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

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.

## 7 April, 2021

Okay! Sorry for the delay, but I finally started reading the paper. So far it was an easy refresher on $$\Flow$$, I've reached the definition of $$\ZF$$-sets on p.19.

So far this is very similar to the last paper. The definition of an ordinal seems to have been corrected, and the definition of $$\ZF$$-sets also seems to be better. But I'll look further into that the next time.

In the general sense, I still have my reservations from the last paper. I mean, if at the end there is a construction of a model of $$\ZFA$$ where $$\Par$$ holds but $$\AC$$ fails, then in principle this construction should be doable in any reasonable meta-theory, not just $$\Flow$$. Unless, of course, $$\Flow$$ somehow admits a natural interpretation of $$\ZFA+\Par+\lnot\AC$$, which renders the whole project kind of moot. It's not uninteresting, it's just changing the names of things and claiming we proved something (in the context of $$\Par$$ and $$\AC$$, that is, obviously there's more to the paper than just that).

## 12 April, 2021

So. Now we get to the new parts of this paper. Atoms. Well, first we have a definition of $$\ZF$$-sets, which is sort of what you'd expect: it's a von Neumann universe defined by recursion. Some basic theorems about $$\ZF$$-sets, nothing exciting there. But then we define $$\ZZ_r(f)$$, for an ordinal $$r$$, by induction:

• $$\ZZ_{\varphi_0}$$ is just $$\ZZ$$. So the first strata is just the $$\ZF$$-sets.
• For a successor step $$r$$, $$\ZZ_r$$ is defined to be those functions whose domains and range are into some $$\ZZ_s$$ for some $$s < r$$, but we specifically want to "hit that top level". So $$\ZZ_{\varphi_3}$$ is going to be functions which include some information about $$\ZZ_{\varphi_2}$$, but they may also include information about $$\ZZ_{\varphi_0}$$.
• For limit steps we want something akin to the unions.

I'm not sure that I'd agree with the definition of successor steps here. Since we're not trying to define separate predicates, really, we're trying to define an increasing hierarchy. But limit steps smooth things up, so it's not a big issue. The point here is that $$\ZF$$-sets are by definition "identity sequences" (i.e. restrictions of $$\frak 1$$), but $$\ZZ_r$$ includes also other functions, as long as the domain and range match.

And indeed, these non-identity functions are called "atoms". The next term is a "brick" which is a function defined on atoms which is an identity function. This obviously leads to the relative von Neumann universe.

This is reminiscent of the construction of models of $$\ZF$$ with atoms by taking a set $$A$$ such that no element of $$A$$ is in the transitive closure of the other, assigning to some $$a_0\in A$$ the role of the empty set, and then defining $$\power^*(X)=\power(X)\setminus\{\varnothing\}\cup\{a_0\}$$. And now iterate $$\power^*$$ starting with $$A$$. This will produce a model of $$\ZF$$ with $$A\setminus\{a_0\}$$ as your set of atoms. Other constructions exist as well, of course. The reason I bring this up here is that there is a discussion after the definition which points out that in $$\Flow$$ the atoms have some internal structure that may interact with each other, and therefore the terms "atom" or "urelement" is a misnomer. But indeed, this is the case when you interpret one theory in another. This should be surprising to anyway. That being said, the situation here is a bit more delicate.

Definition 33 seems to be an attempt to formalise the idea of equivalence classes modulo the "kernel indiscernible" relationship. It seems that we are walking towards permutation models. The last two parts of Definition 33 seems like a corollary from (V), since atoms have rank 0 and Menge do not.

Next, Definition 34 defines a countable union of pairs. It seems very much like we're heading towards the second Fraenkel model or a variation thereof. But all of this will wait for tomorrow.

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