  Does combining a finite collection of objects infinitely many times guarantee
the construction of a particular object? Here we use recursive function theory
to examine the popular scenario of an infinite collection of typing monkeys
reproducing the works of Shakespeare. Our main result is to show that it is
possible to assign typing probabilities in such a way that while it is
impossible that no monkey reproduces Shakespeare's works, the probability of
any finite collection of monkeys doing so is arbitrarily small. We extend our
results to target-free writing, and end with a broad discussion and pointers to
future work.
