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Prooftheoretic investigations on Kruskal's theorem
 Ann. Pure Appl. Logic
, 1993
"... In this paper we calibrate the exact prooftheoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [ ..."
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Cited by 23 (3 self)
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In this paper we calibrate the exact prooftheoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents prooftheoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the prooftheoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of \Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...
Wellordering proofs for MartinLöf Type Theory
 Annals of Pure and Applied Logic
, 1998
"... We present wellordering proofs for MartinLof's type theory with Wtype 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 ..."
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Cited by 18 (11 self)
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We present wellordering proofs for MartinLof's type theory with Wtype 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...
Proof theory of reflection
 Annals of Pure and Applied Logic
, 1994
"... 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. Th ..."
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Cited by 9 (1 self)
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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
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... 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 ma ..."
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Cited by 8 (3 self)
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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'. Ordinaltheoretic 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...
Inductive Definitions and Type Theory: An Introduction
"... MartinLof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a fullscale system for the formalization of constructive mathematics, but has also proved to be a powerful framewo ..."
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Cited by 6 (0 self)
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MartinLof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a fullscale system for the formalization of constructive mathematics, but has also proved to be a powerful framework for programming. The theory integrates an expressive specification language (its type system) and a functional programming language (where all programs terminate). There now exist several proofassistants based on type theory, and many nontrivial examples from programming, computer science, logic, and mathematics have been implemented using these. In this series of lectures we shall describe type theory as a theory of inductive definitions. We emphasize its open nature: much like in a standard functional language such as ML or Haskell the user can add new types whenever there is a need for them. We discuss the syntax and semantics of the theory. Moreover, we present some examples ...
Ordinal Systems
 SETS AND PROOFS
, 2001
"... 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 Schütte Klammer symbols, up to the BachmannHoward ordinal. oeordinal systems, which are natu ..."
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Cited by 6 (1 self)
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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 Schütte Klammer symbols, up to the BachmannHoward ordinal. oeordinal 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 Schütte Klammer symbols", the exact strength of oeordinal systems.
Ordinal Systems, Part 2: One Inaccessible
 In Logic Colloquium ’98
, 2000
"... . We develop an alternative approach to wellordering proofs beyond the BachmannHoward ordinal using transfinite sequences of ordinal notations and use it in order to carry out wellordering proofs for oeordinal systems. We extend the approach of ordinal systems as an alternative way of presentin ..."
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Cited by 4 (1 self)
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. We develop an alternative approach to wellordering proofs beyond the BachmannHoward ordinal using transfinite sequences of ordinal notations and use it in order to carry out wellordering proofs for oeordinal 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 KripkePlatek set theory with one recursivly inaccessible. The upper bound is determined by giving wellordering proofs, which use the technique of transfinite sequences. We derive from the new approach the traditional approach to wellordering 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 ...
Derivatives for ordinal functions and the Schütte brackets
"... I develop the notion of a derivative, an operator which converts a normal function into a (faster) normal function. I show how these can be constructed using higher order xed point extractors, and I develop some eveluation techniques. As an illustrations I show how the Schutte brackets produce a la ..."
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Cited by 3 (3 self)
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I develop the notion of a derivative, an operator which converts a normal function into a (faster) normal function. I show how these can be constructed using higher order xed point extractors, and I develop some eveluation techniques. As an illustrations I show how the Schutte brackets produce a large family of derivatives. Contents 1
A Type Theory for Iterated Inductive Definitions
, 1994
"... We introduce a type theory FA # , which has at least the strength of finitely iterated inductive definitions ID<# . This type theory has as ground types trees with finitely many branching degrees (so called free algebras). We introduce an equality in this theory, without the need for undecidable pri ..."
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Cited by 2 (1 self)
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We introduce a type theory FA # , which has at least the strength of finitely iterated inductive definitions ID<# . This type theory has as ground types trees with finitely many branching degrees (so called free algebras). We introduce an equality in this theory, without the need for undecidable prime formulas. Then we give a direct wellordering proof for this theory by representing a ordinal denotation system in the iteration of Kleene's O. This can be easily done, by introducing functions on the trees, which correspond to the functions in the ordinal denotation system. The proof shows, that FA # proofs transfinite induction up to D 0 D n 0, which shows, that the strength of FA # is at least ID<# . It seems to be obvious, that this bound is sharp. 1 Definition of the type theory FA # Definition 1.1 The type theory FA # is defined as follows: (a) The ground types are defined inductively by: If n # 0 and # 1 , . . . , # n are ground types, then (# 1 , . . . , # n ) is a ground type. The type (# 1 , . . . , # n ) should be the type of wellfounded trees with branching degrees # 1 , . . . , # n . (b) Ground types are types, and if #, # are types then (# # #) is a type. We will omit brackets, using the usual conventions. (c) If # = (# 1 , . . . , # n ) is a ground type, then for i = 1, . . . , n C # i is a constant of type (# i # #) # # (C # i are the constructors for this type) and if # is as before and # a type, then we have the recursion constant R #,# of type ((# 1 # #) # (# 1 # #) # #) # # ((# n # #) # (# n # #) # #) # # # # We will write (C 1 : # 1 , . . . , C n : # n ) for (# 1 , . . . , # n ) to indicate, that C i are names for C # i . 1 (d) The terms are the typed lambda terms, built using the constants of (c). We write t ...