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

By Katrin Tent, Martin Ziegler

This concise advent to version concept starts with commonplace notions and takes the reader via to extra complex themes resembling balance, simplicity and Hrushovski buildings. The authors introduce the vintage effects, in addition to more moderen advancements during this shiny quarter of mathematical common sense. Concrete mathematical examples are integrated all through to make the strategies more straightforward to stick with. The e-book additionally comprises over 2 hundred routines, many with strategies, making the booklet an invaluable source for graduate scholars in addition to researchers.

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By Katrin Tent, Martin Ziegler

This concise advent to version concept starts with commonplace notions and takes the reader via to extra complex themes resembling balance, simplicity and Hrushovski buildings. The authors introduce the vintage effects, in addition to more moderen advancements during this shiny quarter of mathematical common sense. Concrete mathematical examples are integrated all through to make the strategies more straightforward to stick with. The e-book additionally comprises over 2 hundred routines, many with strategies, making the booklet an invaluable source for graduate scholars in addition to researchers.

Show description

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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 .

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