General Prerequisites: There are no formal prerequisites, but familiarity with some basic mathematical objects and notions such as: the rational and real number fields; the idea of surjective, injective and bijective functions, inverse functions, order relations; the notion of a continuous function of a real variable, sequences, series, and convergence, and the definitions of basic abstract structures such as fields, vector spaces, and groups (all covered in Mathematics I and II in Prelims) will be helpful at points.
Course Overview: Introduce sets and their properties as a unified way of treating mathematical structures. Emphasise the difference between an intuitive collection and a formal set. Define (infinite) cardinal and ordinal numbers and investigate their properties. Frame the Axiom of Choice and its equivalent forms and study their implications.
Learning Outcomes: Students will have a sound knowledge of set theoretic language and be able to use it to codify mathematical objects. They will have an appreciation of the notion of infinity and arithmetic of the cardinals and ordinals. They will have developed a deep understanding of the Axiom of Choice, Zorn's Lemma and the Well-Ordering Principle.
Course Synopsis: What is a set? Introduction to the basic axioms of set theory. Ordered pairs, cartesian products, relations and functions. Axiom of Infinity and the construction of the natural numbers; induction and the Recursion Theorem.
Cardinality; the notions of finite and countable and uncountable sets; Cantor's Theorem on power sets. The Tarski Fixed Point Theorem. The Schröder-Bernstein Theorem. Basic cardinal arithmetic.
Well-orders. Comparability of well-orders. Ordinal numbers. Transfinite induction; transfinite recursion [informal treatment only]. Ordinal arithmetic.
The Axiom of Choice, Zorn's Lemma, the Well-ordering Principle; comparability of cardinals. Equivalence of WO, CC, AC and ZL. Cardinal numbers.