Abstract

In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established. 1 Introduction In classical set theory an ordinal # is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function f from # to #, there exists a regular cardinal less than # so that {f(#) : # < } # . The statement that there exists a Mahlo ordinal is a powerful set existence axiom going beyond theories like ZFC. It also outgrows the existence of inaccessible cardinals, hyper inaccessibles, hyperhyperinaccessible and the like. There is also an obvious recursive analogue of Mahlo ordinal. Typically, an ordinal # is baptized recursively Mahlo, if it is admissible and reflects every # 2 sentence on a smaller admissible ordinal. The corresponding formal theory KPM has been proof-theoretically analyzed by Rathjen [14, 15]. KPM is a highly impredicative theory, and its proof-the...

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