PreprintCC BY 4.0arXiv
On a slight weakening of Kripke-Platek Set Theory
Abstract
The weak set theory ReR is obtained from Kripke-Platek Set Theory (KP) by replacing the bounded collection scheme with the bounded replacement scheme. We show that ReR proves TCo, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to ReR and KP have the same consequences. Our proof of TCo relies on the availability of a fragment of class foundation in ReR. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of ZF that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which TCo fails.
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BibTeX
@misc{mckenzie2026slight,
title = {{On a slight weakening of Kripke-Platek Set Theory}},
author = {Zachiri McKenzie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23398},
eprint = {2608.23398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23398},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a slight weakening of Kripke-Platek Set Theory
AU - Zachiri McKenzie
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.23398
UR - https://arxiv.org/abs/2608.23398
AB - The weak set theory ReR is obtained from Kripke-Platek Set Theory (KP) by replacing the bounded collection scheme with the bounded replacement scheme. We show that ReR proves TCo, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to ReR and KP have the same consequences. Our proof of TCo relies on the availability of a fragment of class foundation in ReR. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of ZF that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which TCo fails.
ER -