Operational closure and stability

TitleOperational closure and stability
Publication TypeJournal Article
Year of Publication2013
AuthorsJäger, G
JournalAnnals of Pure and Applied Logic
AbstractIn this article we introduce and study the notion of operational closure: a transitive set d is called operationally closed iff it contains all constants of $\mathsf{OST}$ and any operation $f\in d$ applied to an element $a\in d$ yields an element $fa\in d$ , provided that $f$ applied to a has a value at all. We will show that there is a direct relationship between operational closure and stability in the sense that operationally closed sets behave like $\Sigma_1$ substructures of the universe. This leads to our final result that $\mathsf{OST}$ plus the axiom $(\mathsf{OLim})$ , claiming that any set is element of an operationally closed set, is proof-theoretically equivalent to the system $\mathsf{KP}+(\Sigma_1\text{-}\mathsf{Sep})$ of Kripke-Platek set theory with infinity and $\Sigma_1$ separation. We also characterize the system $\mathsf{OST}$ plus the existence of one operationally closed set in terms of Kripke-Platek set theory with infinity and a parameter-free version of $\Sigma_1$ separation.