--- a/src/Universes.v	Mon Oct 22 13:48:45 2012 -0400
+++ b/src/Universes.v	Mon Oct 22 14:23:52 2012 -0400
@@ -644,7 +644,7 @@
 
 (** On the subject of the law of the excluded middle itself, this axiom is usually quite harmless, and many practical Coq developments assume it.  It has been proved metatheoretically to be consistent with CIC.  Here, "proved metatheoretically" means that someone proved on paper that excluded middle holds in a _model_ of CIC in set theory%~\cite{SetsInTypes}%.  All of the other axioms that we will survey in this section hold in the same model, so they are all consistent together.
 
-   Recall that Coq implements%\index{constructive logic}% _constructive_ logic by default, where excluded middle is not provable.  Proofs in constructive logic can be thought of as programs.  A [forall] quantifier denotes a dependent function type, and a disjunction denotes a variant type.  In such a setting, excluded middle could be interpreted as a decision procedure for arbitrary propositions, which computability theory tells us cannot exist.  Thus, constructive logic with excluded middle can no longer be associated with our usual notion of programming.
+   Recall that Coq implements%\index{constructive logic}% _constructive_ logic by default, where the law of the excluded middle is not provable.  Proofs in constructive logic can be thought of as programs.  A [forall] quantifier denotes a dependent function type, and a disjunction denotes a variant type.  In such a setting, excluded middle could be interpreted as a decision procedure for arbitrary propositions, which computability theory tells us cannot exist.  Thus, constructive logic with excluded middle can no longer be associated with our usual notion of programming.
 
    Given all this, why is it all right to assert excluded middle as an axiom?  The intuitive justification is that the elimination restriction for [Prop] prevents us from treating proofs as programs.  An excluded middle axiom that quantified over [Set] instead of [Prop] _would_ be problematic.  If a development used that axiom, we would not be able to extract the code to OCaml (soundly) without implementing a genuine universal decision procedure.  In contrast, values whose types belong to [Prop] are always erased by extraction, so we sidestep the axiom's algorithmic consequences.