Iterated failures of choice

TL;DR

This paper develops a framework for constructing models where various forms of the axiom of choice fail, illustrating the widespread nature of such failures across different mathematical structures.

Contribution

It introduces a novel method to iterate constructions that contradict the axiom of choice, producing models with extensive counterexamples to choice-related principles.

Findings

Proper class of non-isomorphic algebraic closures of rationals

Every set is an image of a Dedekind-finite set

Failure of all weak choice axioms of the form AC_X^Y

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.