Samuel R. Buss.
"On Herbrand's Theorem,"
In Logic and Computational Complexity,
Lecture Notes in Computer Science #960, 1995, Springer-Verlag, pp. 195-209.
Abstract: We firstly survey several forms of
Herbrand's theorem. What is commonly called ``Herbrand's theorem'' in many textbooks is
actually a very simple form of Herbrand's theorem which applies only to
$\forall\exists$-formulas; but the original statement of Herbrand's theorem applied to
arbitrary first-order formulas. We give a direct proof, based on cut-elimination, of what
is essentially Herbrand's original theorem. The ``nocounterexample theorems'' recently
used in bounded and Peano arithmetic are immediate corollaries of this form of Herbrand's
Secondly, we discuss the results proved in Herbrand's 1930 dissertation.
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Errata. The paper contains an error in the proof of Theorem 3, the general form of Herbrand's theorem. I am grateful to Richard McKinley for discovering this, and for also provided a corrected proof of Theorem 3, based on "deep contraction". The corrected proof is available in the manuscript "A sequent calculus demonstration of Herbrand's Theorem" by Richard McKinley.
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