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

120 Years of Choice: Registration is open

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.

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120 Years of Choice: First Announcement

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!

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What is best in science?

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?"

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I trained neural nets to do forcing and symmetric extensions

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.

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How to prove theorems?

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 when I started, and I look back and find them odd. Let me also clarify, that when I ask the person what do they mean, they usually say the same thing "how do you know that you've proved a theorem and not a proposition or a corollary?", which I usually understand as "what results are worth publishing?"

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Safe spaces

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 "girls can't do math" nonsense was a common way of thinking, or people of colour were not given access to good education, you end up with a culture where people just don't see themselves as someone who might be able to be a mathematician, or a scientist, or whatever. By hiring good role models we are sending the message "no, that's all a massive pile of bullshit, of course you can do this!" and that's good. Hiring more diversely also helps to expand point of views, it helps to expand ways in which people around you think, and it also introduces you, or your staff, or your colleagues, or your students, to someone from a different background, which is an incredibly valuable thing outside of acadmia as well.

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Everything Wrong With The Princess Bride

Nothing. Nothing is wrong with that movie. If you haven't watched it in the last couple of weeks, go and watch it tonight.

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Multiple Choice

Which answer is the most correct?

  1. All of the above.
  2. All of the above.
  3. All of the above.
  4. All of the above.
  5. None of the above.
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Question on Dually Dedekind-infinite sets

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.

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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 this point. My two postdocs are coming with, which is great, and there might be a Ph.D. studentship call sometime later this spring, watch out for announcements. But we'll see about that. The first step is to sette in and get back to work.

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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.

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Flow and the Partition Principle II (2 updates)

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 that this time it will go faster.

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Factorial algorithms and recursive thoughts

Recursion: (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 computing the factorial of a number by recursion. Let's review the standard way recursive algorithm first:

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Hindsight: 2020

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 closes down and so on. But it's not all bad. I got a huge fellowship, which might not seem as much maybe, but it shows that set theory is not dead, and in fact very much alive and well in the UK. I hired two very smart postdocs, and I started supervising some very smart students (and hopefully more to come soon). I even had two interviews, one for the fellowship, which was great, and another for a permanent job in another university which also went great (even though I didn't get it), which I mainly wanted to go through as an experience. I already had my fellowship at that point.

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Prikry Forcing Online

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:

  1. We'll do this online on December 14th;
  2. there are four invited speakers: Alejandro Poveda (HUJI), Tom Benhamou (TAU), Sittinon Jirattikansakul (CMU), and Chris Lambie-Hanson (VCU);
  3. there are two sessions of mini-talks (10 minutes each), open for any early career researcher wanting to give a quick overview or present a very short proof on any set theoretic topic.
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Flow and the Partition Principle: Conclusions

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 impossible to read and understand in a reasonable way). This post is to serve as a more coherent and cohesive conclusion to this process.

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Flow and the Partition Principle (6 updates)

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.

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Zornian Functional Analysis coming to arXiv!

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 note as a source for the proof that the Baire Category Theorem is equivalent to Dependent Choice. After doing that, I stumbled upon an errata by Theo Bühler and Dietmar A. Salamon to their Functional Analysis book, which refers to my write-up.

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YouTubers Wanted

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, but not really a proper idea what it is that I do. So I explained the very basics of set theory and cardinals, ordinals, and the axiom of choice.

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Guest comic: Foundations of Forcing

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.

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Want jobs?

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.

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Speak up

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.

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What can you learn about writing papers from cooking pasta?

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 writing part of papers (I do love the research part, of course).

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On "trivial" statements

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 every subset of \(\alpha\) of size \(\kappa\), contains a subset of size \(\kappa\) which is bounded.

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Yes! Future Leaders Fellowship!

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 announced, so it was unclear what are the success rates, or the "desired candidates" might be.

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Choose your own adventure (redux)

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!

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Countable sets of reals

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 need to accept that all sets are measurable as an axiom):

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Bonus questions

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, \(\beta\)-function)

In a single paragraph, analyse the squire's approach to the existential crisis of the knight, Antonius Block, in Ingmar Bergman's "The Seventh Seal". What is the role of the knight's wife in support and contrast to the squire's?

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Looking Back at the 2010s

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 since that one covered very very basic set theory). I've studied with the wonderful Matti Rubin, and it was a fantastic course. Too bad that it focused almost solely on the axioms (i.e. how the axioms are not provable from others, etc.) and that we only spent a short time dealing with actual set theoretic topics (e.g. Solovay's theorem on partitions of stationary sets, etc.)

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Methods in Higher Forcing Axioms: The inevitable conclusion

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.

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Methods in Higher Forcing Axioms

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 least, but I believe in most mathematical areas) which means a very small conference where almost all the participants are also speaking.

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7 Easy Hacks to Improve Your Math Skills

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 PhD in mathematics it will help you better your ability to study more mathematics and improve your skills.

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New notes online!

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 taught with Azriel Levy (yes, that Azriel Levy, and yes it was quite an awesome experience) and there I managed to cover some fairly nontrivial things in that course, these notes might feel as if there are some gaps there, or that I skip here and there over some information.

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In praise of Replacement

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 power (the others being power set and infinity, of course).

You'd think that people in category theory would like it, from a foundational point of view, it literally tells you that functions exists if you can define them! And category theory is all about the functions (yes, I know it's not, but I'm trying to make a point).

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Preserving Properness

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 Improper Forcing" XVII Observation 2.12 (p.826). Take \(\kappa\) to be uncountable, by forcing with \(\Add(\omega,1)\ast\Col(\omega_1,2^\kappa)\) and appealing to the gap lemma, \((2^{<\kappa})\) is a tree with only \(\aleph_1\) branches. It can therefore be specialized by a ccc forcing in that model. The iteration of these three forcing (Cohen real, collapse, specialize) is clearly proper. But now by forcing with \(\Add(\kappa,1)\) we must in fact violate the properness of this forcing, which was defined in the ground model, since the new branch is also generic for the tree and will therefore collapse \(\omega_1\).

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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 rest, a bunch of pickles for me. Thanks, Viennese people!

So naturally I could not skip on this video.

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Cohen's Oddity

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 well-ordered in his models. Cohen's work, however, directly breaks that. The Dedekind-finite set added is a set of reals. In particular, the reals cannot be well-ordered no more.

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Five Star Theorems

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 fully verified, not fully vetted (at the very least by ourselves).

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Definable Models Without Choice

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.

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Critical Cardinals

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.

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Open Problems!

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 only type of problems that I will add there.

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New website!

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, 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!

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In praise of failure

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 wasted their time trying an impossible task.

Well. Actually no. When I was a dewy eyed freshman, I had taken all my classes with 300 students from computer science and software engineering (Ben-Gurion University has changed that since then). Our discrete mathematics professor was someone who was renowned as somewhat careless when it comes to details in questions and stuff like this (my older brother took calculus with the same professor about ten years before, one day he didn't show up to class, when my brother and two others went to see if he is at his office, he was surprised to find out that today is Tuesday).

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Trust me, I'm a doctor!


Six months after I had turned in my dissertation, I have finally received the approval on the damn thing.

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Some thoughts about teaching introductory courses in set theory

Dianna Crown, the physics woman on YouTube, has posted a video where she is interviewed by her editor about why and how she found herself majoring in physics in MIT.

Here is the video:

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Dangerous knowledge in the Information Age

Back in the days of yore, if one wanted to know mathematics, one would have to go to the university and take a course; or hire a tutor; or go to the library and open a book and learn on their own.

And that was fine. All three options are roughly equivalent, in the sense that they present you the material in a very structured way (or they at least intend to). You don't reach the definition of \(\aleph_0\) because you defined what is equipotency and cardinality. You don't reach the definition of a derivative before you have some semblance of notion of continuity. Knowledge was built in a very structural way. Sometimes you use crutches (e.g. some naive understanding of the natural numbers before you formally introduce them later on as finite ordinals), but for the most part there is a method to the madness.

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The transitive multiverse

There are many discussions on the multiverse of set theory generated by a model. The generic multiverse is given by taking all the generic extensions and grounds of some countable transitive model.

Hamkins' multiverse is essentially taking a very ill-founded model and closing it to forcing extensions, thus obtaining a multiverse which is more of a philosophical justification, for example every model is a countable model in another one, and every model is ill-founded by the view of another model. The problem with this multiverse is that if we remove the requirement for genericity, then everything else can be satisfied by the same model. Namely, \(\{(M,E)\}\) would be an entire multiverse. That's quite silly. Moreover, we sort of give up on a concrete notion of natural numbers that way, and this seems a bit... off putting.

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Strong coloring

I am sitting in the 6th European Set Theory Conference in Budapest, and watching all these wonderful talks, and many of them use colors for emphasis of some things. But yesterday one of the talks was using "too many colors", enough to make me make a comment at the end of the talk after all the questions were answered. Since I received some positive feedback from other people here, I decided to write about it on my blog, if only to raise some awareness of the topic.

There is a nontrivial percentage of the population which have some sort of color vision deficiency. Myself included. Statistically, I believe, if you have 20 male participants, then one of them is likely to have some sort of color vision issues. Add this to the fairly imperfect color fidelity of most projectors, and you get something that can be problematic.

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Moment of Zen

When one is ascending a difficult path uphill, it is a good idea to keep your eyes on the path as you move forward. However, it is not a bad idea to stop sometimes, look back, and appreciate the beauty of the ground you have already covered.
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What a long strange trip it's been...

As some of you may have noticed, I don't use this blog to write about my papers in the "traditional way" math bloggers summarize and explain their recent work. I think my papers are prosaic enough to do that on their own. I do use this blog as an outlet when I have to complain about the arduous toil of being a mathematician (which has an immensely bright light side, of course, so in the big picture I'm quite happy with it).

This morning I woke up to see that my paper about the Bristol model was announced on arXiv. But unbeknownst to the common arXiv follower, this also marks the end of my thesis. The Hebrew University is kind enough to allow you to just stitch a bunch of your papers (along with an added introduction) and call it a thesis. And by "stitch" I mean literally. If they were published, you're even allowed to use the published .pdf (on the condition that no copyright infringement occurs).

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Stationary preserving permutations are the identity on a club

This is not something particularly interesting, I think. But it's a nice exercise in Fodor's lemma.

Theorem. Suppose that \(\kappa\) is regular and uncountable, and \(\pi\colon\kappa\to\kappa\) is a bijection mapping stationary sets to stationary sets. Then there is a club \(C\subseteq\kappa\) such that \(\pi\restriction C=\operatorname{id}\).

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Got jobs?

Good news! I'm about to finish my dissertation. Hopefully, come summer I will be Dr. Asaf Karagila.

So the next order of business is finding a position for next year. So far nothing came up. But I'm open to hearing from the few readers of my blog if they know about something, or have some offers that might be suitable for me.

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Farewell, Matti

My mentor, teacher, mathematical confidant and generally good friend, Matti Rubin passed away this morning. Many of the readers here know him for his mathematical work, many knew him as a friend as well, or as a teacher.

Matti was a kind teacher, even if sometimes over-pedantic.

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Mathematical philosophy on YouTube!

If you follow my blog, you probably know that I am a big fan of Michael Stevens from the VSauce channel, who in the recent year or so released several very good videos about mathematics, and about infinity in particular. Not being a trained mathematician, Michael is doing an incredible task.

Non-mathematicians often tend to be Platonists "by default", so they will assume that every question has an answer and sometimes it's just that we don't know that answer. But it's out there. It's a fine approach, but it can somewhat fly in the face of independence if you are not trained to think about the difference between true and provable.

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Some thoughts about teaching advanced set theory

I've been given the chance to teach the course in axiomatic set theory in Jerusalem this semester. Today I gave my first lecture as a teacher. It went fine, I even covered more than I expected to, which is good, I guess. I am also preparing lecture notes, which I will probably post here when the semester ends. These predicated on some rudimentary understanding in logic and basic set theory, so there might be holes there to people unfamiliar with the basic course (at least the one that I gave with Azriel Levy for the past three years).

Yesterday, however, I spent most of my day thinking about how we---as a collective of set theorists---teach axiomatic set theory. About that usual course: axioms, ordinals, induction, well-founded sets, reflection, \(V=L\) and the consistency of \(\GCH\) and \(\AC\), some basic combinatorics (clubs, Fodor's lemma, maybe Solovay or even Silver's theorem). Up to some rudimentary permutation.

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Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice

Back in the fall semester of 2015-2016 I had taken a course in functional analysis. One of the reasons I wanted to take that course (other than needing the credits to finish my Ph.D.) is that I was always curious about the functional analytic results related to the axiom of choice, and my functional analysis wasn't strong enough to sift through these papers.

I was very happy when the professor, Matania Ben-Artzi, allowed me to write a final paper about the usage of the axiom of choice in the course, instead of taking an exam.

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In praise of some history

Teaching pure mathematics is not a trivial thing. You have to overcome the several barriers that were constructed by the K12 education that mathematics is a bunch of "fit this problem into that mold".

I recently had a chat with James Cummings about teaching. He said something that I knew long before, that being a good teacher requires a bit of theatricality. My best teacher from undergrad, Uri Onn, had told me when I started teaching, that being a good teacher is the same as being a good storyteller: you need to be able and mesmerize your audience and keep them on the edge of their seats, wanting more.

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Constructive proof that large cardinals are consistent

I am not a Platonist, as I keep pointing out. Existence, even not in mathematics, is relative and confusing to begin with, so I don't pretend to try and understand it in a meaningful way.

However, we have a proof, a constructive proof that large cardinals are consistent. And they exist in an inner model of our universe.

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Some thoughts about "automated theorem searching"

Let me begin by giving a spoiler warning. If you haven't watched "The Prisoner" you might be spoiled about one of the episodes. Not that matters, for a show from nearly 40 years ago, but you should definitely watch it. It is a wonderful show. And even if you haven't watched it, it's just one episode, not the whole show. So you can keep on reading.

So, I'm fashionably late to the party (with some good excuse, see my previous post), but after the recent 200 terabytes proof for the coloring of Pythagorean triples, the same old questions are raised about whether or not at some point computers will be better than us in finding new theorems, and proving them too.

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Iterating Symmetric Extensions

I don't usually like to write about new papers. I mean, it's a paper, you can read it, you can email me and ask about it if you'd like. It's there. And indeed, for my previous papers, I didn't even mention them being posted on arXiv/submitted/accepted/published. This paper is a bit different; but don't worry, this is not your typical "new paper" post.

If you don't follow arXiv very closely, I have posted a paper titled "Iterating Symmetric Extensions". This is going to be the first part of my dissertation. The paper is concerned with developing a general framework for iterating symmetric extensions, which oddly enough, is something that we didn't really know how to do until now. There is a refinement of the general framework to something I call "productive iterations" which impose some additional requirements, but allow greater freedom in the choice of filters used to interpret the names. There is an example of a class-length iteration, which effectively takes everything that was done in the paper and uses it to produce a class-length iteration—and thus a class length sequence of models—where slowly, but surely, Kinna-Wagner Principles fail more and more. This means that we are forcing "diagonally" away from the ordinals. So the models produced there will not be defined by their set of ordinals, and sets of sets of ordinals, and so on.

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Syntactic T-Rex: Irregularized

One of my huge pet peeves is with people who think that writing \(1+2+3+\ldots=-\frac1{12}\) is a reasonable thing without context. Convention dictates that when no context is set, we interpret infinite summation as the usual convergence of a series, namely the limit of the partial sums, if it exists (and of course that \(1+2+3+\ldots\) does not converge to any real number). However, a lot of people who are [probably] not mathematicians per se, insist that just because you can set up a context in which the above equality holds, e.g., Ramanujan summation or zeta regularization, then it is automatically perfectly fine to write this out of nowhere without context and being treated as wrong.

But those people forget that \(0=1\) is also very true in the ring with a single element; or you know, just in any structure for a language including the two constant symbols \(0\) and \(1\), where both constants are interpreted to be the same object. And hey, who even said that \(0\) and \(1\) have to denote constants? Why not ternary relations, or some other thing?

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MM70: YouTube links!

During the first day of the conference we realized that it might be a good idea to get the lectured videoed, so we quickly set up the videos for the second and third day. With the exception of one speaker who asked not to be videoed, you can find all the lectures from the second and third day of the conference in this YouTube Playlist.


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Quick update from Norwich

It's been a while, quite a while, since I last posted anything. Even a blurb.

I'm visiting David Asperó in Norwich at the moment, and on Sunday, the 12th, I will return home. It seems that the pattern is that you work most of the day, then head for a few drinks and dinner. Mathematics is eligible for the first two beers, philosophy of mathematics for the next two, and mathematical education for the fifth beer. Then it's probably a good idea to stop. Also it is usually last call, so you kinda have to stop.

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Vsauce on cardinals and ordinals

To the readers of my blog, it should come as no surprise that I have a lot of appreciation to what Michael Stevens is doing in Vsauce. In the past Michael, who is not a mathematician, created an excellent video about the Banach-Tarski paradox, as well another one on supertasks. And now he tackled infinite cardinals and ordinals.

You can find the video here:

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What I realized recently

I recently learned that diamonds are cut and polished with the dust of other diamonds. And I recently realized that success is cut and polished with the dust of failures.

In particular a successful mathematical idea is polished with the dust of the many failed ideas that preceded it.

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The Five WH's of Set Theory

I was asked to write a short introduction to set theory for the European Set Theory Society website. I attempted to give a short answer to what is set theory, why study it, when and how to study it and where to find resources.

You can find the article on the ESTS' website "Resources" page, or in the Papers section of my website.

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MM70: Travel Grants for Students!

The registration to Menachem Magidor's 70th Birthday Conference is still open!

If you happen to be a student and a member of the Association for Symbolic Logic, you can apply for an ASL travel award. For more information as to how, please see here. There's just enough time to still submit your request!

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Goodbye, Oren.

I recently heard the news that Oren Kolman passed away a couple of weeks ago.

Some of you may have known him through MathOverflow as "Avshalom" where he often appeared in the comments with generally useful references, and some of you may have known him in real life as a teacher or a colleague, or a student. Some of you may have even knew him as Eoin Coleman.

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Michael, you're awesome.

After so many terrible YouTube videos about math, about four months ago Michael Stevens made a really nice video about the Banach-Tarski (Banach-T-Rex) paradox. This video was made surprisingly well by someone who has little to none formal mathematical education, but certainly the desire and [at least basic] prowess to understand that perhaps things are not as simple in mathematics - especially when infinite objects are involved - and perhaps you can't just drop something on your audience in hope they view you as a magician. Instead, Michael tried to educate the viewers, in a fairly reasonable way, about infinite objects and the preliminaries needed for the Banach-Tarski paradox.

You can find that video right here:

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MM70: Registration is now open!

I am happy to announce, on behalf of the organizing committee, that the registration for Menachem Magidor's 70th Birthday Conference is now open!

MM70 Poster

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Cofinality and the axiom of choice

What is cofinality of a[n infinite] cardinal? If we think about the cardinals as ordinals, as we should in the case the axiom of choice holds, then the cofinality of a cardinal is just the smallest cardinality of an unbounded set. It can be thought of as the least ordinal from which there is an unbounded function into our cardinal. Or it could be thought as the smallest cardinality of a partition whose parts are all "small".

Not assuming the axiom of choice the definition of cofinality remains the same, if we restrict ourselves to ordinals and \(\aleph\) numbers. But why should we? There is a rich world out there, new colors that were not on the choice-y rainbow from before. So anything which is inherently based on the ordering properties of the ordinals should not be considered as the definition of an ordinal. So first let's recall the two ways we can order cardinals without choice.

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Don't worry about it

In a recent Math.SE question about the foundations of category theory without set theory, someone made a claim that \(\ZF\) makes it hard to learn mathematics, because in \(\ZF\) the questions "is \(\RR\subseteq\pi\)?" and "is \(\RR\in\pi\)?" can be phrased. They continued to argue that there are questions like whether or not hom-sets are disjoint or not, which are hard to explain to people who are "drunk on ZF's kool-aid".

So I raised a question in the comment, and got replies from two other people who kept repeating the age old silly arguments of what are the elements of \(\RR\times\RR\) or what are these or that elements. And supposedly the correct pedagogical answer is "It does not matter what are the elements of \(\RR\times\RR\)." With that I strongly agree, and when I taught my students about ordered pairs on the very first class of the semester, I made it very clear that there are other ways to define ordered pairs and that we only do that because we want to show that there is at least one way in which ordered pairs can be realized as sets; but ultimately we couldn't care less about what way they encode ordered pairs into sets, as long it is a "legal" way.

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How do you read a paper?

Some time ago I was talking to some people about how they read a paper. And I learned that I am somewhat significantly different from a lot of people. I spent some time thinking about it, and I arrived at some interesting conclusions.

So here is how I read a paper, and I'd like to ask you to think about how you read a paper, and why you read it this way.

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Banach-Tarski Banach-T-Rex

I had already written about anti-anti-Banach-Tarski arguments. But now the Mathematical T-Rex has something to say too.


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The Thing Explainer Challenge

Randall Munroe of the xkcd fame has a new book coming up where he explains various concepts using a small repository of "simple" words (this is based on this xkcd comic). He recently posted this blog post, where he reveals a word checker program that he wrote to help him with the task.

So I figured, why not use this for explaining mathematical theorems.

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Name that number

In the best TV show ever produced, Patrick McGoohan plays the mysterious No. Six. He lives in The Village, where former spies are held. The people there are essentially captive, and they all have numbers instead of names. But he is not a number! He is a free man!

We find a similar concept in Zelda's poem "Every man has a name" (לכל איש יש שם), which in Israel is closely associated with the Holocaust and with assigning numbers to people. But alas, we are all numbers in some database. Our ID numbers, employer number, the index under which you appear in the database. You are your phone number, and your bank account number. You are the aggregation of all these numbers. And more.

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How to solve your problems

Anyone who peruses mathematical Q&A sites, or had students come to office hours or send questions via other means (email, designated forums, carrier pigeons, or written on a note tied to a brick tossed into your office) knows the following statement: "I don't know where to begin", or at least one of its variants.

Richard Feynman, who was this awesome guy who did a lot of cool things (and also some physics (but I won't hold it against him today)), has a famous three-steps algorithm for solving any problem.

  1. Write the problem down.
  2. Think. Real. Hard.
  3. Write the solution down.

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Mathematical T-Rex

Ever explained to a "working mathematician" about the undecidability of the continuum hypothesis? I bet you too had felt like this T-Rex.

T-Rex: CH

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Is the Continuum Hypothesis a definite problem?

I am not a Platonist.

In general, while I do find it entertaining to think about god, afterlife, or a concrete mathematical universe, I find more comfort in the uncertainty of existence than I do in the likelihood that my belief is wrong, or in the terrifying conviction that comes along with believing in something (and everyone else is wrong).

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I don't like social media very much. I never really subscribed to the whole Friendster, MySpace, Facebook, Twitter, Google Wave, Google+, and what have you social network sort of approach that you need to have "friends" and "followers" and "follow" other people.

I always preferred to be the master of my domain. The king of my castle. But literally, not the Seinfeld euphemisms sense. In any case. I've been thinking about a page where I can post short thoughts about math, life and otherwise. The blog is not suitable, since I'm not going to add a post each time I have a new thought. So instead I've started a blurbs page. Each blurb has a number, and an anchored link that you can use in case you want to share it.

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Young Set Theory 2015

Have you heard? Young Set Theory 2015 will take place in Jerusalem! How exciting is that? Tomorrow (Monday, July 20th) is the last day for registration. This means that you have only a few hours to get yourself together and send an application!

If you are not on this list, you better hurry up to this application form and register! Come on, what are you waiting for???

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The rules of research

Here are the rules of research. Feel free to add your own.

  1. If it seems obvious, it's probably false as stated.
  2. If it seems obvious and true, it's probably false without additional hypotheses.
  3. If you think that you wrote a proof, you probably missed something obvious. See (1) and/or (2).
  4. You missed something obvious, see (1).
  5. When you go to see your advisor, suddenly all your thoughts align, and you find the solution.
  6. Two hours after finally talking with your advisor, you realize that your solution is obvious, therefore (1) or (2) apply.
  7. If you use forcing to prove the argument, then you probably missed some object being encoded generically.
  8. If you use forcing, and you didn't miss some crucial object, then you missed some other crucial object not being coded by the generic.
  9. When the truth is found to be lies, and all the joy within you dies...
  10. It's not false if you can force it.
  11. It's not true if you used the axiom of choice more than three times in the proof.
  12. It's not cheating if you asked a visitor to the university whose visit did not span longer than two weeks from the moment you asked them.
  13. If your question was about inner models, you may extend the above timespan to a month. Equally, if the question is about the axiom of choice, it should be shortened to a week.
  14. It's not considered unethical to make sacrifice in order to appease Mayan and Aztec gods. Just in case we got it wrong, and they're in charge of the mathematical universe.
  15. If it still seems obvious, you're probably right. It's still false, though.
  16. If you need six technical lemmas, whose proof is reduced to a single line (or just one lemma with an actual proof), then it's probably obvious. Unfortunately, see (1) and (2).
  17. If by some chance something is obvious, but you wrote out the proof, and it checks out, then it wasn't obvious at all.
  18. Remember what the dormouse said: feed your head.
  19. If you haven't watched Futurama in a while, then you're doing something wrong.
  20. Whatever happens, it's the other guy's fault. Also, see (1).
  21. I just work here, you know? I don't.
  22. Rolling a D20 die to determine the truth value of a statement is the original algorithm behind proof verification software.
  23. When you hit the wall, and you're about to give up and decide that whatever you're trying to prove is false, see (4).
  24. The only proofs that write themselves are obvious proofs. If your proof is obvious, see (2) and (3).
  25. To be honest, it needs more cowbell.
  26. Seriously, you're gonna want that cowbell in your proof.
  27. See (1), (2) and (4).
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When the box means nothing

When assuming the axiom of choice the product topology and box the topology are quite different when considering infinite products. For example the Tychonoff product of countably many sets of three elements is compact, metrizable an all in all a very nice space. On the other hand, the box product is not separable or second countable at all.

But without the axiom of choice the world is indeed a strange place. This was posted as answer on math.SE earlier today.

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I need your help!

The account has been suspended, I'd like to thank everyone who helped! I have removed the comments posted by "Isa Bria" after the real Isa Bria has contacted me and asked to remove them.

We have verified, in the meantime, that the same person impersonating me on Quora is the one who used Isa's name in those comments.

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A problem and a possible solution

So closing in on my third year, and in theory I should finish my dissertation by next summer. This means that I should probably start the writing process around April (I'm a fast writer, what with having a quality keyboard and knowing LaTeX quite well).

But if I want to be sure that I can finish next year, I should probably omit one of the problems I originally wanted to solve; and keep that for later, unless it turns out to be particularly simple when I finish the rest.

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The Torture of Mathematical Research

In a manner more befitting to Edgar Allan Poe, Mathematics is a cruel and unforgiving mistress.

Mathematics will often dangle in front of you some ideas, and you will work them out, to find a mistake. Then you will go back to the beginning, find new ideas that she had in store, work those out and proceed only to find a mistake much later. Then you go back to the beginning, and you find yet another minor idea that was missing, and now when everything works you continue. But then you find another gap, and you have to go back to the beginning and hope to find yet another idea. And don't get me started on those ideas that you find not to work during all these searches.

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Turning Green!

Well, it's that special day of the year again. The holiest of days. The day we celebrate the patron of alcohol enthusiasts, Saint Patrick. So raise your whiskey glasses (my recommendation is Jameson 12 for those with deep pockets; Kilbeggan for those with shallow pockets), your Guinness pints, and wear green. Because tomorrow is all about soaking your brains in ethanol while listening to Irish folk songs, Irish punk rock (Thin Lizzy and Flogging Molly, for example) and other drinking songs.

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Existentialism II, like Colonel Kurtz

Last night I posted a strange story about a gecko and a moth.

It occurred to me today that this is a very Kurtzian story, if we take the Brando interpretation of Mistah Kurtz (he dead) in Apocalypse Now! (the Redux version is one of my favorite movies, I guess). In the movie Harrison Ford plays a tape where Kurtz is describing a snail crawling along the straight edge of a razor, crawling slithering, this is his dream, this is his nightmare.

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Spring has begun in Israel.

Yesterday was the first day where you could say that the weather is characteristically spring; and today (as well tomorrow) we are expected for a daytime heatwave and a nighttime cold weather (e.g. Beer-Sheva is expecting a whopping 31 degrees centigrade during the day, and 13 during the night).

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Much needed terminology, that isn't going to happen any time soon

One of the reasons I love set theory so much, and specifically choice related research, is that this is an extremely fertile ground for amusing terminology. We have forcing, cardinals, collapsing, we have all sort of gems and rodents at our disposal... we even have a swamp thing.

Here are a few terminological ideas that I doubt are going to be developed by anyone. But if you plan on doing something similar (or if my terminology inspires some proof) feel free to use these terms, and please let me know!

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On the Partition Principle

Last Wednesday I gave a talk about the Partition Principle in our students seminar. This talk covered the historical background of the oldest open problem in set theory, and two proofs that for a long time I avoided learning. I promised to post a summary of the talk here. So here it is. The historical data was taken from the paper by Banaschewski and Moore, "The dual Cantor-Bernstein theorem and the partition principle." (MR1072073) as well Moore's wonderful book "Zermelo’s Axiom of Choice" (which has a Dover reprint!).

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To Colloops a cardinal

This is nothing new, but it's a choice-y way of thinking about it. Which is really what I enjoy doing.

Definition. Let \(V\) be a model of \(\ZFC\), and \(\PP\in V\) be a notion of forcing. We say that a cardinal \(\kappa\) is "colloopsed" by \(\PP\) (to \(\mu\)) if every \(V\)-generic filter \(G\) adds a bijection from \(\mu\) onto \(\kappa\), but there is an intermediate \(N\subseteq V[G]\) satisfying \(\ZF\) in which there is no such bijection, but there is one for each \(\lambda\lt\kappa\).

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Huge cardinals are huge!

In a previous post, I gave a humorous classification of large cardinals, dividing them to large large cardinals and small large cardinals, and so on. In particular huge cardinals were classified as large large large large large cardinals. But how large are they? Not surprisingly, very large.

In case you forgot, \(\kappa\) is a huge cardinal if there is an elementary embedding \(j\colon V\to M\), where \(M\) is a transitive class containing all the ordinals, with \(\kappa\) critical, and \(M\) is closed under sequences of length \(j(\kappa)\).

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Anti-anti Banach-Tarski arguments

Many people, more often than not these are people from analysis or worse (read: physicists, which in general are not bad, but I am bothered when they think they have a say in how theoretical mathematics should be done), pseudo-mathematical, non-mathematical, philosophical communities, and from time to time actual mathematicians, would say ridiculous things like "We need to omit the axiom of choice, and keep only Dependent Choice, since the axiom of choice is a source for constant bookkeeping in the form of non-measurable sets".

People often like to cite the paradoxical decomposition of the unit sphere given by Banach-Tarski. "Yes, it doesn't make any sense, therefore the axiom of choice needs to be omitted".

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Ramsey cardinals are large large small large cardinals

There is no well defined notion for what is a large cardinal. In some contexts those are inaccessibles, in others those are critical points of elementary embeddings, and sometimes \(\aleph_\omega\) is a large cardinal.

But we can clearly see some various degrees of largeness by how much structure the existence of the cardinal imposes. Inaccessible cardinals prove there is a model for second-order \(\ZFC\), and Ramsey cardinals imply \(V\neq L\). Strongly compact cardinals even imply that \(\forall A(V\neq L[A])\).

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My love-hate relationship with forcing

Forcing is great. Forcing is an amazing method. If you can think about it, then you can probably force to make it happen. All it requires is some creativity and rudimentary understanding of the objects that you are working with.

Forcing is horrible. If you can think about it, you can encode it into generic objects. If you can't think about it, you can encode it into generic objects. If you think that you can't encode it into generic objects, then you are probably wrong, and you can still encode it into generic objects.

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How Fields Became "Nobel"

Here is some interesting piece of mathematical history: How the Fields medal went from "Soviet award" to "Mathematical Nobel".

How Math Got Its 'Nobel'.

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This is not a blog post.

This is not a blog post.


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No uniform ultrafilters

Earlier this morning I received an email question from Yair Hayut. Is it consistent without the axiom of choice, of course, that there are free ultrafilters on the natural numbers but none on the real numbers?

Well, of course that the answer is negative. If \(\cal U\) is a free ultrafilter on \(\omega\) then \(\{X\subseteq\mathcal P(\omega)\mid X\cap\omega\in\cal U\}\) is a free ultrafilter on \(\mathcal P(\omega)\). But that doesn't mean that the question should be trivialized. What Yair asked was actually slightly subtler than that: is it consistent that there are free ultrafilters on \(\omega\), but no uniform ultrafilters on the real numbers?

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Why Carl Sagan was better than Neil deGrasse Tyson, and from the most of us too

I've recently watched the finale of Cosmos, the new version, presented by Neil deGrasse Tyson. It was a very nice series which seem to push forward the fact that science is based on not knowing, rather than knowing, and the will to know. No, not will, the need to know. We need to know, and this is why we go on searching the answers to questions that haunt us.

Neil deGrasse Tyson pushed a lot on the point that we really push the planet to its limits, and we might be close to the point of no return from which there is only a terrible Venus-like fate to this planet. And that is an important issue, no doubt.

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Forcing. This Has To Stop.

Most, if not all, set theorists at one point or another were asked by a fellow mathematician to explain how forcing works. And many chose to give as an opening analogy field extensions. You can talk about how the construction of an algebraic closure is a bit similar, since the generic filter is a bit like the maximal ideal you use to make this construction; or you can talk about adding a transcendental number and the things that change as you add it.

But both these analogies would be wrong. They only take you so far, and not further. And if you wish to give a proper explanation to your listener, there will be no escape from the eventual logic and set theory of it all. I stopped, or at least I'm doing my best, using these analogies. I do, however, use the analogy of "How many roots does \(x^{42}-2\) has?" as an example for everyday independence (none in \(\mathbb Q\), two in \(\mathbb R\) and many in \(\mathbb C\)). But this is to motivate a different part of the explanation: the use of models of set theory (e.g. "How can you add a real number??", well how can you add a root to a polynomial?) and the fact that we don't consider the universe per se. Of course, in a model of \(\ZFC\) we can always construct the rest of mathematics internally, but this is not the issue now. Just like we have a model of one theory, we can have a model for another.

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Debates About The Climate

John Oliver (and his team of writers, I suppose) makes a particularly sharp point about the role of the media in the debate about climate changes.

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Downward Löwenheim-Skolem Theorems and Choice Principles

I have posted a new note on the Papers page.

It's a short little proof that the classic downward Löwenheim-Skolem theorem is equivalent to \(\DC\), and that for a well-ordered \(\kappa\), the downward Löwenheim-Skolem asserting the existence of models of cardinality \(\leq\kappa\) is in fact equivalent to the conjunction of \(\DC\) and \(\AC_\kappa\).

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...And we're back!

Okay, I took the time to make some changes to my homepage.

Clearly, the theme is different now. I also changed the content of the Papers page. I removed the abstracts (for some reason I thought this is going to be a cool thing to have, but with time it grew to annoy me greatly). I will definitely post a few things there in the coming time, some notes and eventually some nice papers -- I hope!

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Equivalent to the axiom of choice that I didn't know about

First I must apologize. I wanted to write a second post about forcing and preserving choice principles (I gave a nice talk in the student seminar about a week after the previous post), and I had a lot of things to say. I just ended up not writing it, and for absolutely no good reason. And somehow things continued that way and I felt more and more awkward to post anything because of that, but the vicious cycle must break somewhere.

I recently tried to figure out the consequence of some forcing in \(\ZF\). This has led me to the following statement:

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The cardinal trichotomy: finite, countable, and uncoutnable.

There is a special trichotomy for cardinality of sets. Sets are either finite, or countably infinite, or uncountable. It's an interesting distinction, and it has a very deep root -- at least in my perspective -- in the role of first-order logic.

Finite objects can be characterized in full using first-order logic. The fact that you can write down how many elements a set have, is a huge thing. For example, every finite structure of a first-order logic language has a categorical axiomatization. If the language is finite, then the axiomatization is finite as well.

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Strong chain conditions and preservation of choice principles

I recently returned from a wonderful week in Italy, where I attended the Young Set Theorist 2013 conference. I met a lot of new people, some old acquaintances, baffled people with oversized pickles, and most importantly shared and learned some great ideas.

One of the nicer things I'd done was to work with Thomas Johnstone on some preservation theorem related to forcing and choice principles (see also this announcement by Victoria Gitman). In order to clean up a bit the proof, I'll introduce a new definition which is going to slightly extend the ideas originally discussed in Italy. So without further jibber jabber, let's talk mathematics.

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Provable Equality Of Exponentiation

It's an almost trivial theorem of cardinal arithmetics in \(\ZF\) that given four cardinals, \(\frak p,q,r,s\) such that \(\frak p<q,\ r<s\) we have \(\frak p^r\leq q^s\).

In a recent question on math.SE some user has asked whether or not we always have a strict inequality. Everyone sufficiently familiar with the basics of independence results would know that it is consistent to have \(2^{\aleph_0}=2^{\aleph_1}=\aleph_2\), in which case taking \(\mathfrak{p=r}=\aleph_0,\ \mathfrak{q=s}=\aleph_1\) gives us equality. But it's also trivial to see that we can always pick cardinals whose difference is large enough to keep the inequality true.

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Choice Principles: What are they?

What does the phrase "\(\varphi\) is a choice principle" mean? This is something that I have spent quite a lot of my time thinking about. Directly and indirectly. What are choice principles as we know them? And who gets to decide?

For a set theorist, at least a "classical" set theorist (working within the confines of \(\ZF\) and its extensions to \(\ZFC\) and so on), a choice principle can aptly be defined as "Sentence \(\varphi\) in the language of set theory which is provable from \(\ZFC\) but independent from \(\ZF\)". Indeed that is how I think of choice principles, and how I referred to them in my masters thesis (albeit I prefaced that definition by pointing out its naivety).

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Infinite dimensions and the axiom of choice

In a recent math.SE question, Thomas Andrews asked whether or not the existence of an infinite linearly independent set in a vector space which is not finitely generated requires the axiom of choice.

The answer is positive. It does require the axiom of choice. The counterexample is due to Läuchli who constructed a model in which there was a vector space which was not finitely generated, but every proper subspace is finitely generated. Given such vector space it is obvious that no infinite set can be linearly independent.

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Vector Spaces and Antichains of Cardinals in Models of Set Theory

I finally uploaded my M.Sc. thesis titled “Vector Spaces and Antichains of Cardinals in Models of Set Theory”.

There are several changed from the printed and submitted version, but those are minor. The Papers page lists them.

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The Philosophy of Cardinality: Pathologies or not?

What are numbers? For the layman numbers are those things we use for counting and measuring. The complex numbers are on the edge of being numbers, but that's only because they are taught in high-schools and many people still consider them imaginary (despite them having some reasonably applicative uses).

But a mathematician knows that a number is basically a notion which represents a quantity. We have so many numbers that I don't even know where to begin if I wanted to list them. Luckily most of the readers (I suppose) are mathematicians and so I don't have to.

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On Leinster's "Rethinking set theory"

There has been a lot of recent discussions regarding Tom Leinster's paper "Rethinking set theory" (arXiv). Being an opinionated person, I only found it natural that I had an opinion on the paper. Now that I have a blog, I have a place to write this opinion as well.

The paper challenges the hegemony of \(\ZFC\) as the choice set theory. It offers an alternative in the form of \(\newcommand{\ETCS}{\axiom{ETCS}}\newcommand{\ETCSR}{\axiom{ETCS+R}}\ETCS\), a categories based set theory. The problem with \(\ETCS\) is that it is slightly weaker than \(\ZFC\). But we also know how much weaker: it lacks the expressibility of the full replacement schema. In this case we can just add a replacement schema-like list of axioms to have \(\ETCSR\).

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The Axiom of Choice and Self-Dual Vector Spaces

I have uploaded a note titled The Axiom of Choice and Self-Duality of Vector Spaces. Here is a short summary and background.

It is a well known fact (in \(\ZFC\) at least) that if \(V\) is a vector space, and \(V^\ast\) is the algebraic dual of \(V\) then \(V\cong V^{\ast\ast}\) if and only if \(\dim V<\infty\).

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First Post

Well... This is my first post on this blog, and I have absolutely no idea how to start it.

Should I make it about myself? about my life? about my academic status? How I about I tell cool stories from my life, perhaps inebriated adventures? army experiences? Maybe I should write about mathematics. Perhaps some nice proof or some nice theorem?

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