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

Logic

By Katrin Tent, Martin Ziegler

This concise creation to version idea starts off with normal notions and takes the reader via to extra complicated issues similar to balance, simplicity and Hrushovski structures. The authors introduce the vintage effects, in addition to newer advancements during this brilliant zone of mathematical common sense. Concrete mathematical examples are integrated all through to make the innovations more uncomplicated to keep on with. The publication additionally comprises over two hundred workouts, many with options, making the publication an invaluable source for graduate scholars in addition to researchers.

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Proof. We define an increasing sequence ∅ = C0 ⊆ C1 ⊆ · · · of new constants by assigning to every L(Ci )-formula ϕ(x) a constant cϕ(x) and Ci+1 = cϕ(x) ϕ(x) L(Ci )-formula . Let C be the union of the Ci and T H the set of all Henkin axioms ∃xϕ(x) → ϕ(cϕ(x) ) for L(C )-formulas ϕ(x). It is easy to see that one can expand every Lstructure to a model of T H . Hence T ∪ T H is a finitely satisfiable Henkin theory. Using the fact that the union of a chain of finitely satisfiable theories is also finitely satisfiable, we can apply Zorn’s Lemma and get a maximal finitely satisfiable L(C )-theory T ∗ which contains T ∪ T H .

Let Th(C) = {ϕ | A |= ϕ for all A ∈ C} be the theory of C. Then M is a model of Th(C) if and only if M is elementarily equivalent to an ultraproduct of elements of C. 2. Show that C is an elementary class if and only if C is closed under ultraproducts and elementary equivalence. 3. Assume that C is a class of finite structures containing only finitely many structures of size n for each n ∈ . Then the infinite models of Th(C) are exactly the models of Tha (C) = {ϕ | A |= ϕ for all but finitely many A ∈ C}.

8. Let T1 and T2 be two theories. Then the following are equivalent: a) There is an ∀∃-sentence which separates T1 from T2 . 30 3. Quantifier elimination b) No model of T2 is the union of a chain (or of a directed family) of models of T1 . Proof. a) ⇒ b): Assume ϕ is a ∀∃-sentence which separates T1 from T2 , (Ai )i∈I is a directed family of models of T1 and B the union of the Ai . 6 B is also a model of ϕ. Since B |= ϕ, B cannot be a model of T2 . b) ⇒ a): If a) is not true, T1 and T2 have models which cannot be separated by an ∀∃-sentence.

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