Interesting Esoterica

Iterated failures of choice

Article by Asaf Karagila
  • 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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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}
}