##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
##Okay, then give me a fixed-point combinator, already!##
-Many fixed-point combinators have been discovered. (And given a fixed-point combinator, there are ways to use it as a model to build infinitely many more, non-equivalent fixed-point combinators.)
+Many fixed-point combinators have been discovered. (And some fixed-point combinators give us models for building infinitely many more, non-equivalent fixed-point combinators.)
Two of the simplest:
<pre><code>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))</code></pre>
-Θ′ has the advantage that <code>f (Θ′ f)</code> really *reduces to* <code>Θ′ f</code>.
-
-<code>f (Y′ f)</code> is only convertible with <code>Y′ f</code>; that is, there's a common formula they both reduce to. For most purposes, though, either will do.
+<code>Θ′</code> has the advantage that <code>f (Θ′ f)</code> really *reduces to* <code>Θ′ f</code>. Whereas <code>f (Y′ f)</code> is only *convertible with* <code>Y′ f</code>; 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 eta-redexes inside them: why can't we simplify the two `\n. u u f n` inside <code>Θ′</code> to just `u u f`? And similarly for <code>Y′</code>?