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

## 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.

## 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.

## 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):