## Posts tagged partition principle

## 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.

Continue reading...## 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.

Continue reading...## 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):

Continue reading...## 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!).

Continue reading...