First Steps in Synthetic Computability Theory

Computability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. Consequently, there have been a number of presentati...

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Bibliographic Details
Published inElectronic notes in theoretical computer science Vol. 155; pp. 5 - 31
Main Author Bauer, Andrej
Format Journal Article
LanguageEnglish
Published Elsevier B.V 12.05.2006
Subjects
constructive mathematics
synthetic computability theory
constructive mathematics
synthetic computability theory
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Summary:Computability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. Consequently, there have been a number of presentations of computability theory that aimed to present the subject in an abstract and conceptually pleasing way. We build on two such approaches, Hyland's effective topos and Richman's formulation in Bishop-style constructive mathematics, and develop basic computability theory, starting from a few simple axioms. Because we want a theory that resembles ordinary mathematics as much as possible, we never speak of Turing machines and Gödel encodings, but rather use familiar concepts from set theory and topology.
ISSN:1571-0661
1571-0661
DOI:10.1016/j.entcs.2005.11.049