On Relating Type Theories and Set Theories

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 Martin-Lof's constructive type theory, Martin-Lof 84]. Intuitively there is an easy way to determine an upper bound on the proof theoretic strength. This is to use thèobvious' types-assets interpretation of these type theories in a strong enough classical axiomatic set theory. The elementary forms of type of Martin-Lof'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 Martin-Lof's type theory, even with its W types and its hierarchy of universes, is not fully impredicative and has proof theoretic strength way below that of second order arithmetic. So it is not clear that the strongly inaccessible cardinals used in our upper bound are really needed. Of course the impredicative type of propositions does give a fully impredicative type theory, which certainly pushes up the proof theoretic strength to a set theory 3 , Z ? , whose strength is well above that of second order arithmetic. The hierarchy of type universes will clearly This paper was written while on sabbatical leave from Manchester University. I am grateful to my two departments for making this possible. I am also grateful to Nijmegen University Computer Science Department for supporting my visit there. Some of the ideas for this paper were developed during that visit. 1 The same motivation may be found in Werner 97]. More or less the same tools are used there as here; i.e. the TS and ST interpretations. But that paper focuses on slightly diierent results to the ones obtained here. 2 Here we will ignore the use of any rules for putting types other than types into the impredicative type of propositions 3 The theory Z ? is obtained from Zermelo set theory, Z, by only using formulae with restricted quantiiers in the …