# Chapter 7. Basic Set Theory

Set theory provides a formal language for speaking about collections, membership, functions, relations, size, and order, and it serves as one of the standard foundations in which ordinary mathematical objects can be represented.

The chapter begins with sets, relations, and functions, where the basic operations of mathematics are described in terms of membership and ordered pairs, giving a common framework for algebra, analysis, topology, and logic.

Cardinality and countability are then introduced to compare the sizes of sets, including finite sets, countably infinite sets such as the natural numbers, and uncountable sets such as the real numbers.

The chapter next studies ordinals and well ordering, where sets are used to represent ordered stages, and this provides a precise language for induction, recursion, and transfinite construction.

Cardinal arithmetic is introduced to describe how sizes of infinite sets behave under operations such as addition, multiplication, and exponentiation, where the infinite case differs sharply from the finite case.

Finally, the Zermelo-Fraenkel axioms, with and without the Axiom of Choice, are presented as a formal foundation for set theory, specifying the basic principles that govern sets and the universe built from them.