Abstract

. We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Fraenkel set theory and the opposite way. More precisely, we actually construct two families of encodings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r.t. expressive power and thus are essentially equivalent. Both encodings are quite elementary: type theory is interpreted in set theory through a generalization of Coquand 's simple proof-irrelevance interpretation. Set theory is encoded in type theory using a variant of Aczel's encoding; we have formally checked this last part using the Coq proof assistant. 1 Introduction This work is an attempt towards better understanding of the expressiveness of powerful type theories. We here investigate the Calculus of Inductive Constructions (CIC); this formalism is, with some variants, the one implemen...

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