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

Posts tagged general

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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