Results 1 
9 of
9
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 ..."
Abstract

Cited by 29 (11 self)
 Add to MetaCart
(Show Context)
1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott (&quot;Constructive Validity&quot;) [31] and MartinL&quot;of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL&quot;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, n ..."
Abstract

Cited by 27 (5 self)
 Add to MetaCart
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+# , whi ..."
Abstract

Cited by 21 (11 self)
 Add to MetaCart
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...
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, name ..."
Abstract

Cited by 15 (11 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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.
Predicative functionals and an interpretation of c ID<ω
 Ann. Pure Appl. Logic
, 1998
"... In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s univers ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinalfree characterization of the <Γ0recursive functions.
Published In Predicative Functionals and an Interpretation of ÎD<ω∗
, 1997
"... In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s uni ..."
Abstract
 Add to MetaCart
(Show Context)
In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifierfree theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative ” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinalfree characterization of the <Γ0recursive functions. 1