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

By Katrin Tent, Martin Ziegler

This concise advent to version idea starts off with typical notions and takes the reader via to extra complex themes similar to balance, simplicity and Hrushovski buildings. The authors introduce the vintage effects, in addition to more moderen advancements during this brilliant region of mathematical common sense. Concrete mathematical examples are integrated all through to make the techniques more uncomplicated to stick with. The publication additionally comprises over 2 hundred routines, many with ideas, making the e-book an invaluable source for graduate scholars in addition to researchers.

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22 2. Elementary extensions and compactness b) ϕ = ¬ . As T ∗ is finitely complete, ϕ ∈ T ∗ ⇐⇒ the induction hypothesis we have A∗ |= ϕ ⇐⇒ A∗ |= ∈ T ∗ holds and by ∈ T ∗ ⇐⇒ ϕ ∈ T ∗ . ⇐⇒ c) ϕ = ( 1 ∧ 2 ). As T ∗ contains all sentences which follow from a finite subset of T ∗ , ϕ belongs to T ∗ if and only if 1 and 2 belong to T ∗ . Thus A∗ |= ϕ ⇐⇒ A∗ |= i (i = 1, 2) ⇐⇒ i ∈ T ∗ (i = 1, 2) ⇐⇒ ϕ ∈ T ∗ . d) ϕ = ∃x (x). We have A∗ |= ϕ ⇔ A∗ |= (c) for some c ∈ C ⇔ (c) ∈ T ∗ for some c ∈ C ⇔ ϕ ∈ T ∗ . The second equivalence is the induction hypothesis and for the third we argue as follows: if ϕ ∈ T ∗ , we choose c satisfying ϕ → (c) ∈ T ∗ .

This means that there is a tuple b such that (B, b) |= (a). 2 applied to T = Th(B) shows that A ⇒Δ B if and only if there exists a map f and a structure B ≡ B such that f : A →Δ B . 3. Let T1 and T2 be two theories. Then the following are equivalent: a) There is a universal sentence which separates T1 from T2 . b) No model of T2 is a substructure of a model of T1 . Proof. a) ⇒ b): Let ϕ be a universal sentence which separates T1 from T2 . Let A1 be a model of T1 and A2 a substructure of A1 . 16 A2 is also model of ϕ.

If f ∈ L is an n-ary function symbol (n ≥ 0) and a1 , . . , an is from A, we consider the formula . ϕ(x) = f(a1 , . . , an ) = x. Since ϕ(x) is always satisfied by an element of A, it follows that A is closed under f B . Now we show, by induction on , that A |= ⇐⇒ B |= for all L(A)-sentences . This is clear for atomic sentences. The induction steps for = ¬ϕ and = (ϕ1 ∧ ϕ2 ) are trivial. 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 |= .

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

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