This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory and Martin-Löf's intuitionistic theory of types. This paper treats Mahlo's π-numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non-datur, is equivalent to ZF plus (...) the assertion that there are inaccessibles of all transfinite orders. Finally, the theorems of that extension of CZF are interpreted in an extension of Martin-Löf's intuitionistic theory of types by a universe. (shrink)

One objective of this paper is the determination of the proof-theoretic strength of Martin-Löf'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 Δ 2 1 comprehension and bar induction. As Martin-Löf 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. (shrink)

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).