Iterating the recursively Mahlo operations

Abstract

In this paper we address a problem: How far can we iterate lower recursively Mahlo operations in higher reflecting universes? Or formally: How much can lower recursively Mahlo operations be iterated in set theories for higher reflecting universes? It turns out that in ${\Pi }_N$-reflecting universes the lowest recursively Mahlo operation can be iterated along towers of ${\Sigma }_1$-exponential orderings of height $N-3$, and that all we can do is such iterations. Namely the set theory for ${\Pi }_N$-reflecting universes is proof-theoretically reducible to iterations of the operation along such a tower.