Background on the Herbrand Theorem and Herbrand's error is given in §1 below. §2 presents Godei's correction. As it turns out, his proof is extremely close to that devised by Dreben and his co-workers; published in full in Dreben and Dentón, 1966 (and, somewhat condensed, in van Heijenoort, 1967, 572-576 and Herbrand, 1971,193-199). §3 concerns counterexamples to Herbrand's false lemma and numerical bounds for the corrected lemma. In particular, I show that the bound obtained by Gödel's argument cannot be significantly improved
"Herbrand's error and Gödel's correction." Mod. Log. 3 (2) 103 - 118, February 1993.