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Archive for Mathematical Logic

, Volume 38, Issue 1, pp 19–60 | Cite as

Denotational semantics for intuitionistic type theory using a hierarchy of domains with totality

  • Geir Waagbø

Abstract.

A modified version of Normann's hierarchy of domains with totality [9] is presented and is shown to be suitable for interpretation of Martin-Löf's intuitionistic type theory. This gives an interpretation within classical set theory, which is natural in the sense that \(\Sigma\)-types are interpreted as sets of pairs and \(\Pi\)-types as sets of choice functions. The hierarchy admits a natural definition of the total objects in the domains, and following an idea of Berger [3] this makes possible an interpretation where a type is defined to be true if its interpretation contains a total object. In particular, the empty type contains no total objects and will therefore be false (in any non-empty context). In addition, there is a natural equivalence relation on the total objects, so we derive a hierarchy of topological spaces (quotient spaces wrt. the Scott topology), and give a second interpretation using this hierarchy.

Keywords

Equivalence Relation Topological Space Choice Function Type Theory Quotient Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Geir Waagbø
    • 1
  1. 1. Department of Mathematics, University of Oslo, Postboks 1053, Blindern, N-0316 Oslo, Norway NO

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