Gödel's second incompleteness theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA is Σ n+1 -definable and T is Σ n -sound, then T dose not prove the sentence Σ n -Sound(T ) that expresses the Σ n -soundness of T . The optimality of the generalization is shown by presenting a Σ n+1 -definable (indeed a complete ∆ n+1 -definable) and Σ n−1 -sound theory T such that PA ⊆ T and Σ n−1 -Sound(T ) is provable in T . It is also proved that no recursively enumerable and Σ 1 -sound theory of arithmetic, even very weak theories which do not contain Robinson's Arithmetic, can prove its own Σ 1 -soundness.