McKenzie, ZachiriEnayat, Ali2025-04-142025-04-142022-04-15McKenzie, Z., & Enayat, A. (2022). End extending models of set theory via power admissible covers. Annals of Pure and Applied Logic, 173(8), article-number 103132. https://doi.org/10.1016/j.apal.2022.1031320168-007210.1016/j.apal.2022.103132http://hdl.handle.net/10034/629354© 2022 The Author(s). Published by Elsevier B.V.Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalising model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powerset-preserving end extensions and rank extensions of countable models of subsystems of ZFC . The canonical extension KP P of Kripke-Platek set theory KP plays a key role in our work; one of our results refines a theorem of Rathjen by showing that Σ 1 P -Foundation is provable in KP P (without invoking the axiom of choice).enhttps://creativecommons.org/licenses/by/4.0/End extensionModels of set theoryRank extensionAdmissible setsPower admissible setsArticle1873-2461Annals of Pure and Applied Logic2025-04-13173