# Is the theory Flow actually consistent?

Recently the paper

• Adonai S. Sant'Anna, Otavio Bueno, Marcio P. P. de França, Renato Brodzinski, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664

appeared on the arXiv, which claims that the authors' new theory, Flow, can prove the Partition Principle ("if $$A$$ is nonempty and surjects onto $$B$$, $$B$$ injects into $$A$$") but not the Axiom of Choice. If correct, this would resolve a long-open question in Set Theory (whether PP implies AC).

However, nowhere in the paper is the mentioned theory proved consistent (even relative to some LCA, which would be necessary as the paper claims their theory can interpret "ZF + there is a Grothendieck Universe") as far as I can tell. The axioms/principles that make up the theory seems to be dispersed throughout the paper. Definitions 7 and 10 (p13/p14) also seems to be self-referential, and no way of resolving self-referential definitions is discussed.

Therefore, I wonder if anyone on MO knows is the mentioned theory Flow consistent (relative to some LCA)?

• The theory has been discussed somewhat on Twitter also, so some relevant information may appear there: twitter.com/AndresECaicedo1/status/1314431294676197377 – Jem Oct 9 at 16:33
• I think that considering that this is a theory that is in the wild less than 24 hours, it's a bit unreasonable to expect an answer. – Asaf Karagila Oct 9 at 16:52
• Edited to be "consistent relative to some LCA" (I don't particularly care which, as long as it's not known to be inconsistent with ZFC and already exists) - and I agree that a quick answer is unlikely, but there may be something relatively obvious that I've missed that helps resolve it quickly. – Jem Oct 9 at 16:57
• @Nik Weaver Specifically, they prove [axioms F1-F11, F11T imply PP] and [axioms F1-F11, F11T and postulates Hyper-ZF-Sets and Hyperfunctions imply ¬AC], but do not prove that these systems themselves are consistent or provide models. – Jem Oct 9 at 17:04
• @Jem then it seems we are lacking two consistency proofs here, not just one. – Nik Weaver Oct 9 at 18:41

• Second, is the main idea of the proof seems to observe we cannot carry on transfinite recursion to define an injection $\psi\to o$ for all ordinals $o$. Does it ensure there is no injection from $\psi$ to $o$? There could be a weird way to impose a well-order of $\psi$, other than using transfinite recursion. – Hanul Jeon Oct 11 at 8:41
• The point of Hanul Jeon, which I think is right, is that the following argument is unreasonable: - If a function $f$ wellordering $\psi$ cannot be defined by a transfinite induction, then there is no such function. The proof of theorem 55 seems to be based on this. – Rodrigo Freire Oct 11 at 21:07