Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-2qt69 Total loading time: 0.245 Render date: 2022-08-08T08:25:51.446Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Order in open intervals of computable reals

Published online by Cambridge University Press:  01 February 1999

JAN VON PLATO
Affiliation:
University of Helsinki, Unioninkatu 40 B, 00014 University of Helsinki, Finland. E-mail: vonplato@helsinki.fi

Abstract

In constructive theories, an apartness relation is often taken as basic and its negation used as equality. An apartness relation should be continuous in its arguments, as in the case of computable reals. A similar approach can be taken to order relations. We shall here study the partial order on open intervals of computable reals. Since order on reals is undecidable, there is no simple uniformly applicable lattice meet operation that would always produce non-negative intervals as values. We show how to solve this problem by a suitable definition of apartness for intervals. We also prove the strong extensionality of the lattice operations, where by strong extensionality of an operation f on elements a, b we mean that apartness of values implies apartness in some of the arguments: f(a, b)≠f(c, d) ⊃acbd.

Most approaches to computable reals start from a concrete definition. We shall instead represent them by an abstract axiomatically introduced order structure.

Type
Research Article
Copyright
1999 Cambridge University Press

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.)
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org 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.

Order in open intervals of computable reals
Available formats
×

Save article to Dropbox

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

Order in open intervals of computable reals
Available formats
×

Save article to Google Drive

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

Order in open intervals of computable reals
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *