XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week3.mdwn;h=477284245c5f48b88919c6b2a24ee18d9bd7d57d;hp=a1e79f5c0538e4d913d8e3e6360f908c8590cdfc;hb=aaadc20de94685e29c35abbd865e5f35d194e7b9;hpb=e9421d05c554e8abfa13c11eb62789d6c0a19d59
diff git a/week3.mdwn b/week3.mdwn
index a1e79f5c..47728424 100644
 a/week3.mdwn
+++ b/week3.mdwn
@@ 126,7 +126,10 @@ With sufficient ingenuity, a great many functions can be defined in the same way
##However...##
Some computable functions are just not definable in this way. The simplest function that *simply cannot* be defined using the resources we've so far developed is the [[!wikipedia Ackermann function]]:
+Some computable functions are just not definable in this way. We can't, for example, define a function that tells us, for whatever function `f` we supply it, what is the smallest integer `x` where `f x` is `true`.
+
+Neither do the resources we've so far developed suffice to define the
+[[!wikipedia Ackermann function]]:
A(m,n) =
 when m == 0 > n + 1
@@ 333,16 +336,14 @@ Isn't that cool?
##Okay, then give me a fixedpoint combinator, already!##
Many fixedpoint combinators have been discovered. (And given a fixedpoint combinator, there are ways to use it as a model to build infinitely many more, nonequivalent fixedpoint combinators.)
+Many fixedpoint combinators have been discovered. (And some fixedpoint combinators give us models for building infinitely many more, nonequivalent fixedpoint combinators.)
Two of the simplest:
Θ′ ≡ (\u f. f (\n. u u f n)) (\u f. f (\n. u u f n))
Y′ ≡ \f. (\u. f (\n. u u n)) (\u. f (\n. u u n))
Θ′
has the advantage that f (Θ′ f)
really *reduces to* Θ′ f
.

f (Y′ f)
is only convertible with Y′ f
; that is, there's a common formula they both reduce to. For most purposes, though, either will do.
+Θ′
has the advantage that f (Θ′ f)
really *reduces to* Θ′ f
. Whereas f (Y′ f)
is only *convertible with* Y′ f
; that is, there's a common formula they both reduce to. For most purposes, though, either will do.
You may notice that both of these formulas have etaredexes inside them: why can't we simplify the two `\n. u u f n` inside Θ′
to just `u u f`? And similarly for Y′
?