Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal. 1

. We propose a representation of interactive systems in dependent

We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in Martin-Lof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of sets in V, there exists a universe closed under f . We show that the proof theoretical strength of MLM is /\Omega 1\Omega M+! . Therefore we reach a strength slightly greater than jKPMj and V can be considered as a Mahlo-universe. Together with [Se96a] it follows jMLMj = /\Omega 1(\Omega M+! ). 1 Introduction An ordinal M is recursively Mahlo iff M is admissible and every M-recursive closed unbounded subset of M contains an admissible ordinal. Equivalently, this is the case iff M is admissible and for all \Delta 0 formulas OE(x; y; ~z), and all ~z 2 LM such that 8x 2 LM :9y 2 LM :OE(x; y; ~z) there exists an admissible ordinal fi ! M such that 8x 2 L fi 9y 2 L fi :OE(x; y; ~z) holds. ...

In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recusion-theoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.

This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and proof-theoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...

We present a type theory T T M, extending Martin-Löf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a Kripke-Platek style set theory KP M +, is given and we show that the proof theoretical strength of T T M is ≤ |KP M + | = ψΩ1 (ΩM+ω). By [Se96a], this bound is sharp. 1

Ordinal systems are structures for describing ordinal notation systems, which extend the more predicative approaches to ordinal notation systems, like the Cantor normal form, the Veblen function and the Schutte Klammer symbols, up to the Bachmann-Howard ordinal. oe-ordinal systems, which are natural extensions of this approach, reach without the use of cardinals the strength of the theories for transfinitely iterated inductive definitions ID oe in an essentially predicative way. We explore the relationship with the traditional approach to ordinal notation systems via cardinals and determine, using "extended Schutte Klammer symbols", the exact strength of oe-ordinal systems. 1 Introduction 1.1 Motivation The original problem, which motivated the research in this article, seemed to be a pedagogical one. We have been trying to teach ordinal notation systems above the BachmannHoward ordinal several times. The impression we had was that we were able to teach the technical develo...

The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be

We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard D-interpretation, which was used before and was shown to interpret only subsystems proof-theoretically weaker than T0 , our interpretation can reach the full strength of T0 . The R-interpretation is an adaptation of Kleene's recursive realizability, and is applicable only to intuitionistic theories. Introduction Systems of Explicit Mathematics were introduced by S. Feferman in the 70-es as a logical framework for Bishop-style constructive mathematics (see [Fef75], [Fef79]). In [Fef79] he gave an embedding of the basic theory T 0 into a subsystem # 1 2 -CA+BI of 2-nd order arithmetic and conjectured that the converse also holds. In [Ja83] G. Jager carried out a necessary well-ordering proof in T 0 , which together with [JP82] completed its proof-theoretical analysis and established prooftheoretic equivalence of the system of Explicit Mathematics T 0 , system o...

We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-Lof type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-Lof type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.