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

Blog posts from 2020

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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