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**Additional resources for A Course in Model Theory (Lecture Notes in Logic)**

**Sample text**

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 . A structure A is T -ec exactly if it is T ∀ -ec. This is clear for condition a) since every model B of T ∀ can be embedded in a model M of T . For b) this follows from the fact that A ⊆ B ⊆ M and A ≺1 M implies A ≺1 B.

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 ). Proof. We construct A as the union of an ascending sequence S0 ⊆ S1 ⊆ · · · of subsets of B. We start with S0 = S. If Si is already deﬁned, we choose an element aϕ ∈ B for every L(Si )-formula ϕ(x) which is satisﬁable in B and deﬁne Si+1 to be Si together with these aϕ . It is clear that A is the universe of an elementary substructure. It remains to prove the bound on the cardinality of A.

Note that if A is an elementary extension of A, then SA (B) = SA (B) and tpA (a/B) = tpA (a/B). We will use the notation tp(a) for tp(a/∅). Similarly, maximal ﬁnitely satisﬁable sets of formulas in x1 , . . , xn are called n-types and Sn (B) = SA n (B) denotes the set of n-types over B. For an n-tuple a from A, there is an obvious deﬁnition of tpA (a/B) ∈ SA n (B). Very much in the same way, we can deﬁne the type tp(C/B) of an arbitrary set C over B. This will be convenient in later chapters. In order to do this properly we allow free variables xc indexed by c ∈ C and deﬁne tp(C/B) = {ϕ(xc1 , .