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
According to Gödel's second incompleteness theorem, the sentence expressing that a theory likePis consistent is undecidable byP, supposingPis consistent. The full proof of this result is beyond the scope of a book on the level of the present one, but the overall structure of the proof and main ingredients that go into the proof will be indicated in this short chapter. In place of problems there are some historical notes at the end.
Officially we defined T to be inconsistent if every sentence is provable from T, though we know this is equivalent to various other conditions, notably that for some sentence S, both S and ~S are provable from T. If T is an extension of Q, then since 0 ≠ 1 is the simplest instance of the first axiom of Q, 0 ≠ 1 is provable from T, and if 0 = 1 is also provable from T, then T is inconsistent; while if T is inconsistent, then 0 = 1 is provable from T, since every sentence is. Thus T is consistent if and only if 0 = 1 is not provable from T. We call ~PrvT(), which is to say ~∃y PrfT(, y), the consistency sentence for T. Historically, the original paper of Gödel containing his original version of the first incompleteness theorem (corresponding to our Theorem 17.9) included towards the end a statement of a version of the following theorem.
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.
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.
