Necessary and Sufficient Conditions for Proving Choice in Zermelo-Fraenkel Set Theory
Valentyn Khokhlov
TL;DR
The paper addresses the problem of proving the existence of choice functions within $ZF$ without assuming the Axiom of Choice ($AC$). It introduces a non-constructive, separation-based approach, starting with families of well-ordered sets and extending to families of sets equipped with a partial order possessing a least element. The main result is a necessary-and-sufficient condition: a choice function exists for a non-empty family in $ZF$ without $AC$ if and only if every set in the family admits a partial order with a least element. The work further applies these ideas to intervals in $\mathbb{R}$, hyper-intervals in higher dimensions, and classes of contractible or path-connected spaces, demonstrating broad applicability and offering a framework that does not depend on explicit canonical elements. The contributions include a generalization from well-ordered to partially ordered contexts, a formalization of the separation-based construction of candidate choice relations, and implications that extend to topology and countability results within $ZF$ without $AC$, highlighting the significance of order structures in establishing choice in the absence of the full Axiom of Choice.
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.