x=x\). But what if we remove that option. Namely, if the inner model is not the entire universe, then choice holds. Continue reading...
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…
I've decided to have a list of open problems on my site. I am no Erdős, nor Hilbert, nor Knuth. But I want my own problems page, and it's my site. So to celebreate the new website, I created just that. For the first couple of problems, I've chosen to focus on the axiom of choice. And I don't think that I have much choice, but to keep that interest running. But I can promise that this is not the…
Welcome to my new website! It is a static website, because I am tired of the WordPress format for a long long time now. So for the occasion, I also got a new domain, karagila.org . Isn't this nice? The only domain and all the links should work, at least for the foreseeable future. So there's nothing to worry about linkrot for now. But please do update your links! Continue reading...
I had a recent back and forth on Math.SE with a user that asked whether or not some exercise he found in some textbook is correct. The OP asked not to provide a proof, but rather to confirm if this statement is at all provable. When I asked why not just try and prove the damn thing, the reply was that if there is a typo or a mistake and the statement is in fact not provable, then they would have…
Finally! Six months after I had turned in my dissertation, I have finally received the approval on the damn thing. Continue reading...