We study the extension of Feferman’s operational set theory provided by adding operational versions of unbounded existential quantification and power set and determine its proof-theoretic strength in terms of a suitable theory of sets and classes.

This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms of the reflection principle, (...) are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 , Theorem 107 and Theorem 108. (shrink)

Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the settheoretic universe is as an indefinite (or “potential”) totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (or “completed”) totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, in the (...) case of predicative analysis, the natural numbers are considered to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. Various such semiconstructive systems of analysis and set theory are formulated here and their proof-theoretic strength is characterized. Interestingly, though the logic is weakened, one can in compensation strengthen certain principles in a way that could be advantageous for mathematical applications. (shrink)

In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm 0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the (...) paper Jager and Strahm [11] are indeed sharp. (shrink)

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.

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.

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 discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.