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Extending the System T_0 of explicit mathematics: the limit and Mahlo axioms
"... In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive def ..."
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In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.
Lectures on proof theory
 in Proc. Summer School in Logic, Leeds 67
, 1968
"... This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of purely ni ..."
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Cited by 12 (5 self)
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This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of purely nitary statements. One of the main approaches that turned out to be the most useful in pursuit of this program was that due to Gerhard Gentzen, in the 1930s, via his calculi of \sequents" and his CutElimination Theorem for them. Following that we trace how and why prima facie in nitary concepts, such as ordinals, and in nitary methods, such as the use of in nitely long proofs, gradually came to dominate prooftheoretical developments. In this rst lecture I will give anoverview of the developments in proof theory since Hilbert's initiative in establishing the subject in the 1920s. For this purpose I am following the rst part of a series of expository lectures that I gave for the Logic Colloquium `94 held in ClermontFerrand 2123 July 1994, but haven't published. The theme of my lectures there was that although Hilbert established his theory of proofs as a part of his foundational program and, for philosophical reasons whichwe shall get into, aimed to have it developed in a completely nitistic way, the actual work in proof theory This is the rst of three lectures that I delivered at the conference, Proof Theory: History
On Bounded Set Theory
"... We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation ..."
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Cited by 10 (1 self)
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We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation between BST, possibly with AntiFoundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory.
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...
Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... 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 mathemat ..."
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Cited by 8 (5 self)
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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 prooftheoretically 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 prooftheoretic 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...
An upper bound for the proof theoretical strength of MartinLöf Type Theory with Wtype and one universe
, 1996
"... (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 Primit ..."
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Cited by 7 (6 self)
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(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
The strength of MartinLöf type theory with a superuniverse. Part II
 PART I. ARCHIVE FOR MATHEMATICAL LOGIC 39, ISSUE
, 2000
"... Universes of types were introduced into constructive type theory by MartinLof [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 pap ..."
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Cited by 7 (2 self)
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Universes of types were introduced into constructive type theory by MartinLof [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 socalled superuniverse due to Palmgren (cf. [4, 5, 6]). It is proved that MartinLof type theory with a superuniverse, termed MLS, is a system whose prooftheoretic ordinal resides strictly above the FefermanSchutte ordinal \Gamma 0 but well below the BachmannHoward ordinal. Not many theories of strength between \Gamma 0 and the BachmannHoward 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.
Functional interpretation and inductive definitions
 Journal of Symbolic Logic
"... Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1. ..."
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Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1.
A Bounded Set Theory with AntiFoundation Axiom and Inductive Definability
 Computer Science Logic, 8th Workshop, CSL'94 Kazimierz
, 1995
"... this paper let \Deltaformulas and \Deltaterms be those defined as \Delta R with the construct theleast omitted. (They define predicates and operations also known as basic [Gan74] or rudimentary [Jen72] ones.) Note, that provablytotal \Sigma definable operations in KP 0 coincide with those defin ..."
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Cited by 6 (1 self)
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this paper let \Deltaformulas and \Deltaterms be those defined as \Delta R with the construct theleast omitted. (They define predicates and operations also known as basic [Gan74] or rudimentary [Jen72] ones.) Note, that provablytotal \Sigma definable operations in KP 0 coincide with those definable by \Deltaterms [Saz85, Saz85a, Saz87]. Therefore, we may use the name KP 0 also for the subtheory of KPR 0 , based on this \Deltalanguage, which does not involve both the term construct theleast and the corresponding axiom. Analogously, \Delta R (\Delta