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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+# , whi ..."
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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...
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.
Proof theory of MartinLöf type theory. An overview
 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 equiconsistent ..."
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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 MartinLöf type theory with Wtype and one microscopic universe containing only two finite sets in carried out. Then we look at the analysis MartinLöf 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.
Some Decision Problems of Enormous Complexity
 In Proc., LICS’99
, 1999
"... We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to # 0 time completeness to #(# # , ..."
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Cited by 3 (0 self)
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We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to # 0 time completeness to #(# # ,0) time completeness to #(# # ,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated complexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of this manuscript to be published elsewhere. 1. Iterated exponential time  universal relational sentences Let F be a function from A* into B*, where A,B are finite alphabets. We say that F is iterated exponential time computable if and only if there is a multitape Turing machine TM (which processes inputs from A* and outputs from B*) and an integer constant c > 0 such that TM computes F(x) with run time at most...
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 ..."
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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.
The Higher Infinite in Proof Theory
 LOGIC COLLOQUIUM '95. LECTURE NOTES IN LOGIC
, 1995
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