Lecture Notes Compactness and Completeness of Propositional Logic and First-Order Logic

Another reason, no less compelling, for starting with Compactness is didactic: It makes it easier to grasp topological aspects of the notion (and the origin of its name). We can go deep in a topological direction, by explaining Compactness purely in terms of notions such as ultrafilters, ultraproducts, Hausdorff spaces, the finite-intersection property, and others, but that would take us further afield from the focus of this course.2