OpenAI, the Partition Principle, and mathematics

Oct 08 2026, 10:25

By the time I got out of bed yesterday, I had three people ask me if I'd seen the OpenAI announcement that the Partition Principle does not imply the Axiom of Choice. The number kept increasing during the day, and I can see why. I am famously interested in this problem.

Indeed, at least two people asked if I plan on sending Sam Altman a bottle of whisky, as promised in my Problems page. The answer to that is no. And let me explain to you why, and why you should be very angry at OpenAI.

Flow and the Partition Principle II (2 updates)

Mar 31 2021, 08:01

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.

Flow and the Partition Principle: Conclusions

Oct 26 2020, 20:10

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.

Flow and the Partition Principle (6 updates)

Oct 09 2020, 14:30

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.

Countable sets of reals

Apr 06 2020, 13:15

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):

On the Partition Principle

Dec 20 2014, 20:45

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!).