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EXISTENTIALLY COMPLETE STRUCTURES AND
MODELCOMPLETIONS AND MODELCOMPANIONS
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algebraically closed fields approximating chains biregular chapter class of existentially complete division algebra complete theory condition consistent constant symbols contains Corollary definition denoted Diag(M division rings division subalgebra elementarily equivalent elementary class elementary extension elementary substructure elements existential formula existential sentence existential sentence defined existential type existential type defined existentially complete division existentially complete models existentially complete structures existentially universal structures finite forcing companion finitely generic models following are equivalent formally real fields Godel numbers Henkin theory holds inductive class infinitely generic structures isomorphic joint embedding property maximal existential type model-companion mutually model-consistent n-type natural numbers one-one reducible order theory partial recursive functions persistently complete polynomials prenex normal form Proof Proposition quantifiers real closed fields realized regular models Robinson satisfies a0 second order arithmetic set of Godel set of sentences subfield Suppose Theorem true type of aQ vn-l vn+l vr+l