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

By combining tree representation of sets with the method introduced in the previous three papers I–III [39], [35], [37] in the series, we give a new ${\Pi }_2^1$-preserving interpretation of $\mathrm{KP}{\omega }_r+\left({\Pi }_{n+2}\text{-}\mathrm{Found}\right)+\theta$ (Kripke–Platek set theory with the foundation schema restricted to ${\Pi }_{n+2}$, and augmented by θ) in ${\Sigma }_1^1\text{-}{\mathrm{AC}}_0+\left({\Pi }_{n+2}^1\text{-}\mathrm{TI}\right)+\theta$ for any ${\Pi }_2^1$ sentence θ, 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 ${\Pi }_2^1$ theorem of $\mathrm{BI}\equiv {\Sigma }_1^1\text{-}{\mathrm{AC}}_0+\left({\Pi }_{\infty }^1\text{-}\mathrm{TI}\right)$ does not increase the consistency strength of KPω. Among such ${\Pi }_2^1$ 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. The reader's familiarity to the previous works I–III in the series might help, but is not necessary.