Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In Mendelson's Introduction to Mathematical Logic, the proof of Godel's Theorem for S (his axiomatic arithmetic) goes via proving that a sentence that can be interpreted as "This statement has no proof in S" cannot be proved either false or true in S, if S is consistent.

According to the completeness of the Predicate Calculus, any logically valid wf of a theory K is a theorem of K. The statement I interpret as "This statement has no proof in S" cannot be proved either false or true, so I presume that there are models of arithmetic (or rather of axiom system S) in which it is false, and models in which it is true. Is this correct?

A model of arithmetic in which it is true seems sane enough. Are there models of S in which a proof of what is interpreted as "This statement has no proof in S" turns up as some sort of non-standard number, or have I got completely confused? I have a vision in my mind of a non-standard number encoding "1 is not a proof of S. 2 is not a proof of S. 3 is not a proof of S...." or in some other way satisfying the equation that asserts that X is a proof of Y, if not the mathematician posing the equation :-)

share|cite|improve this question

1 Answer 1

up vote 4 down vote accepted

A model of arithmetic in which the G"odel sentence "I am unprovable" is false is necessarily a non-standard model. It contains an infinite element which satisfies, in the model, the formula expressing the property of being a proof of the G"odel sentence --- a formula that is not satisfied by any standard natural number (not even in a non-standard model).

share|cite|improve this answer
You can type o with an umlaut as ö but not in a comment, apparently :( – Nate Eldredge Sep 27 '10 at 21:22
(ö)/ There's no reason to be diacritical. – Eric Tressler Sep 27 '10 at 23:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.