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

TL;DR

This paper introduces a new formal framework called Flow, demonstrating that the Partition Principle and the Axiom of Choice are independent in ZFU, thus addressing a longstanding open problem in set theory.

Contribution

The paper presents Flow, a novel formal theory that models set-theoretic principles and shows the independence of the Partition Principle from the Axiom of Choice in ZFU.

Findings

Flow can embed ZF, ZFC, and ZFU as consequences.

A model of ZFU is constructed where the Partition Principle holds but the Axiom of Choice fails.

The framework addresses the open problem of whether the Partition Principle entails the Axiom of Choice.

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.