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.
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.
Published online by Cambridge University Press: 05 June 2012
At the end of this short chapter, we introduce the pivotal idea of a p.r. adequate theory of arithmetic, i.e. one that can appropriately capture all p.r. functions, properties and relations. Then, in the next chapter, we will show that Q and hence PA are p.r. adequate.
However, we haven't yet explained the idea of capturing a function as opposed to capturing a property or relation. So we must start with that.
Capturing a function
Suppose f is a one-place numerical function. Now define the relation Rf by saying that m has the relation Rf to n just in case f(m) = n. We'll say Rf is f's corresponding relation. Functions and their corresponding relations match up pairs of things in exactly the same way: f and Rf have the same extension, namely the set of ordered pairs 〈m, f(m)〉.
And just as the characteristic function trick (Section 11.6) allows us to take ideas defined for functions and apply them to properties and relations, this very simple tie between functions and their corresponding relations allows us to carry over ideas defined for relations and apply them to functions (total functions, as always in this book.)
For a start, consider how we can use this tie to define the idea of expressing a function using an open wff.
To save this book to your Kindle, first ensure no-reply@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.
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.
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.
