MATH40510

Set theory was initiated by Georg Cantor in the late 19th century, with his revolutionary ideas about the nature of infinity. His work was greeted with much scepticism initially, but today set theory is a cornerstone of modern mathematics, and provides a foundational framework for all of standard mathematics.

Cantor's ideas won acceptance in the early 20th century, particularly after they were adopted by David Hilbert, who included Cantor's 'Continuum Hypothesis' as the first of his famous list of 23 problems posed in 1900, which greatly influenced the direction of 20th century mathematics. Set theory was taken further, notably by Zermelo, who developed a system of axioms that, roughly speaking, determine which objects are allowed to be called sets.

In this module, we study these axioms of set theory, particularly (a) the Axiom of Infinity, which we use to construct the natural numbers 0, 1, 2, 3, ... , the Principle of Mathematical Induction and the arithmetic of the natural numbers (from which the rest of standard mathematics follows); and (b) the Axiom of Choice and the equivalent statement Zorn's Lemma, and their applications to set theory and to other areas of mathematics, both the good (e.g. every vector space has a basis) and the downright peculiar (e.g. the Banach-Tarski Paradox). Time allowing, we will consider ordinal and cardinal numbers (the two main types of infinite numbers used in set theory), and `undecidable' statements (precise mathematical statements, one of which being Cantor's Continuum Hypothesis, that turn out to be neither provably true nor false).

We will cover the topics listed below.

1. infinity, paradoxes and axioms;
2. the Axiom of Infinity;
3. partially ordered sets;
4. the Axiom of Choice;
5. equinumerosity and dominance;
6. ordinal and cardinal numbers (time allowing);
7. metamathematics and undecidability (in brief, time allowing).