# Functions

### Topics

Set theory is a branch of mathematics that lets mathematicians make math even more abstract than it already is. It's like the Jackson Pollack of mathematics. In set theory, not even the numbers (0, 1, 2, 3, 4, . . . ) are taken for granted. They're built from scratch, just like your grandma's raisin oatmeal cookies.

Starting with only the curly brackets and a comma, we can build the whole numbers. Here's how it works. Pretend 0, 1, 2, 3, 4, . . . are symbols, but we don't know what they mean yet. We have to define them. First, we define 0 be the empty set:

0 = {}

then, we define 1 to be the set containing 0:

1 = {0}

then, 2 as the set containing 0 and 1:

2 = {0, 1}.

See where this is going? Each whole number can be defined as the set of all whole numbers that came before it:

*n* = {0, 1, . . . , (*n* – 1)}.

From here, there are ways to define the rest of the numbers (integers, rational numbers, and real numbers, for example) and the arithmetic operations, all in terms of sets. Unfortunately, set theory can't help you to define your calves. Sorry, you're going to need to go to the gym. There's no way around it.

There are many different kinds of sets, and some sets have other sets contained within them as elements. However, there's no such thing as a set containing all the sets there are, since that one biggest set wouldn't be able to contain itself. Did we blow your mind yet? This concept is known as Russell's Paradox, and is often presented as a story about a barber. This site gives a nice presentation of Russell's Paradox and explains the connection between the story and the set theory.