Chain Conditions and the Axiom of Choice

TL;DR

This paper explores the implications of the Axiom of Choice in set theory, establishing equivalences related to countability in various mathematical contexts without relying on the axiom.

Contribution

It identifies categorical principles that underpin equivalences of countability assertions in metric spaces, probability, and Hilbert spaces without the Axiom of Choice.

Findings

Equivalence of countability assertions in metric spaces

Categorical principles underlying these equivalences

Implications for set theory without the Axiom of Choice

Abstract

Within the framework of Zermelo-Fraenkel set theory without the Axiom of Choice, we establish equivalents to the assertion "the union of a countable collection of finite sets is countable" in the context of metric spaces, probability theory, and Hilbert spaces. The categorical principle underlying these equivalences is identified.