Flow: the Axiom of Choice is independent from the Partition Principle

TL;DR

The paper introduces a new framework called Flow that models set theory and demonstrates that the Partition Principle does not imply the Axiom of Choice, addressing a long-standing open problem.

Contribution

It develops the Flow framework, showing how ZF, non-well founded ZF, and ZFC can be embedded, and constructs a model where PP holds without AC.

Findings

Flow encompasses ZF, non-well founded ZF, and ZFC.

A model is constructed where PP holds but AC does not.

Strongly inaccessible cardinals are derived from Flow's axioms.

Abstract

We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of strongly inaccessible cardinals is entailed from our axioms. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest open problem in set theory: if PP entails AC.