Herbrand's Theorem

There are two important theorems known as Herbrand's theorem.

The first arises in ring theory. Let an ideal class be in  if it contains an ideal whose th power is principal. Let  be an odd integer  and define  by . Then . If  and , then .

The Herbrand theorem in logic states that a formula  is unsatisfiable iff there is a finite set of ground clauses of  that is unsatisfiable in propositional calculus. It is assumed that elements of the Herbrand base are treated as propositional variables. Since unsatisfiability is dual to validity ( is unsatisfiable iff the negation  is valid), the Herbrand theorem establishes that the Herbrand universe alone is sufficient for interpretation of first-order logic. This theorem also reduces the question of unsatisfiability in first-order logic to the question of unsatisfiability in propositional calculus.

REFERENCES

Ireland, K. and Rosen, M. "Herbrand's Theorem." §15.3 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 241-248, 1990.