In the preface of my book on hypercomputation I have stated that all models of computation described in the book assume the axiom of choice. Instead of explaining explicitly why it is needed. I give an excerpt from Gregory H. Moore's prologue to Zermelo's Axiom of Choice in the hope that readers will understand why it is needed.
Yet without the Axiom, mathematics today would be quite different. The very nature of modern mathematics would be altered and, if the Axiom's most severe constructivist critics prevailed, mathematics would be reduced to a collection of algorithms. Indeed, the Axiom epitomizes the fundamental changes—mathematical, philosophical, and psycological—that took place when mathematicians seriously began to study infinite collections of sets.