By Dr. Joram Hirschfeld, Dr. William H. Wheeler (auth.)
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Additional info for Forcing, Arithmetic, Division Rings
Let ~" be a language (for ~ ) variables is a set among Vo, be used to indicate others introduced the language for ~. , Vn_ I. in this section An existential formulas The notation the free variables. ~(~), to handle be a class of similar structures, appropriate A type is introduced of and let n-type ~(~) with free A(Vo,... , Vn_l) Although are relative this concept to the class ~ will and all and this dependency will not be stated explicitly in all instances. , for each free occurrence Vn_l).
Construct of all the sentences a chain of structures defined from ~ as follows: (i) ~ = B + i M B I~ is a successor ~B' then let extension Let (ii) ~ of MB class. M 1 I~ ~ . M . then either in in ~ such that Let this ~B or there is an M' I~ ~B" M~ = B<~ M B. ~ or is an extension M1 ~ MI~M Mn+l I~ of M 2 ~ ~ ~ in since ~ M, either obtaining is M1 ~ a chain ... is a sentence ~¢. in is in defined construction, such that if Mn+l I~ Now let ~-, and Hw defined Mw = ~ in M n, M n. is infinitely generic ~.
F' Since ~- K and that satisfies subfield in is defined is as follows. , is a field extension L M domain The proof K, where such that type which of any universal class of fields. M is said to be existentially in exist whenever M ~ formula type also. is an class. Proposition member of ~ member of ~. IS. Assume is a substructure ~ is an inductive of an existentially class. Then each universal 32 Proof. The proof is almost and so will be omitted. 3 is that existential in the argument. that for any structure defined card(M) identical One also needs types the the number of existential to " card(~(~)) The class of existentially characterization universal structures has a similar to that of the class of existentially complete structures.