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Large cardinals, small violations of choice
Abstract
We study the relationship between large cardinals and Blass' Small Violation of Choice (SVC), which states that the Axiom of Choice holds in a set-generic extension of the universe. Our main theorem is that in the presence of a supercompact cardinal, either the Axiom of Choice fails significantly, or else SVC holds in L(𝒫(Ord)). We show that a supercompact is not strong enough to reflect SVC, while an extendible cardinal is. If we assume Usuba's Loś Property holds for the supercompact, then we can outright derive SVC in the universe.
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BibTeX
@misc{duncan2026large,
title = {{Large cardinals, small violations of choice}},
author = {Hope Duncan and Gabriel Goldberg and Asaf Karagila},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05140},
eprint = {2611.05140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05140},
}
RIS
TY - UNPB
TI - Large cardinals, small violations of choice
AU - Hope Duncan
AU - Gabriel Goldberg
AU - Asaf Karagila
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05140
UR - https://arxiv.org/abs/2611.05140
AB - We study the relationship between large cardinals and Blass' Small Violation of Choice (SVC), which states that the Axiom of Choice holds in a set-generic extension of the universe. Our main theorem is that in the presence of a supercompact cardinal, either the Axiom of Choice fails significantly, or else SVC holds in L(𝒫(Ord)). We show that a supercompact is not strong enough to reflect SVC, while an extendible cardinal is. If we assume Usuba's Loś Property holds for the supercompact, then we can outright derive SVC in the universe.
ER -