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.
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