Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T20:21:57.158Z Has data issue: false hasContentIssue false

9 - Extension of λC with definitions

Published online by Cambridge University Press:  05 November 2014

Rob Nederpelt
Affiliation:
Technische Universiteit Eindhoven, The Netherlands
Herman Geuvers
Affiliation:
Radboud Universiteit Nijmegen
Get access

Summary

Extension of λC to the system λD0

In the present chapter we investigate the formal aspects of adding definitions to a type system. In this we follow the pioneering work of N.G. de Bruijn (cf. de Bruijn, 1970). As the basic system we take λC, the most powerful system in the λ-cube. System λC is suitable for the PAT-interpretation, because it encapsulates λP. But it also covers the nice second order aspects of λ2. Therefore, λC appears to be enough for the purpose of ‘coding’ mathematics and mathematical reasonings and is an excellent candidate for the natural extension we want, being almost inevitable for practical applications: the addition of definitions.

We start with an extension leading from λC to a system called λD0. This system contains a formal version of definitions in the usual sense, the so-called descriptive definitions, so it can be used for a great amount of applications in the realm of logic and mathematics. But λD0 does not yet allow a satisfactory representation of axioms and axiomatic notions; these will be considered in the following chapter, in which a small, further extension of λD0 leads to our final system λD. (We have noticed before that we do not consider inductive and recursive definitions, since we can do without them; see Section 8.2.)

In order to give a proper description of λD0, we first extend our set of expressions, as given in Definition 6.3.1 for λC.

Type
Chapter
Information
Type Theory and Formal Proof
An Introduction
, pp. 189 - 210
Publisher: Cambridge University Press
Print publication year: 2014

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 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.

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
×