Advanced Logic

Module description

The course will cover the basics of set theory, the mathematical study of the infinite. Starting from the paradoxes of set-membership such as Russell’s, we will introduce the basics of naïve and axiomatic set theory, including the ideas of relations, partial orders, functions, partitions and equivalence classes, ordinal and cardinal numbers. We justify the foundational role of set theory by showing how other mathematical objects, such as natural or rational numbers, can be represented as sets. We conclude by deep philosophical puzzles on the nature of the notion of set, property instantiation, and infinity.

Assessment details

Summative assessment: 1 x 2 hour examination (100%)

Formative assessment: 4 sets of exercises

Educational aims & objectives

To introduce students to: 1) The class paradoxes, which revived philosophy of maths. 2) The theory of the transfinite. 3) Representing the natural numbers in the realm of sets, a gateway to foundations of maths. 4) Transferrable tools: Omni-usable concepts, eg function, partial/total ordering, equivalence relation. 5) Advanced tools: recursive definition, mathematical induction.

Learning outcomes

By the end of the module the students will be able to demonstrate intellectual, transferable and practicable skills appropriate to a level 6 module and in particular will be able to demonstrate: 1) An understanding of the basic notion of mathematical set theory. 2) A knowledge of fundamental paradoxes of naive set theory. 3) An understanding of the various kinds of infinities. 4) An ability to make use of the set theoretic construction of number systems. 5) An understanding of the philosophical significance of set theory.

Teaching pattern

One one-hour weekly lecture and one one-hour weekly seminar over ten weeks. 

Suggested reading list

• Lecture Notes.
• Hrbacek, K., & Jech, T. (1999). Introduction to Set Theory, 220 of Pure and Applied Mathematics: A Series of Monographs and Textbooks.
• Jech, T. (2003). Set theory: The third millennium edition, revised and expanded. Springer Berlin Heidelberg.

Module description disclaimer

King’s College London reviews the modules offered on a regular basis to provide up-to-date, innovative and relevant programmes of study. Therefore, modules offered may change. We suggest you keep an eye on the course finder on our website for updates.

Please note that modules with a practical component will be capped due to educational requirements, which may mean that we cannot guarantee a place to all students who elect to study this module.

Please note that the module descriptions above are related to the current academic year and are subject to change.