Infinity in computable probability

TL;DR

This paper explores the probability of infinite objects being produced by infinite repetitions of finite processes, showing that certain outcomes can be almost impossible despite their inevitability in theory.

Contribution

It introduces a novel probabilistic framework using recursive function theory to analyze infinite collections of processes and their ability to produce specific objects.

Findings

It is possible to assign probabilities so that the chance of any finite subset reproducing a target object is arbitrarily small.

The results extend to scenarios involving target-free writing.

The study provides a theoretical foundation for understanding infinite probabilistic processes.

Abstract

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.