On the proof-theoretic strength of monotone induction in explicit mathematics

Abstract We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle (MID) asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ 1 2 -axiom of choice and Π 1 2 -comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems.