Necessary and Sufficient Conditions for Proving Choice in Zermelo-Fraenkel Set Theory

TL;DR

This paper presents a novel method using the axiom schema of separation to establish necessary and sufficient conditions for the existence of choice functions in ZF set theory without relying on the Axiom of Choice, extending to topological spaces.

Contribution

It introduces a new approach employing separation to prove choice functions without explicit canonical elements, and establishes a necessary and sufficient condition involving partial orders with least elements.

Findings

A new method for proving choice functions in ZF without AC.

Necessary and sufficient condition: each set admits a partial order with a least element.

Application to topological spaces like hyper-intervals and spheres.

Abstract

This paper introduces an alternative approach to proving the existence of choice functions for specific families of sets within Zermelo-Fraenkel set theory (ZF) without assuming any form on the Axiom of Choice (AC). Traditional methods of proving choice, when it is possible without AC, are based on explicit constructing a choice function, which relies on being able to identify canonical elements within the sets. Our approach, instead, employs the axiom schema of separation. We begin by considering families of well-ordered sets, then apply the schema of separation twice to build a set of possible candidates for the choice functions, and, finally, prove that this set is non-empty. This strategy enables proving the existence of choice function in situations where canonical elements cannot be identified explicitly. We then extend our method beyond families of well-ordered sets to families of sets, over which partial orders with a least element exist. After exploring possibilities for further generalization, we establish a necessary and sufficient condition: in ZF, without assuming AC, a choice function exists for a non-empty family if and only if each set admits a partial order with a least element. Finally, we demonstrate how this approach can be used to prove the existence of choice functions for families of contractible and path-connected topological spaces, including hyper-intervals in $\mathbb{R}^n$, hyper-balls, and hyper-spheres.