Iterated failures of choice

Abstract

We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of counterexamples. For example, the rational numbers have a proper class of non-isomorphic algebraic closures, every partial order embeds into the cardinals of the model, every set is the image of a Dedekind-finite set, every weak choice axiom of the form ${\mathsf{AC}}_X^Y$ fails with a proper class of counterexamples, every field has a vector space with two linearly independent vectors but without endomorphisms that are not scalar multiplication, etc.

Complete rewrite of the paper; 22 pages

References in corpus (3)

• Dependent Choice, Properness, and Generic Absoluteness
• Combinatorial Properties and Dependent choice in symmetric extensions based on Lévy Collapse
• Generalizing Hartogs' Trichotomy Theorem