 
Summary: On relating type theories and set theories \Lambda
Peter Aczel
Departments of Mathematics and Computer Science
Manchester University
petera@cs.man.ac.uk
August 1, 1998
Introduction
The original motivation 1 for the work described in this paper was to determine the proof
theoretic strength of the type theories implemented in the proof development systems
Lego and Coq, [Luo and Pollack 92, Barras et al 96]. These type theories combine the
impredicative type of propositions 2 , from the calculus of constructions, [Coquand 90],
with the inductive types and hierarchy of type universes of MartinLof's constructive type
theory, [MartinLof 84]. Intuitively there is an easy way to determine an upper bound
on the proof theoretic strength. This is to use the `obvious' typesassets interpretation
of these type theories in a strong enough classical axiomatic set theory. The elementary
forms of type of MartinLof's type theory have their familiar set theoretic interpretation,
the impredicative type of propositions can be interpreted as a two element set and the
hierarchy of type universes can be interpreted using a corresponding hierarchy of strongly
inaccessible cardinal numbers. The assumption of the existence of these cardinal numbers
goes beyond the proof theoretic strength of ZFC. But MartinLof's type theory, even
