This course aims to provide a general introduction to some advanced topics in logic. The main topic that is picked for this semester is set theory. We will cover set theory in two parts. In the first part, we will focus on the history and philosophy of set theory. The second part will mainly be about the technical aspects of set theory.

Part I: History and Philosophy of Set Theory

In daily life, we usually use the notion of a set for referring to a collection of objects, but the notion of a set in mathematics and logic goes far beyond such daily life contexts. This notion, together with its theory, gives us the ability to talk about infinity in an intelligent manner and allows us to conduct investigations in metaphysics. In that respect, set theory has a strong connection with philosophy. In this part of the course, we will briefly survey the history of the notion of infinity and its implications in metaphysics. Then, we will briefly cover the history of the early development of set theory.

Part II: Set Theory

In addition to its philosophical implications, set theory also serves as the foundation of mathematics in the sense that almost all mathematical objects can be defined in terms of a set. In this part of the course, we will learn about the basics of axiomatic set theory. We will start with what is known as ‘naïve set theory’. It is called naïve because it does not provide explicit safeguards against paradoxical results such as the famous Russell’s paradox. While covering naive set theory, we will also be learning proof methods for set theory. Set-theoretic proofs have significant differences from the proofs that we do in formal logic courses. After naive set theory, we will continue with Zermelo and Fraenkel’s set theory with the axiom of choice (ZFC). We will cover relations and functions within the axiomatic framework, and then we will finish with a general introduction to transfinite arithmetic.