An Introduction to Mathematical Logic
(2023)cite arxiv:2310.09921.Abstract
This introduction begins with a section on fundamental notions of
mathematical logic, including propositional logic, predicate or first-order
logic, completeness, compactness, the Löwenheim-Skolem theorem, Craig
interpolation, Beth's definability theorem and Herbrand's theorem. It continues
with a section on Gödel'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
$I\Sigma_1$), computability (fundamental notions and an analysis of
K\Honig'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.
Description
[2310.09921] An Introduction to Mathematical Logic