Formal languages, structures and theories are introduced. Definability issues are presented. The Compactness Theorem is given, with many applications. The compactness is proved using ultrafilters. Elements of model theory are presented: quantifier elimination (as an example: QE for algebraically closed fields, with applications to commutative algebra), elementary substructures, Löwenheim–Skolem theorems and ω-categoricity (back-and-forth method).
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