-Moreover, the recipes that enable us to name fixed points for any
-given formula aren't *guaranteed* to give us *terminating* fixed
-points. They might give us formulas X such that neither `X` nor `f X`
-have normal forms. (Indeed, what they give us for the square function
-isn't any of the Church numerals, but is rather an expression with no
-normal form.) However, if we take care we can ensure that we *do* get
-terminating fixed points. And this gives us a principled, fully
-general strategy for doing recursion. It lets us define even functions
-like the Ackermann function, which were until now out of our reach. It
-would also let us define arithmetic and list functions on the "version
-1" and "version 2" implementations, where it wasn't always clear how
-to force the computation to "keep going."
+Let's pick a way of defining the successor function and reason about it.
+Here is one way that is compatible with the constraints given in
+homework 2: `succ := \nfz.f(nfz)`. This takes a Church
+number, and returns the next Church number. For instance,
+
+ succ 2 == succ (\fz.f(fz))
+ == (\nfz.f(nfz)) (\fz.f(fz))
+ ~~> \fz.f((\fz.f(fz))fz)
+ ~~> \fz.f(f(fz))
+ == 3
+
+Using logic similar to the discussion above of the fixed point for K,
+we can say that for any Church number argument to the successor
+function, the result will be the next Church number. Assume that
+there is some Church number `n` that is a fixed point. Then
+`succ n <~~> n` (because `n` is a fixed point) and `succ n <~~> n + 1`
+(since that's what the successor function does). By the Church Rosser
+theorem, `n <~~> n + 1`. What kind of `n` could satisfy that
+requirement?
+
+Let's run the recipe:
+
+ H := \f . succ (ff)
+ == \f . (\nfz.f(nfz)) (ff)
+ ~~> \h . (\nfz.f(nfz)) (hh)
+ ~~> \hfz.f(hhfz)
+
+ H H == (\hfz.f(hhfz)) (\hfz.f(hhfz))
+ ~~> \fz.f((\hfz.f(hhfz))(\hfz.f(hhfz))fz)
+ ~~> \fz.f(f((\hfz.f(hhfz))(\hfz.f(hhfz))fz))
+ ~~> \fz.f(f(f((\hfz.f(hhfz))(\hfz.f(hhfz))fz))
+
+We can see that the fixed point generates an endless series of `f`'s.
+In terms of Church numbers, this is a way of representing infinity:
+if the size of a Church number is the number `f`'s it contains, and
+this Church number contains an unbounded number of `f`'s, then its
+size is unbounded.
+
+We can also see how this candidate for infinity behaves with respect
+to our other arithmetic operators.
+
+ add 2 (HH) == (\mnfz.mf(nfz)) (\fz.f(fz)) (H H)
+ ~~> \fz.(\fz.f(fz)) f ((HH)fz)
+ ~~> \fz.\z.f(fz) ((HH)fz)
+ ~~> \fz.f(f((HH)fz))
+ == \fz.f(f(((\hfz.f(hhfz)) (\hfz.f(hhfz)))fz))
+ ~~> \fz.f(f((\fz.f((\hfz.f(hhfz)) (\hfz.f(hhfz))))fz))
+ ~~> \fz.f(f(f((\hfz.f(hhfz)) (\hfz.f(hhfz)))))
+
+So 2 + (HH) <~~> (HH). This is what we expect from arithmetic infinity.
+You can check to see if 2 * (HH) <~~> (HH).
+
+So our fixed point recipe has delivere a reasonable candidate for
+arithmetic infinity.
+
+One (by now obvious) upshot is that the recipes that enable us to name
+fixed points for any given formula aren't *guaranteed* to give us
+*terminating* fixed points. They might give us formulas X such that
+neither `X` nor `f X` have normal forms. (Indeed, what they give us
+for the square function isn't any of the Church numerals, but is
+rather an expression with no normal form.) However, if we take care we
+can ensure that we *do* get terminating fixed points. And this gives
+us a principled, fully general strategy for doing recursion. It lets
+us define even functions like the Ackermann function, which were until
+now out of our reach. It would also let us define arithmetic and list
+functions on the "version 1" and "version 2" implementations, where it
+wasn't always clear how to force the computation to "keep going."