One consequence of the second incompleteness theorem is that no consistent arithmetic theory of sufficient strength can express its own consistency. One way of

One consequence of the second incompleteness theorem is that no consistent arithmetic theory of sufficient strength can express its own consistency. One way of proving such a theory inconsistent is therefore to find a proof within that theory of its own consistency. A complete theory is one in which, for any statement φ in the language of that theory, either φ is provable or ¬φ is provable within that theory. Note that this is a different sense of completeness than that proven in Gödel's Completeness Theorem, which states that any sentence satisfied by all models of a theory is provable. A consistent theory is one which contains no contradictions. Because mathematics generally employs classical logic it is explosive [1] and any contradiction allows one to derive any sentence whatsoever in the language of the theory as a theorem. Because of this an alternative way to say that a theory is inconsistent is to say that all the sentences in the language of the theory are theorems. [1] http://en.wikipedia.org/wiki/Principle_of_explosion http://en.wikipedia.org/wiki/Principle_of_explosion