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If ϕ is a negation or a conjunction, the claim follows directly from the induction hypothesis. If ϕ(x) = ∃y (x, y), then ϕ(a) holds in A exactly if there exists b ∈ A with A |= (a, b). As the family is directed, there always exists a j ≥ i with b ∈ Aj . By the induction hypothesis we have A |= (a, b) ⇐⇒ Aj |= (a, b). Thus ϕ(a) holds in A exactly if it holds in an Aj (j ≥ i). Now the claim follows from Ai ≺ Aj . 1. Let A be an L-structure and (Ai )i∈I a chain of elementary substructures of A. Show that i∈I Ai is an elementary substructure of A.

It remains to consider the case = ∃xϕ(x). If holds in A, there exists a ∈ A such that A |= ϕ(a). The induction hypothesis yields B |= ϕ(a), thus B |= . For the converse suppose holds in B. Then ϕ(x) is satisﬁable in B and by Tarski’s test we ﬁnd a ∈ A such that B |= ϕ(a). By induction A |= ϕ(a) and A |= holds. We use Tarski’s Test to construct small elementary substructures. 3. Suppose S is a subset of the L-structure B. Then B has an elementary substructure A containing S and of cardinality at most max(|S|, |L|, ℵ0 ).

Quantiﬁer elimination 35 Digression: existentially closed structures and the Kaiser hull. Let T be an L-theory. 2 that the models of T ∀ are the substructures of models of T . The conditions a) and b) in the deﬁnition of “model companion” can therefore be expressed as T ∀ = T ∀∗ . Hence the model companion of a theory T depends only on T ∀ . 10. An L-structure A is called T -existentially closed (or T ec), if a) A can be embedded in a model of T . b) A is existentially closed in every extension which is a model of T .