An Introduction to Mathematical Logic

Abstract: This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's definability theorem and Herbrand's theorem. It continues with a section on G\"odel's incompleteness theorems, which includes a discussion of first-order arithmetic and primitive recursive functions. This is followed by three sections that are devoted, respectively, to proof theory (provably total recursive functions and Goodstein sequences for $\mathsf{I\Sigma}_1$), computability (fundamental notions and an analysis of K\H{o}nig's lemma in terms of the low basis theorem) and model theory (ultraproducts, chains and the Ax-Grothendieck theorem). We conclude with some brief introductory remarks about set theory (with more details reserved for a separate lecture). The author uses these notes for a first logic course for undergraduates in mathematics, which consists of 28 lectures and 14 exercise sessions of 90 minutes each. In such a course, it may be necessary to omit some material, which is straightforward since all sections except for the first two are independent of each other.