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

Infinity in computable probability