End extending models of set theory via power admissible covers

TL;DR

This paper introduces the power admissible cover construction for ill-founded models of set theory, enabling new results on end extensions and rank extensions of models of subsystems of ZFC, refining existing theorems.

Contribution

It develops the power admissible cover machinery for ill-founded models, extending Barwise's admissible covers, and applies it to obtain new results in set theory model extensions.

Findings

Refines Rathjen's theorem on $ ext{KP}^ ext{P}$

Proves $ ext{Sigma}_1^ ext{P}$-Foundation in $ ext{KP}^ ext{P}$ without choice

Provides new insights into powerset-preserving end extensions

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).