# Iterated failures of choice

- Published in 2019
- Added on

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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.

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## Other information

- key
- Iteratedfailuresofchoice
- type
- article
- date_added
- 2024-06-10
- date_published
- 2019-06-10

### BibTeX entry

@article{Iteratedfailuresofchoice, key = {Iteratedfailuresofchoice}, type = {article}, title = {Iterated failures of choice}, author = {Asaf Karagila}, 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.}, comment = {}, date_added = {2024-06-10}, date_published = {2019-06-10}, urls = {http://arxiv.org/abs/1911.09285v3,http://arxiv.org/pdf/1911.09285v3}, collections = {fun-maths-facts}, url = {http://arxiv.org/abs/1911.09285v3 http://arxiv.org/pdf/1911.09285v3}, year = 2019, urldate = {2024-06-10}, archivePrefix = {arXiv}, eprint = {1911.09285}, primaryClass = {math.LO} }