PreprintCC BY 4.0arXiv
Compactness beyond choice and HOD
Abstract
We characterise exacting, rank-Berkeley, Berkeley, and club Berkeley cardinals by compactness properties of logics and, equivalently, by compactness properties of certain topological spaces. This raises the strength known to be obtainable by statements about compactness into the realm of choiceless large cardinal axioms, the strongest known statements in terms of consistency strength. Moreover, it shows that compactness assumptions can directly violate the axiom of choice and the axiom V=HOD.
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BibTeX
@misc{osinski2026compactness,
title = {{Compactness beyond choice and HOD}},
author = {Jonathan Osinski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25291},
eprint = {2609.25291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25291},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compactness beyond choice and HOD
AU - Jonathan Osinski
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.25291
UR - https://arxiv.org/abs/2609.25291
AB - We characterise exacting, rank-Berkeley, Berkeley, and club Berkeley cardinals by compactness properties of logics and, equivalently, by compactness properties of certain topological spaces. This raises the strength known to be obtainable by statements about compactness into the realm of choiceless large cardinal axioms, the strongest known statements in terms of consistency strength. Moreover, it shows that compactness assumptions can directly violate the axiom of choice and the axiom V=HOD.
ER -