A new model construction by making a detour via intuitionistic theories IV: A closer connection between KPω and BI

Abstract

By combining tree representation of sets with the method introduced in the previous papers [39,35,37] in the series, we give a new interpretation of KPωr+(Πn+2-Found)+θ (Kripke–Platek set theory with the foundation schema restricted to Πn+2 augmented by θ) in Σ11-AC0+(Πn+21-TI)+θ for any Π21 sentence θ, in such a way that Π21 sentences are preserved, where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard interpretation. Thus the addition of any Π21 theorem of BI≡Σ11-AC0+(Π∞1-TI) does not increase the consistency strength of KPω. Among such Π21 theorems are several fixed point principles for positive arithmetical operators and ω-model reflection (the cofinal existence of coded ω-models) for theorems of BI.