Notes on random reals

TL;DR

This paper provides an accessible introduction to the elementary aspects of the theory of random real numbers, focusing on Kolmogorov- and ML-randomness, aiming to motivate further study.

Contribution

It offers a simplified overview of key concepts in the theory of random reals, serving as an entry point for students into more advanced treatments.

Findings

Introduces basic notions of Kolmogorov- and ML-randomness

Provides simple, illustrative arguments for foundational concepts

Encourages further exploration of the rich theory of random reals

Abstract

The theory of random real numbers is exceedingly well-developed, and fascinating from many points of view. It is also quite challenging mathematically. The present notes are intended as no more than a gateway to the larger theory. They review just the most elementary part of the theory (bearing on Kolmogorov- and ML-randomness). We hope that the simple arguments presented here will encourage the enterprising student to examine richer treatments of the subject available elsewhere, notably, in Downey and Hirschfeldt (2010).