Sets in Coq, Coq in Sets - Bruno Barras (Journal of Formalized Reasoning, 2010)
This work is about formalizing models of various type theories of the Calculus of Constructions family. Here we focus on set theoretical models. The long-term goal is to build a formal set theoretical model of the Calculus of Inductive Constructions, so we can be sure that Coq is consistent with the language used by most mathematicians.One aspect of this work is to axiomatize several set theories: ZF possibly with inaccessible cardinals, and HF, the theory of hereditarily finite sets. On top of these theories we have developped a piece of the usual set theoretical construction of functions, ordinals and fixpoint theory. We then proved sound several models of the Calculus of Constructions, its extension with an infinite hierarchy of universes, and its extension with the inductive type of natural numbers where recursion follows the type-based termination approach.The other aspect is to try and discharge (most of) these assumptions. The goal here is rather to compare the theoretical strengths of all these formalisms. As already noticed by Werner, the replacement axiom of ZF in its general form seems to require a type-theoretical axiom of choice (TTAC).