End extending models of set theory via power admissible covers

2021/08/05 by McKenzie, Zachiri, Enayat, Ali
#03C62 #03C70 #03E30 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2108.02677

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 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 Σ₁^P-Foundation is provable in KP^P (without invoking the axiom of choice).

Related

• Partially-elementary end extensions of countable models of set theory
• Power Kripke-Platek set theory and the axiom of choice
• Pseudo-countable models
• What is the theory ZFC without power set?
• Models of Set Theory: Extensions and Dead-ends