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

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. Continue reading...

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. Continue reading...

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. Continue reading...

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, 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...

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). Continue reading...

Trust me, I'm a doctor!

Finally!

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

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: Continue reading...

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. Continue reading...

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. Continue reading...

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. Continue reading...

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. Continue reading...

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. Continue reading...

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. Continue reading...

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? Continue reading...

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. Continue reading...

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: Continue reading...

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. Continue reading...

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. Continue reading...

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! Continue reading...

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!

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.

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. Continue reading...

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??? Continue reading...

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! Continue reading...

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". Continue reading...

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. Continue reading...

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. Continue reading...

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.


First Post

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