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  1. Universes Over Frege Structures.Reinhard Kahle - 2003 - Annals of Pure and Applied Logic 119 (1-3):191-223.
    In this paper, we study a concept of universe for a truth predicate over applicative theories. A proof-theoretic analysis is given by use of transfinitely iterated fixed point theories . The lower bound is obtained by a syntactical interpretation of these theories. Thus, universes over Frege structures represent a syntactically expressive framework of metapredicative theories in the context of applicative theories.
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  • The Proof-Theoretic Analysis of Σ11 Transfinite Dependent Choice.Christian Rüede - 2003 - Annals of Pure and Applied Logic 122 (1-3):195-234.
    This article provides an ordinal analysis of Σ11 transfinite dependent choice.
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  • Reflections on Reflections in Explicit Mathematics.Gerhard Jäger & Thomas Strahm - 2005 - Annals of Pure and Applied Logic 136 (1-2):116-133.
    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.
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  • A New Model Construction by Making a Detour Via Intuitionistic Theories II: Interpretability Lower Bound of Feferman's Explicit Mathematics T 0.Kentaro Sato - 2015 - Annals of Pure and Applied Logic 166 (7-8):800-835.