description

Name: ZFC
Title: An encoding of Zermelo-Fraenkel Set Theory in Coq
Author: Benjamin Werner
Institution: INRIA Rocquencourt
Description:
  The encoding of Zermelo-Fraenkel Set Theory is largely inspired by
Peter Aczel's work dating back to the eighties. A type Ens is defined,
which represents sets. Two predicates IN and EQ stand for membership
and extensional equality between sets. The axioms of ZFC are then
proved and thus appear as theorems in the development.

  A main motivation for this work is the comparison of the respective
expressive power of Coq and ZFC.

  A non-computational type-theoretical axiom of choice is necessary to
prove the replacement schemata and the set-theoretical AC.

  The main difference between this work and Peter Aczel's is that
propositions are defined on the impredicative level Prop. Since the
definition of Ens is, however, still unchanged, I also added most of
Peter Aczel's definition. The main advantage of Aczel's approach is a
more constructive vision of the existential quantifier (which gives
the set-theoretical axiom of choice for free).

Keywords: Set Theory, Zermelo-Fraenkel, Calculus of Inductive Constructions
Category: Mathematics/Logic/Set theory