Normal forms for derivations in Arai's $AI^{-}_\xi$

Abstract

In this paper, we shall consider normal forms for derivations in ${\rm AI}_{\xi}^{-}$, where ${\rm AI}_{\xi}^{-}$ is a system introduced by Arai (cf. [4]) and its consistency implies the consistency of Feferman's ${\rm ID}_{\xi}$ (cf. [6]). We shall give two normal form theorems for derivations in ${\rm AI}_{\xi}^{-}$. One (Theorem 1) implies the consistency of ${\rm AI}_{\xi}^{-}$. The other (Theorem 2) implies the $\omega$-consistency of ${\rm AI}_{\xi}^{-}$.