Transfinite constructions in classical type theory

Abstract

We study a transfinite construction we call tower construction in classical type theory. The construction is inductive and applies to partially ordered types. It yields the set of all points reachable from a starting point with an increasing successor function and a family of admissible suprema. Based on the construction, we obtain type-theoretic versions of the theorems of Zermelo (well-orderings), Hausdorff (maximal chains), and Bourbaki and Witt (fixed points). The development is formalized in Coq assuming excluded middle.