Category theory is a highly abstract branch of mathematics that involves looking at what ideas and constructions are common throughout all of mathematics. In this class, we'll take a more concrete look at some of the ideas of category theory through the most basic category of all, the category of sets. Roughly speaking, studying the category of sets means studying sets by only talking about functions between sets and forgetting the sets themselves! We'll see that many things you can do with sets can be expressed in this language. In particular, we'll show that some basic properties of arithmetic are really general principles of category theory!

In this class we will play Mao, a card game about learning rules. In most games, first you learn how to play and then you play. In Mao, you learn the rules by playing! You will be told only a minimal amount of information about the rules of Mao before play begins, and then will be penalized if you break rules. By observing when you (and other players) are penalized, you can figure out how the rules work. When you "win", you get to add a new rule to the game which the other players have to figure out. You can come to as many sections as you like, though the game will start over for each new section.

What does it mean for two sets to have the same number of elements (a set is just any collection of objects)? Well, if the sets are finite, we just count how many elements there are and see whether we get the same number. But if the sets are infinite (for example, the set of all whole numbers), what do we do? Can we "count to infinity"? Is there some other way we can compare two sets without counting them? In this class we'll see how to answer these questions. We'll find that in fact, there is more than one "size" of infinity. That is, "infinity" is not a single "number" but many different ones, and some infinities are infinitely bigger than other infinities!

Intuitively, real numbers are all the points on a number line, or all rational or irrational numbers. But what does this actually mean? It turns out that the defining property of the real numbers are that they are $${\textit complete}$$--that is, there are no "holes" in the real numbers in the way that irrational numbers are "holes" in the rational numbers. I'll show how to make this precise and give an abstract construction of the real numbers from the rational numbers.

A "measure" is a notion of length or area or volume. For example, we know what the length of a line segment is, but what does it mean to talk about the "length" of an arbitrary subset of a line? It turns out that there is a notion of length called Lebesgue measure that is defined for a very large class of subsets of a line. However, there are some sets whose length is impossible to define in a consistent way. I'll sketch the basic ideas behind Lebesgue measure and show why we can't define it on all subsets of a line.