## MATH4ST Set Theory

First Semester |

We will study the axioms of set theory, what they tell us about the universe of sets, and how we can use the universe of sets to study the universe of mathematics.

We start by looking at one of the constructions of the real numbers in terms of rational numbers. We will observe that the rational numbers are constructed from the natural numbers, and that the natural numbers are constructed using very simple sets (such as the empty set). So everything comes down to sets in the end. We then consider paradoxes which illustrate the need for axiomatic set theory. We study the nine axioms of Zermelo-Fraenkel set theory. Next we consider the axiom of choice and its consequences - the good (for example, every vector space has a basis), the bad (for example, the existence of a non-measurably set in the theory of Lebesgue integration) and the ugly (for example, the Banach-Tarski paradox: we can partition the unit ball into finitely many pieces, move them around using only translations and rotations, and reassemble to produce two unit balls). As time permits we will then study ordinal and cardinal arithmetic (with and without the axiom of choice).

Some of the paper will be guided independent study using the book "Classic Set Theory: for guided independent study" by Derek Goldrei.

### Prerequisites

MATH201 Real Analysis, MATH202 Linear Algebra

### Lecturer

Astrid an Huef (Room 232A, phone 479 7760, email: astrid@maths.otago.ac.nz)

### Internal Assessment

3 written assignments

### Final exam

There will be a two-hour exam.