A Course in Model Theory (Lecture Notes in Logic) by Katrin Tent, Martin Ziegler

By Katrin Tent, Martin Ziegler

This concise creation to version conception starts off with ordinary notions and takes the reader via to extra complex issues similar to balance, simplicity and Hrushovski buildings. The authors introduce the vintage effects, in addition to more moderen advancements during this shiny zone of mathematical common sense. Concrete mathematical examples are integrated all through to make the suggestions more uncomplicated to stick with. The e-book additionally comprises over 2 hundred workouts, many with options, making the booklet an invaluable source for graduate scholars in addition to researchers.

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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 , .

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