By TH. Skolem

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Interpretations GENTZEN'S Hauptsatz or HERBRAND'S theorem establish an interesting connection between the predicate calculus II and the propositional calculus Ilo ; the contrast between this work and the translation discussed in the last section will make our criticisms much clearer. I consider a formula (x)(Ey)(z)A(x, y, z) (or m:) where A(x, y, z) is quantifier-free. (i) m: can be proved in III precisely if there are quantifier-free terms yo(a), ... (a, a l ... a,,) of III such that A[a, yo(a), Ut] VA[a, Yl (a; Ut), a 2 ] V ..

3Yk) V(YI' ... , Yk) satisfies the conditions of the theorem. ORDERED STRUCTURES AND RELATED CONCEPTS 55 Indeed, V(Yv ... , Yk) has the required form since V(~, ... , a k) is a conjunction of elements of N'. • , a k ) holds in that structure. On the other hand it is not possible that X hold also in M. For in that case there would exist constants bv ... , bk in M such that V(b v ... , bk ) holds in M. Since V does not contain any quantifiers it would then follow further that V(b v ... , bk ) and hence X = (3Yl) ...

13] G. KREISEL, On the concepts of completeness and interpretation of formal systems, Fundamenta Mathematicae, 39 (1952), 103-127. [14] , Some concepts concerning formal systems of number theory Mathematische Zeitschrift, 57 (1952), 1-12. [15] W. ACKERMANN, Zur Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, 117 (1940), 162-194. Department of Mathematics, University of Reading, England. ABRAHAM ROBINSON ORDERED STRUCTURES AND RELATED CONCEPTS § 1. Introduction The work which is described in the present report owes its existence to the following circumstances.