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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
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.
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 ...
Proof Theory of MartinLof Type Theory  An
 Mathematiques et Sciences Humaines, 42 année, n o 165:59 – 99
, 2004
"... 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 equiconsisten ..."
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Cited by 4 (2 self)
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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 MartinLof type theory with Wtype and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of MartinLof type theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive 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.
An Ordinal Analysis of Stability
 ARCHIVE FOR MATHEMATICAL LOGIC
, 2005
"... This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves th ..."
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Cited by 3 (1 self)
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This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek 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.