Abstract
Feferman [9] defines an impredicative system T0 of explicit mathematics, which is proof-theoretically equivalent to the subsystem ▪ of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as T0. To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by [22]. In addition, we give similar results for the system based on Aczel's original Frege structure [1]. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in [24].
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