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

Abstract: We introduce a formal theory called Flow where the intended interpretation of its terms is that of function. We prove ZF, ZFC and ZFU (ZF with atoms) can be immersed within Flow as natural consequences from our framework. Our first important application is the introduction of a model of ZFU where the Partition Principle holds but the Axiom of Choice fails, if Flow is consistent. So, our framework allows us to address the oldest open problem in set theory: if the Partition Principle entails the Axiom of Choice.