Library Coq.Logic.Berardi
This file formalizes Berardi's paradox which says that in
the calculus of constructions, excluded middle (EM) and axiom of
choice (AC) imply proof irrelevance (PI).
Here, the axiom of choice is not necessary because of the use
of inductive types.
@article{Barbanera-Berardi:JFP96,
author = {F. Barbanera and S. Berardi},
title = {Proof-irrelevance out of Excluded-middle and Choice
in the Calculus of Constructions},
journal = {Journal of Functional Programming},
year = {1996},
volume = {6},
number = {3},
pages = {519-525}
}
Excluded middle
Hypothesis EM :
forall P:
Prop,
P \/ ~ P.
Conditional on any proposition.
Axiom of choice applied to disjunction.
Provable in Coq because of dependent elimination.
We assume a type with two elements. They play the role of booleans.
The main theorem under the current assumptions is that T=F
The powerset operator
A piece of theory about retracts
The dependent elimination above implies the axiom of choice:
This lemma is basically a commutation of implication and existential
quantification: (EX x | A -> P(x)) <=> (A -> EX x | P(x))
which is provable in classical logic ( => is already provable in
intuitionnistic logic).
The paradoxical set
Definition U :=
forall P:
Prop,
pow P.
Bijection between U and (pow U)
We deduce that the powerset of U is a retract of U.
This lemma is stated in Berardi's article, but is not used
afterwards.
Encoding of Russel's paradox
The boolean negation.
the set of elements not belonging to itself