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Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathemat ..."
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Cited by 8 (5 self)
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This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are prooftheoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and prooftheoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...
Impredicative Overloading in Explicit Mathematics
, 2000
"... In this article we introduce the system OTN of explicit mathematics based on elementary separation, product, join and weak power types. We present a settheoretical model for OTN, and we develop in OTN a theory of impredicative overloading. Together this yields a solution to the problem of impredica ..."
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Cited by 2 (2 self)
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In this article we introduce the system OTN of explicit mathematics based on elementary separation, product, join and weak power types. We present a settheoretical model for OTN, and we develop in OTN a theory of impredicative overloading. Together this yields a solution to the problem of impredicativity encountered in denotational semantics for overloading and latebinding. Further, our work provides a first example of an application of power types in explicit mathematics. Keywords: Objectoriented constructs, type structure, proof theory. 1 Introduction Overloading is an important concept in objectoriented programming. For example, it occurs when a method is redefined in a subclass or when a class provides several methods with the same name but with di#erent argument types. Theoretically speaking, overloading denotes the possibility that several functions f i with respective types S i # T i may be combined to a new overloaded function f of type {S i # T i } i#I . We then ...
A Semantics for ...: A Calculus With Overloading and LateBinding
, 1999
"... Up to now there was no interpretation available for calculi featuring overloading and latebinding, although these are two of the main principles of any objectoriented programming language. In this paper we provide a new semantics for a stratied version of Castagna's fg , a calculus combini ..."
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Up to now there was no interpretation available for calculi featuring overloading and latebinding, although these are two of the main principles of any objectoriented programming language. In this paper we provide a new semantics for a stratied version of Castagna's fg , a calculus combining overloading with latebinding. The modelconstruction is carried out in EETJ + (Tot) + (FI N ), a system of explicit mathematics. We will prove the soundness of our model with respect to subtyping, typechecking and reductions. Furthermore, we show that our semantics yields a solution to the problem of loss of information in the context of type dependent computations. Keywords: Explicit mathematics, typed calculus, overloading, latebinding, loss of information. 1 Introduction Polymorphism is one of the concepts to which the objectoriented paradigm owes its power. The distinction is made between parametric (or universal) and \ad hoc" polymorphism. Using parametric polymorp...