Introduction to Set Theory

This page collects the handwritten lecture notes I compiled when I taught an introductory set theory course at UCLA in Winter 2022, along with some useful links and copies of the exams I wrote for the class (with solutions).

Useful links

The Open Logic Project: A free open-source collection of teaching materials for introductory logic courses. It includes a fairly nice introductory set theory textbook.
Cantor’s attic: A compedium of interesting ordinals and cardinals (and some fast growing functions on the natural numbers).
Set-Theoretic Foundations: A paper by Penelope Maddy categorizing possible goals of a foundational theory of mathematics.
On the Logic of Number: A paper by Charles Sanders Pierce from 1881 which may contain the first axiomatization of the natural numbers.
The Natural Number Game: A game which you can play in your browser and which allows you to prove statements about the natural numbers from axioms similar to the Peano axioms (using the Lean proof assistant).

Exams

Handwritten lecture notes

Each lecture title links to the notes for that lecture.

Lecture	Topic
1	Introduction: The goals of set theory
2	Introduction: Naive set theory and Russell’s paradox
3	Reconstruction of math: First order logic and first axioms
4	Reconstruction of math: Relations
5	Reconstruction of math: Types of relations
6	Reconstruction of math: Functions
7	Reconstruction of math: Natural numbers
8	Reconstruction of math: Axiom of Infinity
9	Reconstruction of math: Integers and rational numbers
10	Reconstruction of math: Real numbers
11	Cardinality: Introduction
12	Cardinality: Uncountability
13	Cardinality: Cantor-Schroeder-Bernstein and cardinal arithmetic
14	Ordinals: Well-orders
15	Ordinals: Isomorphisms and embeddings
16	Ordinals: Transfinite recursion
17	Ordinals: Constructing the ordinals
18	Cumulative Hierarchy: Stages and ranks
19	Cumulative Hierarchy: All sets are ranked
20	Ordinals: Applications
21	Ordinals: Ordinal arithmetic
22	Axiom of Choice: Introduction
23	Axiom of Choice: Equivalent statements
24	Independence Proofs: Introduction
25	Independence proofs: Examples