We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The proof-theoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of Kripke–Platek set theory.

In this paper we discuss extensions of Feferman's theory T 0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recursion-theoretic models for systems of explicit mathematics which is based on nonmonotone inductive definitions.

The Suslin operator is a type-2 functional testing for the well-foundedness of binary relations on the natural numbers. In the context of applicative theories, its proof-theoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation of the upper bounds in question. Several theories featuring the Suslin operator are embedded into ordinal theories tailored for dealing with non-monotone inductive definitions that enable a smooth definition of the application relation.

This paper introduces theories for arithmetical quasi-inductive definitions (Burgess, 1986) as it has been done for first-order monotone and nonmonotone inductive ones. After displaying the basic axiomatic framework, we provide some initial result in the proof theoretic bounds line of research (the upper one being given in terms of a theory of sets extending Kripke–Platek set theory).