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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 ..."
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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. ..."
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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 ordin ..."
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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 ordinal be ..."
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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.