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 generalizing 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 $\mathsf{ZFC}$. The canonical extension $\mathsf{KP}^\mathcal{P}$ of Kripke-Platek set theory $\mathsf{KP}$ plays a key role in our work; one of our results refines a theorem of Rathjen by showing that $\Sigma_1^\mathcal{P}\text{-}\mathsf{Foundation}$ is provable in $\mathsf{KP}^\mathcal{P}$ (without invoking the axiom of choice).