The strength of some Martin-L�f type theories

Abstract

Abstract. One objective of this paper is the determination of the proof-theoretic strength of Martin-L6f's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with A~ comprehension and bar induction. As Martin-L6f intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts.Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. IntroductionMartin-L6f's intuitionistic theory of types was originally introduced as a system for formalizing intuitionistic mathematics. It is a typed theory of constructions containing types which themselves depend on the constructions contained in previously constructed types. These dependent types enable one to express the general Cartesian product of the resulting families of types, as well as the disjoint union of such a famiIy.Using the Brouwer-Heyting semantics of the logical constants one sees how logical notions are obtained in this theory, This is done by interpreting propositions as types and proofs as constructions [ML 84]. In addition to ground types for the finite sets and the set of natural numbers the theory can be taken to contain types which play * The second author would like to thank the National Science Foundation of the USA for support by grant DMS-9203443

The generalised type-theoretic interpretation of constructive set theory

“…A particularly important role will be played in the following by the natural subtheory MLi + W of MLWi, which has W-types but they are not reflected in U. Many proof-theoretical results concerning these pure type theories are presented in [17]. where (p and y/ are propositions.…”

“…We now develop an analogous system of abbreviations for the type of small subclasses of a type, as defined in (13). For a : Sub{A) we define el(a) =def ol.\ : U. Judgements analogous to those in (16) are easily derivable, so that the definitions in (17) and (18) The problem can be explained as follows. In CZF.…”

“…We introduce versions of quantification over iterative small classes using Lem ma 3.8 and the definitions expressing quantification over a small subclass of a type in (17) and ( Again, these definitions can easily be shown to be compatible.…”

“…The type-theoretic interpretation of Constructive Zermelo-Frankel set theory, or CZF for short, provides an explicit link between constructive set theory and Martin-L?f type theory [1,2,3]. This interpretation is a useful tool in the proof theoretical investigations of constructive formal systems [17] and allows us to relate the set-theoretic and type-theoretic approaches to the development of constructive …”

Wellfounded Trees and Dependent Polynomial Functors

“…Secondly, they play an essential role in the interpretation of constructive set theories in dependent type theories [3]. Finally, from the proof-theoretic point of view, they represent the paradigmatic example of a generalised inductive definition and contribute considerably to the proof-theoretic strength of dependent type theories [8].…”