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

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.

Let me thank all the people who were involved in the process: Adonai Sant'Anna and Marcio P. P. de França who are authors of the paper; Hanul Jeon, Andrés E. Caicedo, Toby Meadow, Iian Smythe, and others who engaged with the paper (and with my analysis) over that period of time; others to whom I apologise for not mentioning by name.

## The conclusions

I will start with the end. The paper, as it stands, does not contain a sound proof of the claimed result. There are several major gaps:

1. The actual proof that $$\AC$$ fails is a bit shoddy. It is not fully explained; nor it really shows that there is no well-order of "hyperfunctions" (which correspond to a certain notion of ill-founded sets). At best, it seems, we get that there is no definable well-order. But that's not enough. Consider $$\ZFC$$+Atoms, then there is no definable well-ordering of the set of atoms, but one does exists in the universe.

2. There is no actual proof why hyperfunctions are consistent with $$\Flow$$. Even if you want to put aside for a moment whether or not $$\Flow$$ is consistent, at the very least you must wonder why can hyperfunctions even exist? This is not at all addressed by the authors.

3. The interpretation of $$\ZF$$ is not working. The definitions of $$\ZF$$ sets and the definition of ordinals are both faulty. The authors have informed me (and announced on Twitter) that they have a correction to this specific issue, but at least the one which appears in the paper is faulty.

4. There is no claims regarding the relative consistency of $$\Flow$$. Ultimately, it is up to the authors to prove that their theory is consistent. If it turns out that their theory is just a reformulation of $$\ZF+\lnot\AC+\axiom{PP}$$, then claiming that it is consistent is just rephrasing the original problem. This whole thing is being compounded by a misuse of the term "Grothendieck universe" to describe something which is akin to a proper class (whereas a Grothendieck universe is by definition a set), which makes it seemingly that $$\Flow$$ proves the consistency of inaccessible cardinals, where in reality this turns out to not be the case.

So these four main points are good enough reasons to believe that the problem was not solved in the paper.

## How to understand $\Flow$, as a set theorist

I don't know. The idea in $$\Flow$$ is to have functions as the object in the universe; they axiomatise these functions in a way that seems to be geared towards interpreting $$\ZF$$. So in some sense, if we have a model of $$\ZF$$, say $$M$$, then the functions in $$\Flow$$ are in some sense partial class functions from $$M$$ to $$M$$. This is not quite the whole story, since these functions act on each other and themselves, but it seems like one can try and make sense out of this.

The concept of "emergent" corresponds to "small", and so hereditarily small is what we expect a set to be. And other concepts also have a reasonably intuitive translation like that.

One particular key point here is that the "Grothendieck universe" defined in the paper corresponds to a proper class, rather than to a set. It is not even clear if $$\Flow$$ has a truth predicate for this class. But that seemingly puts $$\Flow$$ somewhere between $$\NBG$$ and $$\MK$$, as far as I can tell. Although we will need to wait for the corrections of the interpretation before we can say more.

## What do I think of $\Flow$?

I think it's an interesting idea, and I'd like to see it developed more carefully. One critique I have is that it seems "more difficult to use" as a foundation of mathematics, compared to things like $$\ZF$$ or the likes of it. For one, the $$\in$$ relation is very deeply ingrained into our understanding of modern mathematics, and whereas other non-set theoretic foundations of mathematics find a way to represent the membership relation in a fairly simple way (e.g. having a certain type; belonging to a category; being a special type of object which is "simple enough") $$\Flow$$ seems to fail in doing that.

This is not to say that the theory is not useful, and if it turns out to be somehow "a functional interpretation of $$\ZF$$", and it has applications elsewhere, then that would be very good for everyone.

What I don't think can happen at the moment, is for $$\Flow$$ to take hold as a foundation of mathematics. The use of "functions" where composition is not associative is already a bit confusing. This means that a lot of what we are used to associate with "functions" is not going to work, moreover, these functions act on themselves, which is not unheard of, but still somewhat confusing.

One of the reasons, perhaps, for the confusion, is that as one of the authors explained it once on Twitter: they view a universe of $$\Flow$$ as something dynamic that changes and develops. Mathematics, and first-order in particular, is dealing with fixed and static universes. We can stratify such a universe, and then we can study how the universe is developing through the different layers, but the universe itself is static. It's there, and what's not there is not there.

## So, what's next?

Well, the authors of the paper are back to the drawing board. And I will proceed with my efforts to prove that the Partition Principle does not imply the Axiom of Choice using more traditional methods.

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