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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 ...
Notes on Constructive Set Theory
, 1997
"... Contents 1 Introduction 11 2 Some Axiom Systems 21 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 CZF 0 . . . . ..."
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Cited by 46 (9 self)
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Contents 1 Introduction 11 2 Some Axiom Systems 21 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 CZF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Constructive Zermelo Fraenkel, CZF . . . . . . . . . . 23 2.3.3 Basic Constructive Set Theory, BCST . . . . . . . . . 23 3 Elementary Mathematics in Constructive Set Theory 31 3.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.2 Cartesian Products of Classes . . . . . . . . . . . . . . 32 3.2.3 Relations and Functions between Classes . . . . . . . . 33 3.3 The class of Natural
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
Wellfounded Trees and Dependent Polynomial Functors
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class ..."
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Cited by 26 (4 self)
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We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.
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...
Intuitionistic Choice and Classical Logic
 Arch. Math. Logic
, 1997
"... this paper we show how to combine the unrestricted countable choice, induction on infinite wellfounded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand t ..."
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Cited by 16 (4 self)
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this paper we show how to combine the unrestricted countable choice, induction on infinite wellfounded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand the extensional version of HA
Presheaf models of constructive set theories
, 2004
"... Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic S ..."
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Cited by 16 (4 self)
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Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects
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.