Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T23:00:22.131Z Has data issue: false hasContentIssue false

6 - Aspects of Chaitin’s Omega

Published online by Cambridge University Press:  07 May 2020

By
George Barmpalias
Edited by
Johanna N. Y. Franklin  and
Christopher P. Porter
Johanna N. Y. Franklin
Affiliation:
Hofstra University, New York
Christopher P. Porter
Affiliation:
Drake University, Iowa
Get access

Summary

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.

Keywords

algorithmic informationalgorithmic randomnessChaitin's Omegacomputability theoryhalting probability
Type
Chapter
Information
Algorithmic Randomness
Progress and Prospects
 , pp. 175 - 205
Publisher: Cambridge University Press
Print publication year: 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×