By Abraham Robinson (Eds.)
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We include the possibility that no constants occur in either K or K’. 4. Let X be any statement which is defined in a set K and let K' be a set which is related t o K . 2. And from that statement X' we may then derive another statement X" which is defined in K by the corresponding procedure. It would not be generally true to say that X " coincides with X . 3 and the application of a rule of substitution of the predicate calculus (rule X in ref. 11) show that X = X" is deducible from K . 5. Let K and K' be two related sets, and let X be a statement which is defined in R.
5 above) any two algebraically closed extensions of 1M which are of equal cardinal, exceeding the cardinal of M , are isomorphic. 1). 6. 1. *,%%)I whose constants belong to M , such that Y holds in an (algebraically closed) extension M’ of M , we have to show that Y holds already in M . 7. 8. possesses a solution. In these formulae the Greek letters stand either for elements of M or for the unknowns yl, ... y,,. 7 by equations, by introducing for each inequality a new unknown y, and by replacing the inequality by the equivalent (o(-p)yi- 1 =0 (&+ B- y)yj- 1 = 0 (&P- y ) y +- 1 = 0 respectively.
2. , x,,,y. 3. [Jw4 - Po(x9 Y)= 4 A X(Y, Y, Y) A P(Y, Y, Y)1 p n ( x , y)= [ ( 3 z )[P,-l(x, 2 ) A P(z,x,y)]], n = l , 2, r Thus, P,,(x, y) expresses the relation y=z*. 4. - *) - 4, y) = [(Xu,)-. (3%)(3vi)-.. ( 3 % ) - * ( 3 W W - l ) [PAY, u1) A P2(% u2) A P3(% u3) A . * A Pn(y,un)A P ( q , ul,vl) A P(x2,u2,v2)A ... A P ( q , un,v,) S(V~ ~ ,2 ~ , 2 A )S(ZO,, ~ 3 ~ , 3 A ) * *. 2 for n=l, 2, 3, ... (except for some obvious formal modifications for n = 1, and for n = 2). 6. The elementary theory of algebraically closed fields is model-complete.