We study constructive set theories, which deal with operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be impredicative . (...) © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We introduce a way of relativizing operational set theory that also takes care of application. After presenting the basic approach and proving some essential properties of this new form of relativization we turn to the notion of relativized regularity and to the system OST that extends OST by a limit axiom claiming that any set is element of a relativized regular set. Finally we show that OST is proof-theoretically equivalent to the well-known theory KPi for a recursively inaccessible universe.