By Petr Vopenka, Petr Hajek

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Y E Ext,, (x). Hence L: E H” Ext,, (x). Now suppose that u E H” ExtR, (x). We have v H y for some J’ such that (J’. e. u E ExtRz(u). (3) Let X be a saturated part of C ( R , ) , let x E X and let u H x ; suppose that ExtR2( u ) = EX*,, ( u ) and that v H y . By (I) and (2) we have Ext,, (x) = (Cnv ( H ) ) ” (ExtR, (u)) = = (Cnv (H))” (ExtR2(u)) = Ext,, ( y ) ; hence y E X and t’ E H ” X . Thus H”X is a saturated part of C(R,). e. that (Vx E C(R,)) (Sat,, (Ed,, ( x)). If u E C(R,) and u H x then Ext,, ( u ) = = H”(ExtR, (x)) and Sat,, (H”((Ext,,(x))).

E. - (pm. e. cp holds in 2132 * %. 1229. Let T and S be theories and let s%R1and 1111, be models of T in S. Ill2 (%R1 < %TI2)*) if for any T-formula cp, qm' is provable in ( S , qml). is equivalent to %TI2,if 2132, < %TI2and 9x2 < 1132, . m, as a model of T in S is weaker than m2since %IT,and W , can be models of other theories as well; see the following lemma. *) To be exact, we should say that 39 1230 LOGICAL FOUNDATIONS [CHAP. I SEC. 2 1230. LEMMA. a) S is stronger than T iff the identical mapping 3b, of T-formulas is a model of T in S.

7). g. s E F”D ( X ) ( 3 y , U) ( ( u , y ) E X & x = F’y) ( 3 y E D ( X ) ) (X = F’y) (3y, fl) ((F’u, F ’ y ) E F”X & x = F’y) = = (30) ( ( u , x) E F”X) = x E D ( F ” X ) . EE 1322. METATIIEOREM. , the formulas U = Fi(X, Y ) are absolute). Demonstration. (a) The absoluteness of X E Y is trivial. Let us proceed i n the theory TC, Vlk (M), dTC/dGt (M). Recall that sets in the sense of Gt (M) are o f the form x n M where x E M. Further, if { u , u ) E M then 11, 1’ E M, ( u , u} n M = {u, u } and { u , u ] = { u n M, u n M}*.

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