Published online by Cambridge University Press: 07 May 2020
The halting probability of a Turing machine was introduced by Chaitin, who also proved that it is an algorithmically random real number and named it Omega. Since his seminal work, many popular expositions have appeared, mainly focusing on the metamathematical or philosophical significance of this number (or debating against it). At the same time, a rich mathematical theory exploring the properties of Chaitin's Omega has been brewing in various technical papers, which quietly reveals the significance of this number to many aspects of contemporary algorithmic information theory. The purpose of this survey is to expose these developments and tell a story about Omega which outlines its multi-faceted mathematical properties and roles in algorithmic randomness.