Abstract
Abstract Until now, speaking of types has been speaking of sets. In this chapter, we shall extend the notion of type to cover the type of sets, the type of propositions, various types of propositional functions, the type of quantifiers, etc. This notion of type, or category, was introduced by Martin-Löf in 1984, pp. 21-23. He soon developed a calculus of types, the higher-level type theory. The type theory of Martin-Löf 1982 and 1984, within which we have been working so far, is accordingly called lower-level type theory, or Martin-Löf’s set theory.Higher-level type theory relates to lower-level type theory in the same way as simple type theory (Church 1940) relates to classical predicate calculus. At the same time, it provides a logical framework, in which many different logical calculi and mathematical theories can be presented.
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