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Besides the proofs already listed, one essentially different treatment which comes to mind is Gentzen's consistency proof of PA, which established that PA can prove the well-ordering of ordinal notations less than $\epsilon_0$ but could not prove the well-ordering of a notation for $\epsilon_0$, and that, in turn, the well-ordering of $\epsilon_0$ would suffice to prove the consistency of PA. Characterizing the proof-theoretic ordinal of a theory yields incompleteness results by an essentially different (and arguably far deeper / more general) route to that of Godel.