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

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-Lof'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.