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Inductive Sets and Families in MartinLöf's Type Theory and Their SetTheoretic Semantics
 Logical Frameworks
, 1991
"... MartinLof's type theory is presented in several steps. The kernel is a dependently typed calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. ..."
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Cited by 76 (13 self)
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MartinLof's type theory is presented in several steps. The kernel is a dependently typed calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. At each step syntax, inference rules, and settheoretic semantics are given. 1 Introduction Usually MartinLof's type theory is presented as a closed system with rules for a finite collection of set formers. But it is also often pointed out that the system is in principle open to extension: we may introduce new sets when there is a need for them. The principle is that a set is by definition inductively generated  it is defined by its introduction rules, which are rules for generating its elements. The elimination rule is determined by the introduction rules and expresses definition by primitive recursion on the way the elements of the set are generated. (In this paper I shall use the term ...
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
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
Setoids in Type Theory
, 2000
"... Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we ..."
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Cited by 30 (4 self)
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Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we argue that a commonly advocated approach to partial setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1
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.
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...
Inaccessibility in Constructive Set Theory and Type Theory
, 1998
"... This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo's numbers whi ..."
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Cited by 16 (4 self)
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This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo's numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of MartinLof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's numbers and relating the history of universes in MartinLof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...
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