X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week2.mdwn;h=a8e3ca4e6968aa393b6de9e0a7a8c1d74724d7be;hp=eedc6de0458e1046e2b9ec5bdee7f4ac6dfd28da;hb=ff25bb904871819368f18ccec236263072cb561f;hpb=82558787dab095c164d4a6d2cdfceebc9b335698
diff --git a/week2.mdwn b/week2.mdwn
index eedc6de0..a8e3ca4e 100644
--- a/week2.mdwn
+++ b/week2.mdwn
@@ -324,13 +324,13 @@ The logical system you get when eta-reduction is added to the proof system has t
> if `M`, `N` are normal forms with no free variables, then M ≡ N
iff `M` and `N` behave the same with respect to every possible sequence of arguments.
-That is, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments L1, ..., Ln
such that:
+This implies that, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments L1, ..., Ln
such that:
M L1 ... Ln x y ~~> x
N L1 ... Ln x y ~~> y
-That is, closed normal forms that are not just beta-reduced but also fully eta-reduced, will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
+So closed beta-plus-eta-normal forms will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
So the proof theory with eta-reduction added is called "extensional," because its notion of normal form makes syntactic identity of closed normal forms coincide with extensional equivalence.
@@ -432,6 +432,7 @@ But is there any method for doing this in general---for telling, of any given co
* [Scooping the Loop Snooper](http://www.cl.cam.ac.uk/teaching/0910/CompTheory/scooping.pdf), a proof of the undecidability of the halting problem in the style of Dr Seuss by Geoffrey K. Pullum
+Interestingly, Church also set up an association between the lambda calculus and first-order predicate logic, such that, for arbitrary lambda formulas `M` and `N`, some formula would be provable in predicate logic iff `M` and `N` were convertible. So since the right-hand side is not decidable, questions of provability in first-order predicate logic must not be decidable either. This was the first proof of the undecidability of first-order predicate logic.
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