`x`_{n}

and `zero`, where `x`_{n}

is the last element of the list. This gives us `successor zero`, or `one`. That's the value we've accumuluted "so far." Then we go apply the function `\x sofar. successor sofar` to the two arguments `x`_{n-1}

and the value `one` that we've accumulated "so far." This gives us `two`. We continue until we get to the start of the list. The value we've then built up "so far" will be the length of the list.
We can use similar techniques to define many recursive operations on lists and numbers. The reason we can do this is that our "version 3," fold-based implementation of lists, and Church's implementations of numbers, have a internal structure that *mirrors* the common recursive operations we'd use lists and numbers for.
As we said before, it does take some ingenuity to define functions like `extract-tail` or `predecessor` for these implementations. However it can be done. (And it's not *that* difficult.) Given those functions, we can go on to define other functions like numeric equality, subtraction, and so on, just by exploiting the structure already present in our implementations of lists and numbers.
With sufficient ingenuity, a great many functions can be defined in the same way. For example, the factorial function is straightforward. The function which returns the nth term in the Fibonacci series is a bit more difficult, but also achievable.
##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 Ackermann function:
A(m,n) =
| when m == 0 -> n + 1
| else when n == 0 -> A(m-1,1)
| else -> A(m-1, A(m,n-1))
A(0,y) = y+1
A(1,y) = y+2
A(2,y) = 2y + 3
A(3,y) = 2^(y+3) -3
A(4,y) = 2^(2^(2^...2)) [where there are y+3 2s] - 3
...
Simpler functions always *could* be defined using the resources we've so far developed, although those definitions won't always be very efficient or easily intelligible.
But functions like the Ackermann function require us to develop a more general technique for doing recursion---and having developed it, it will often be easier to use it even in the cases where, in principle, we didn't have to.
##How to do recursion with lower-case omega##
[TODO]
##Generalizing##
In general, a **fixed point** of a function f is a value *x* such that f`Ψ`

. That is, some function that returns, for any expression `f` we give it as argument, a fixed point for `f`. In other words:
`Ψ f <~~> f (Ψ f)`

Then applying `Ψ`

to the "starting formula" displayed above would give us our fixed point `X` for the starting formula:
`Ψ (\self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) ))`

And this is the fully general strategy for
defining recursive functions in the lambda calculus. You begin with a "body formula":
...self...
containing free occurrences of `self` that you treat as being equivalent to the body formula itself. In the case we're considering, that was:
\lst. (isempty lst) zero (add one (self (extract-tail lst)))
You bind the free occurrence of `self` as: `\self. BODY`. And then you generate a fixed point for this larger expression:
`Ψ (\self. BODY)`

using some fixed-point combinator `Ψ`

.
Isn't that cool?
##Okay, then give me a fixed-point combinator, already!##
Many fixed-point combinators have been discovered. (And given a fixed-point combinators, there are ways to use it as a model to build infinitely many more, non-equivalent fixed-point 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.
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 `Θ′`

to just `u u f`? And similarly for `Y′`

?
Indeed you can, getting the simpler:
```
Θ ≡ (\u f. f (u u f)) (\u f. f (u u f))
Y ≡ \f. (\u. f (u u)) (\u. f (u u))
```

I stated the more complex formulas for the following reason: in a language whose evaluation order is *call-by-value*, the evaluation of `Θ (\self. BODY)`

and `Y (\self. BODY)` will in general not terminate. But evaluation of the eta-unreduced primed versions will.
Of course, if you define your `\self. BODY` stupidly, your formula will never terminate. For example, it doesn't matter what fixed point combinator you use for `Ψ`

in:
`Ψ (\self. \n. self n)`

When you try to evaluate the application of that to some argument `M`, it's going to try to give you back:
(\n. self n) M
where `self` is equivalent to the very formula `\n. self n` that contains it. So the evaluation will proceed:
(\n. self n) M ~~>
self M ~~>
(\n. self n) M ~~>
self M ~~>
...
You've written an infinite loop!
However, when we evaluate the application of our:
`Ψ (\self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) ))`

to some list `L`, we're not going to go into an infinite evaluation loop of that sort. At each cycle, we're going to be evaluating the application of:
\lst. (isempty lst) zero (add one (self (extract-tail lst)))
to *the tail* of the list we were evaluating its application to at the previous stage. Assuming our lists are finite (and the implementations we're using don't permit otherwise), at some point one will get a list whose tail is empty, and then the evaluation of that formula to that tail will return `zero`. So the recursion eventually bottoms out in a base value.
##Fixed-point Combinators Are a Bit Intoxicating##
![tatoo](/y-combinator.jpg)
There's a tendency for people to say "Y-combinator" to refer to fixed-point combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Y-combinator is only one of many fixed-point combinators.
I used `Ψ`

above to stand in for an arbitrary fixed-point combinator. I don't know of any broad conventions for this. But this seems a useful one.
As we said, there are many other fixed-point combinators as well. For example, Jan Willem Klop pointed out that if we define `L` to be:
\a b c d e f g h i j k l m n o p q s t u v w x y z r. (r (t h i s i s a f i x e d p o i n t c o m b i n a t o r))
then this is a fixed-point combinator:
L L L L L L L L L L L L L L L L L L L L L L L L L L
[TODO: Explain how what we've done relates to the version using lower-case ω.]