Building up a toolbox for Martin-Löf’s type theory: subset theory
Beginning in 1970, Per Martin-Löf has developed an intuitionistic type theory (hence-forth type theory for short) as a constructive alternative to the usual foundation of mathematics based on classical set theory. We assume the reader is aware at least of the main peculiarities of type theory, as formulated in Martin-Löf 1984 or Nordström et al. 1990; here we recall some of them to be able to introduce our point of view. The form of type theory is that of a logical calculus, where inference rules to derive judgements are at the same time set-theoretic constructions, because of the “propositions-as-sets” interpretation. The spirit of type theory—expressing our interpretation in a single sentence-—is to adopt those notions and rules which keep total control of the amount of information contained in the different forms of judgement. We now briefly justify this claim. First of all, the judgement asserting the truth of a proposition A, which from an intuitionistic point of view means the existence of a verification of A, in type theory is replaced by the judgement a ∈ A which explicitly exhibits a verification a of A. In fact, it would be unwise, for a constructivist, to throw away the specific verification of A which must be known to be able to assert the existence of a verification! The judgement that A is a set, which from an intuitionistic point of view means that there exists an inductive presentation of A, is treated in type theory in a quite similar way (even if in this case no notation analogous to a ∈ A is used) since the judgement A set in type theory becomes explicit knowledge of the specific inductive presentation of A. In fact, the rules for primitive types and for type constructors are so devised that whenever a judgement A set is proved, it means that one also has complete information on the rules which describe how canonical elements of A are formed. Such a property, which might look like a peculiarity of type theory, is as a matter of fact necessary in order to give a coherent constructive treatment of quantifiers.