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A finite axiomatization of inductiverecursive definitions
 Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science
, 1999
"... Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an in ..."
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Cited by 42 (14 self)
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Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductiverecursive definitions. We prove consistency by constructing a settheoretic model which makes use of one Mahlo cardinal. 1
Interactive Programs in Dependent Type Theory
, 2000
"... . We propose a representation of interactive systems in dependent ..."
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Cited by 36 (9 self)
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. We propose a representation of interactive systems in dependent
Extending MartinLöf Type Theory by One MahloUniverse
 Arch. Math. Log., 39:155
, 1998
"... We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in MartinLof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of set ..."
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Cited by 15 (8 self)
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We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in MartinLof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of sets in V, there exists a universe closed under f . We show that the proof theoretical strength of MLM is /\Omega 1\Omega M+! . Therefore we reach a strength slightly greater than jKPMj and V can be considered as a Mahlouniverse. Together with [Se96a] it follows jMLMj = /\Omega 1(\Omega M+! ). 1 Introduction An ordinal M is recursively Mahlo iff M is admissible and every Mrecursive closed unbounded subset of M contains an admissible ordinal. Equivalently, this is the case iff M is admissible and for all \Delta 0 formulas OE(x; y; ~z), and all ~z 2 LM such that 8x 2 LM :9y 2 LM :OE(x; y; ~z) there exists an admissible ordinal fi ! M such that 8x 2 L fi 9y 2 L fi :OE(x; y; ~z) holds. ...
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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Cited by 13 (8 self)
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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.
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...
A model for a type theory with Mahlo universe
, 1996
"... We present a type theory T T M, extending MartinLöf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a KripkePlatek style set theory KP M +, is given and we show that the proof theoretical strength of T ..."
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Cited by 7 (6 self)
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We present a type theory T T M, extending MartinLöf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a KripkePlatek style set theory KP M +, is given and we show that the proof theoretical strength of T T M is ≤ KP M +  = ψΩ1 (ΩM+ω). By [Se96a], this bound is sharp. 1
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.
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Realization of analysis into Explicit Mathematics
 The Journal of Symbolic Logic
, 2000
"... We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard Dinterpretation, which was used before and was shown to interpret only subsystems prooftheoretically weaker than T0 , our interpretation can reach the full strength of T0 . The ..."
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Cited by 4 (2 self)
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We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard Dinterpretation, which was used before and was shown to interpret only subsystems prooftheoretically weaker than T0 , our interpretation can reach the full strength of T0 . The Rinterpretation is an adaptation of Kleene's recursive realizability, and is applicable only to intuitionistic theories. Introduction Systems of Explicit Mathematics were introduced by S. Feferman in the 70es as a logical framework for Bishopstyle constructive mathematics (see [Fef75], [Fef79]). In [Fef79] he gave an embedding of the basic theory T 0 into a subsystem # 1 2 CA+BI of 2nd order arithmetic and conjectured that the converse also holds. In [Ja83] G. Jager carried out a necessary wellordering proof in T 0 , which together with [JP82] completed its prooftheoretical analysis and established prooftheoretic equivalence of the system of Explicit Mathematics T 0 , system o...
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.