Iterated failures of choice
Logic
2021-07-26 v3
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 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.
Cite
@article{arxiv.1911.09285,
title = {Iterated failures of choice},
author = {Asaf Karagila},
journal= {arXiv preprint arXiv:1911.09285},
year = {2021}
}
Comments
Complete rewrite of the paper; 22 pages