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

A. A. Markov's constructive mathematics: some reflections. A review of constructive mathematics in the Markovian sense is presented, with particular attention to constructive analysis. An account is given of the basic methodological principles of constructive mathematics and of expectations connected with them. A number of results of constructive mathematical analysis, including basic facts of constructive continuum theory and functions on the constructive continuum are presented. Additional questions of constructive analysis, including problems of topology and functional analysis, are considered as well. It is pointed out that although the initial revolutionary intentions are, perhaps, behind us now, the constructive direction still provides a valuable experience of development of mathematics on a purely syntactic and effectivist basis. The last two circumstances are of special importance in the light of recent developments in computer science.