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A General Formulation of Simultaneous InductiveRecursive Definitions in Type Theory
 Journal of Symbolic Logic
, 1998
"... The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursi ..."
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Cited by 65 (9 self)
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The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive definition which is implicit in MartinLöf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous inductionrecursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model ...
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
On universes in type theory
 191 – 204
, 1998
"... The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in ..."
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Cited by 32 (8 self)
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The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in
Inductionrecursion and initial algebras
 Annals of Pure and Applied Logic
, 2003
"... 1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and MartinL"of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL"of's definition of a universe `a la T ..."
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Cited by 28 (11 self)
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1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and MartinL"of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL"of's definition of a universe `a la Tarski [19], which consists of a set U
The Strength of Some MartinLöf Type Theories
 Arch. Math. Logic
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely ..."
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Cited by 25 (5 self)
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One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLof intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. 0 Introduction MartinLof's intuitionistic theory of types was originally introduce...
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...
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. ...
The Strength of Some MartinLöf Type Theories
 ARCHIVE FOR MATHEMATICAL LOGIC
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of MartinLöf's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely th ..."
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Cited by 14 (10 self)
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One objective of this paper is the determination of the prooftheoretic strength of MartinLöf's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLöf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman.
Density Theorems for the DomainsWithTotality Semantics of Dependent Types
 Applied Categorical Structures
, 2000
"... . We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpre ..."
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Cited by 9 (0 self)
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. We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpretation of dependent types, and of D. Normann on universes of wellfounded types with density. Key words: Continuous functionals, Domains, Totality, Dependent types, Universes 1. Introduction In Mathematical Logic and Computer Science there is growing interest in constructive type theories as developed by Martin{Lof [8]. This paper is concerned with a semantics of such theories within the realm of Ershov{Scott domains [5] with totality [10]. Erik Palmgren and Viggo Stoltenberg{Hansen [15], [17] developed a semantics for a partial type theory (modelling partial functions and functionals) based on the notion of a parametrization, i.e. a domain depending on parameters. Since this semantics wa...
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