An extended predicative definition of the Mahlo universe

This article, which will be reviewed by Zentralblatt Math, contains the first predicative definition of the Mahlo universe, by extending the concept of predicativity. This is a break through result, since it introduces a methodology which allows to justify proof theoretically much stronger theories than were known before predicatively.Before this article predicativity was limited to inductive recursive definition, and it was widely believed that it is impossible to go beyond that notion in a predicative way. With this article for the first time this barrier has been passed using a novel approach

Proof theory and Martin-Löf Type Theory

In this article an overview over the work of the author on developing proof theoretic strong extensions of Martin-Loef Type Theory including precise proof theoretic bounds is given. It presents the first publication of the proof theoretically strongest known extensions of Martin-Loef Type Theory, namely the hyper-Mahlo Universe, the hyper-alpha-Mahlo universe, the autononomous Mahlo universe and the Pi_3-reflecting universe. This is part of a proof theoretic program in developing proof theoretic as strong as possible constructive theories in order to obtain a constructive underpinning of strong classical theories with a full proof theoretic analysis

First order theories for nonmonotone inductive definitions: recursively inaccessible and Mahlo

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms à la Richter with recursively inaccessible and Mahlo closure ordinals is give

Broad Infinity and Generation Principles

This paper introduces Broad Infinity, a new and arguably intuitive axiom scheme. It states that "broad numbers", which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal. Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The paper relates these principles under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.Comment: 52 page