@article {jae13,
title = {Operational closure and stability},
journal = {Annals of Pure and Applied Logic},
volume = {164},
number = {7{\textendash}8},
year = {2013},
pages = {813-821},
abstract = {In 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.},
issn = {0168-0072},
doi = {http://dx.doi.org/10.1016/j.apal.2013.01.004},
url = {http://www.iam.unibe.ch/ltgpub/2013/jae13.pdf},
author = {Gerhard J{\"a}ger}
}