Results 1 
7 of
7
An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
Abstract

Cited by 65 (0 self)
 Add to MetaCart
An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
Functional interpretation and inductive definitions
 Journal of Symbolic Logic
"... Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1. ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1.
A Comparison of Two Systems of Ordinal Notations
"... I show how the Bachmann method of generating countable ordinals using uncountable ordinals can be replaced by the use of higher order xed point extractors available in the term calculus of Howard's system of constructive ordinals. This leads to a notion of the intrinsic complexity of a notated ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
I show how the Bachmann method of generating countable ordinals using uncountable ordinals can be replaced by the use of higher order xed point extractors available in the term calculus of Howard's system of constructive ordinals. This leads to a notion of the intrinsic complexity of a notated ordinal analogous to the intrinsic complexity of a numeric function described in Gödel's T.
An Applied λCalculus for Iteration Templates
"... Let H be the term calculus of Howard's system of constructive ordinals. I use this to name iteration gadgets (generalized ordinal notations). I isolate a family of higher order ordinal functions which give a partial semantics of H. I indicate how H provides an intrinsic measure of each ordin ..."
Abstract
 Add to MetaCart
Let H be the term calculus of Howard's system of constructive ordinals. I use this to name iteration gadgets (generalized ordinal notations). I isolate a family of higher order ordinal functions which give a partial semantics of H. I indicate how H provides an intrinsic measure of each ordinal below the Howard ordinal.
Iteration Templates as Generalized Ordinal Notations
"... This document is a survey of the three papers [H], [Sch],[Com]. It gives the relevant background and shows how the three papers t together to form a whole. Material from those papers is repeated only when necessary. As it stands the document is not intended for publication. However, if there is suci ..."
Abstract
 Add to MetaCart
This document is a survey of the three papers [H], [Sch],[Com]. It gives the relevant background and shows how the three papers t together to form a whole. Material from those papers is repeated only when necessary. As it stands the document is not intended for publication. However, if there is sucient interest then I may rewrite it to include [H], [Sch],[Com], and so from a self contained developement. This version was complied on November 19, 2001. Contents 1
Contents
"... Let Ω be an arbitrary frame, and consider the category Psh(Ω) of presheaves on Ω and the and the category Set(Ω) of Ωsets. There are various other associated categories and functors between them. I describe these with emphasis on the functor which separates a presheaf and the functor that sheafifie ..."
Abstract
 Add to MetaCart
Let Ω be an arbitrary frame, and consider the category Psh(Ω) of presheaves on Ω and the and the category Set(Ω) of Ωsets. There are various other associated categories and functors between them. I describe these with emphasis on the functor which separates a presheaf and the functor that sheafifies a presheaf. These are notes from a short course (of 6 hours) given in August and September 2001. They are little more than a first draft, and may contain some errors. In particular the notes contain some comments in this kind of type face indicating that there is something still to be done or sorted out. If you find these notes useful or think there are parts that could be improved, please feel free to contact me.