Axiom of choice and the partition principle

Abstract

We introduce the Partition Principle PP, an axiom introduced by Russell in the context of its similarities and differences with the Axiom of Choice AC. We start by proving some properties of PP, and AC, and show that AC, entails PP. To address the problem of whether the converse holds, we develop the Zermelo-Fraenkel ZF set theory and examine its consistency and build a model in which AC, fails. We follow this with a discussion of forcing, a technique introduced by Paul Cohen to build new models of set theory from existing ones, which have differing properties from the starting model. We conclude by examining candidate models called permutation models where AC, fails, which may be useful as candidate models for forcing a model in which PP, holds but AC, does not. We conjecture that such a model exists, and that PP, does not entail AC.