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We present well-ordering proofs for Martin-Lof's type theory with W-type and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 -CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...

The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1

(2), W (2) and I (3). To make it easier to remember the meaning of the symbols, we give the following hints: r is the (unique) element of an identity type I; n k is the nth element of the finite type N k with k elements, C the Casedistinction for this type; O is the zero, S the Successor, P Primitive recursion or induction over the natural numbers N ; i stands for left inclusion, j for right inclusion, D is the choice in the type A +B of disjoint union of A and B; p 0 and p 1 are the projections, p the pairing for the #-typ

Universes of types were introduced into constructive type theory by Martin-Lof [3]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then "reflects" C. This is the second part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf. [4, 5, 6]). It is proved that Martin-Lof type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schutte ordinal \Gamma 0 but well below the Bachmann-Howard ordinal. Not many theories of strength between \Gamma 0 and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory. In this second part of the paper the concern is with the with upper bounds. 1 Introduction One reason for splitting this paper into two parts was its sheer length. But the main reason ...

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...

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

. We develop an alternative approach to well-ordering proofs beyond the Bachmann-Howard ordinal using transfinite sequences of ordinal notations and use it in order to carry out well-ordering proofs for oe-ordinal systems. We extend the approach of ordinal systems as an alternative way of presenting ordinal notation systems started in [Set98b] and develop ordinal systems, which have in the limit exactly the strength of Kripke-Platek set theory with one recursivly inaccessible. The upper bound is determined by giving well-ordering proofs, which use the technique of transfinite sequences. We derive from the new approach the traditional approach to well-ordering proofs using distinguished sets. The lower bound is determined by extending the concept of ordinal function generators in [Set98b] to inaccessibles. 1 Introduction This article is a followup of [Set98b]. In that article we introduced ordinal systems as an alternative way of describing ordinal notation systems which usually make ...

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.

This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 -comprehension. On the set-theoretic side 1 2 -comprehension corresponds to Kripke-Platek set theory, KP, plus 1 -separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all > , is -stable, i.e. L is a 1 -elementary substructure of L . The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 1 2 -comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on -indescribable cardinals which is then employed for determining an upper bound for the proof{ theoretic strength of the theory KPi + 8 9 is + -stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.