End extending models of set theory via power admissible covers

Abstract

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).