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