+This is not because there is any difficulty typing what the functions
+involved do "from the outside": for instance, the predecessor function
+is a function from numbers to numbers, or τ -> τ, where τ
+is our type for Church numbers (i.e., (σ -> σ) -> σ
+-> σ). (Though this type will only be correct if we decide that
+the predecessor of zero should be a number, perhaps zero.)
+
+Rather, the problem is that the definition of the function requires
+subterms that can't be simply-typed. We'll illustrate with our
+implementation of the predecessor, sightly modified in inessential
+ways to suit present purposes:
+
+ let zero = \s z. z in
+ let snd = \a b. b in
+ let pair = \a b. \v. v a b in
+ let succ = \n s z. s (n s z) in
+ let collect = \p. p (\a b. pair (succ a) a)
+ let pred = \n. n collect (pair zero zero) snd in
+
+Let's see how far we can get typing these terms. `zero` is the Church
+encoding of zero. Using `N` as the type for Church numbers (i.e.,
+<code>N ≡ (σ -> σ) -> σ -> σ</code> for some
+σ, `zero` has type `N`. `snd` takes two numbers, and returns
+the second, so `snd` has type `N -> N -> N`. Then the type of `pair`
+is `N -> N -> (type(snd)) -> N`, that is, `N -> N -> (N -> N -> N) ->
+N`. Likewise, `succ` has type `N -> N`, and `collect` has type `pair
+-> pair`, where `pair` is the type of an ordered pair of numbers,
+namely, <code>pair ≡ (N -> N -> N) -> N</code>. So far so good.
+
+The problem is the way in which `pred` puts these parts together. In
+particular, `pred` applies its argument, the number `n`, to the
+`collect` function. Since `n` is a number, its type is <code>(σ
+-> σ) -> σ -> σ</code>. This means that the type of
+`collect` has to match <code>σ -> σ</code>. But we
+concluded above that the type of `collect` also had to be `pair ->
+pair`. Putting these constraints together, it appears that
+<code>σ</code> must be the type of a pair of numbers. But we
+already decided that the type of a pair of numbers is `(N -> N -> N)
+-> N`. Here's the difficulty: `N` is shorthand for a type involving
+<code>σ</code>. If <code>σ</code> turns out to depend on
+`N`, and `N` depends in turn on <code>σ</code>, then
+<code>σ</code> is a proper subtype of itself, which is not
+allowed in the simply-typed lambda calculus.
+
+The way we got here is that the `pred` function relies on the built-in
+right-fold structure of the Church numbers to recursively walk down
+the spine of its argument. In order to do that, the argument had to
+apply to the `collect` operation. And since `collect` had to be the
+sort of operation that manipulates numbers, the infinite regress is
+established.
+
+Now, of course, this is only one of myriad possible implementations of
+the predecessor function in the lambda calculus. Could one of them
+possibly be simply-typeable? It turns out that this can't be done.
+See the works cited by Oleg for details.
+
+Because lists are (in effect) a generalization of the Church numbers,
+computing the tail of a list is likewise beyond the reach of the
+simply-typed lambda calculus.
+
+This result is surprising. It illustrates how recursion is built into
+the structure of the Church numbers (and lists). Most importantly for
+the discussion of the simply-typed lambda calculus, it demonstrates
+that even fairly basic recursive computations are beyond the reach of
+a simply-typed system.