Today I stumbled upon an interesting article about a ‘paradoxical’ appearance of the axiom of choice in a generalization of the known ‘hats on prisoners’ puzzle.

~~Two~~ Three interesting thing in the linked page:

- The generalization of the game: multiple colours, infinite prisoners. I also guess ultrafilters appear, in disguise, when talking equivalences on infinite sequences.
- The comments thread, with some giants like Trimble arguing ontologically about AC. Weird but cool.
- The comment of Tao (that made me cross that ‘two’ up there) about relating this ‘paradox’ to the Banach-Tarski-type oddities coming from the interplay between measure theory and group theory.

Being speaking (or better, echoing) about AC and having mentioned Banach-Tarski, it seems a good opportunity to talk about a poster of mine I did for the *C0eM∀τ* [0], a simple but exciting congress students arrange each year here at the ULL math department.

When they told be about the initiative I was so happy to have the opportunity to produce something creative (*quick scolding glance at Italian education*[1]) that I instinctively said *yes* even before thinking about a proper topic. In hindsight that was risky! I was still going around all day trying to find a shelter and finishing the paperwork to get enrolled at the university, and I didn’t have a lot of time to do this. Also, I had never made a poster before. But still, I managed, and I think having something pleasant to do helped me passing those days.

Anyway, my original idea was a not-better-specified *something* about category theory. I would have liked to produce something self-contained, at reach for attendants (mainly undergrads) and curious, yet not too trivial (‘hey Rubik’s cube is a group!’ [2]). I quickly realized *cats* were too broad for this. But suddenly and unexpectedly, Banach-Tarski popped into my mind.

At the beginning of the past semester, I read extensively about it. I decided to look into it after becoming a little more savvy about AC and measure theory in the past two years. I already knew the paradox, of course, but I had never *really* understood it, which for a mathematician means I never understood the proof. Actually, I didn’t even try, because when I first learned about it I think I was still in high school and quite unarmed for the subject [3]. Eventually it was very pleasant and fulfilling to come back at it and magically see it unlocking before my mind: the excitement of understanding it was the most intense math-induce emotion I ever experienced. I got really interested in the topic for a while, even getting to read the first three (or something) chapters of Wagon’s outstanding book about it. Then it got too combinatorial for my taste and I dropped it. But still, six months after, my brain regurgitated it and I immediately thought it was the perfect topic for my poster: self-contained yet far-reaching, approachable yet stimulating, and, not less importantly, I was already familiar with it and related topics [4]. *Bonus*: I had some good expositions available (the aforementioned book & a great Wikipedia entry).

That said, after a good amount of procrastination and fights with TikZ (but shout out to tikz-poster for the incredibly ~~constrained~~ helpful package!), I managed to produce a thing. I used Google Translator and the help of one of the organizers to translate it to Spanish, and that’s the version that was exposed and the congress. I had to explain it to an ‘evaluation committee’ and to people, I thought I wasn’t going to be able to do it because my Spanish skills were (and still are) garbage, and so I got a bilingual student by my side to help me with translations. Eventually, I stepped in the flow and thanks to an absurdly amount of similarity between Italian and Spanish (mathematical and not) language, I queried my sidekick just a couple of times, only to hear him say ‘yeah, it’s the same word in Spanish’ [6]. I got embarrassed just for one question, when a teacher pointed out that my sentence ‘there’s no additive and isometrically invariant measure on R^3’ was completely wrong (rudeness mine) being Lebesgue measure additive and isometrically invariant. She’s completely right, by the way, although my use of the term *measure* was ‘classical’, hence I assumed it to be full (in which case Lebesgue fails to be a counterexample). I should have used the more ambiguous word *volume*, thereby shielding myself from any critique [7].

In the days following the expositions I perfected the English version of the poster, and this is the one I want to share with you. Yet for intellectual honesty I must also provide the original Spanish version, now hanged in my room.

**Notes**

[0] Unfortunately this link will be broken in less than a year from now, due to the absence of a permalink. *Gnaffe!* There’s my name on the ‘posters’ page, wow.

[1] But I have to be just here and get some guilt. I’ve been a student rep for ~2 years and never did anything close to this.

[2] My 2 cents on this topic: those things are great to instill curiosity but rapidly fall back into ‘lame’ if you don’t calibrate precisely your aim:

[3] Maybe I knew what a group was, but a group action? Choice-set?

[4] I need to write this somewhere: the ‘savvyness’ I got about AC in the past two years is completely due to the fascinating book of Moore’s about it. I was very lucky to run into his book early in my studies because it bolted my ‘AC senses’ early on, making it much easier to understand where and when and why it’s used. Before understanding it, you’re essentially blind to issues concerning constructability, and it’s nice to sharpen your sense as soon as possible in order to avoid becoming *blind to your own blindness *[5]. The historical perspective he uses is also so good at doing this work, because mathematicians at the beginning of the past century were essentially as clueless as the reader, and so the narrative helps a lot.

[5] A ‘bad boy!’ to undergrad math education which repeats the same things about naïve set theory in every course but fails to give you this intuition (even at a basic level).

[6] That doesn’t mean I’m not grateful to him! He was still a nice safety net to have, and more likely than not his presence made me much more confident and hence helped me being so effective (hyperbole) in the exposition.

[7] My favourite flavour of *proof by intimidation*.