By Abraham Robinson (Eds.)

ISBN-10: 0444533834

ISBN-13: 9780444533838

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**Example text**

The set {@Iis ordered partially in a natural way by the rule that for any two elements @ l and dj2 of the set, @1 3 dj2 if Q2 is a continuation of d j l . We may then employ Zorn's lemma to show that {@I includes a maximal element @O. Let Fl = FY7 and F, = be the corresponding subfields of F' and G: =G;, Gl =Gi the corresponding valuation groups. We propose to show that F; = Fi= F'. Suppose on the contrary that F: is a proper subfield of F', so that the same applies to F i . e. F: and F,* are obtained from F:, Fi by the adjunction of roots of corresponding normal equations).

1. F(4,G ( 4 , E(x,y), V(Z,Y), &‘(x,Y),s’(Z,Y,z),x(x,y,z),P(x,y,z). These denote, in turn, the property of belonging to the field, the property of belonging to the group, the relation of equality, the relation of valuation (“the group element y is the value of the field element z”), the relations of order (“z is smaller than or equal t o y”), and of addition in the group, and the relations of addition and multiplication in the field, respectively. 6). 2. 3 for the relations F,G, V , &’, x’,S , P.

KO is the set of all statements of the form - E ( c , b ) where c is an arbitrary but fixed constant which is not contained in M , and b varies over all the elements of M . Then all the models of Kl are algebraically closed proper extensions of M , and so Y is satisfied in all of them. It follows that Y is deducible from K,, and hence that there exists a finite subset of KO, {- E(c7 Q, - such that the statement E(c, b2)7 * * a - E(c7 h)],say, 43 REAL-CLOSED PIELDS is deducible from K u N . 13. a - E ( y ,b J l l 3 y also is deducible from K u N .

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