The relative consistency of the negation of the Axiom of Choice using permutation models

This document discusses the axiom of choice (AC) and related principles in set theory. It makes the following key points: 1. AC allows proving the existence of sets without explicit construction, which some early mathematicians found controversial. 2. AC is equivalent to several other principles like Zorn's lemma and is needed to prove certain paradoxical theorems. 3. Various weaker forms of choice exist, like the axiom of dependent choice, and some principles like the prime ideal theorem do not imply full AC. 4. Models of set theory without AC, called permutation models, can be constructed, showing the consistency of denying AC is independent of Zermelo-Fraenkel set theory