On universes in type theory

Abstract

The notion of a universe of types was introduced into constructive type theory by Martin-Löf (1975). According to the propositions-as-types principle inherent in type theory, the notion plays two roles. The first is as a collection of sets or types closed under certain type constructions. The second is as a set of constructively given infinitary formulas. In this paper we discuss the notion of universe in type theory and suggest and study some useful extensions. We assume familiarity with type theory as presented in, for example, Martin-Löf (1984). Universes have been effective in expanding the realm of constructivism. One example is constructive category theory where type universes take the roles of Grothendieck universes of sets, in handling large categories. A more profound example is Aczel’s (1986) type-theoretic interpretation of constructive set theory (CZF). It is done by coding ϵ-diagrams into well-order types, with branching over an arbitrary type of the universe. The latter generality is crucial for interpreting the separation axiom. The introduction of universes and well-orders (W-types) in conjunction gives a great proof-theoretic strength. This has provided constructive justification of strong subsystems of second-order arithmetic studied by proof-theorists (see Griffor and Rathjen (1994) and Setzer (1993), and for some early results, see Palmgren (1992)). At present, it appears that the most easily justifiable way to increase the proof-theoretic strength of type theory is to introduce ever more powerful universe constructions. We will give two such extensions in this paper. Besides contributing to the understanding of subsystems of second-order arithmetic and pushing the limits of inductive definability, such constructions provide intuitionistic analogues of large cardinals (Rathjen et al, in press). A third new use of universes is to facilitate the incorporation of classical reasoning into constructive type theory. We introduce a universe of classical propositions and prove a conservation result for ‘Π-formulas’. Extracting programs from classical proofs is then tractable within type theory. The next section gives an introduction to the notion of universe. The central part of the paper is section 3 where we introduce a universe forming operator and a super universe closed under this operator.