Charlie Ahrendts

Whenever mathematicians talk about the foundations of mathematics, set theory is pretty much guaranteed to be a topic in the discussion. This might seem a little odd since it has only been around for just over a century, so why is it such a big deal?

Set theory is a branch of mathematics that was invented around 1880 by the mathematician Georg Cantor. In its most fundamental way it looks at collections of objects, where these objects can be almost everything: numbers, functions, shapes, pretty much anything you can think of. This makes set theory very versatile on the one hand, and fundamental on the other, meaning that most areas of mathematics can be expressed using only the basic rules of the field. We call these rules the Zermelo-Fraenkel axioms, or ZF for short. Let’s take one of them, the axiom of extensionality, and deconstruct it step by step.

∀A ∀B (∀X (X ∈ A ⇔ X ∈ B) ⇒ A = B)

Let us start with the symbols. ∀ can be read as either given any or as for all. The letters A, B and X denote sets. A set can be any collection of one or more objects. We will for now think of these objects just as numbers, although keep in mind that there are many other ways to construct a set. A set can also have infinitely many elements, like that of the counting (or natural) numbers.

The symbol ∈ is short for is a member of. We use it when we want to say that a certain object is in the set. For example, the element apple is a member of the set of all fruits and the number 6 is a member of the set of all even numbers.

⟺ is read as if and only if. This means that if the statement on the left is true, the one on its right has to be true as well, and vice versa. Either both sides are true or both are false, but never only one of them. 

In contrast, this arrow ⟹ means that if the statement on the left of it is true, then we can conclude that what is on the right of it is also true, but not necessarily the other way round. 

Putting this all into words, the extensionality axiom states: 

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B. 

This still sounds very confusing, but in essence it just states that A and B are the same set if both consist of the same elements. X is in our case just a number. So if I have two sets, A and B and I take any number X that is in either A or B, then if I can say with certainty that any such number I can pick is also in the other set, the extensionality axiom says that both have to contain the exact same elements and are therefore in fact the same.

It’s still pretty tricky so let’s use an example: 

Set A consists of the numbers 4, 5 and 6, and set B the numbers 4, 5 ,6 and 7.  We can write these as A = {4, 5, 6} and B = {4, 5, 6, 7}. If we follow the logic of the axiom, the sets are only equal if and only if every element (X) of set A is also in set B. Here we can see the difference between if and if and only if again. Every element of A is indeed also an element of B, but not every element of B is also an element of A. This is why A and B are not equal. If, on the other hand, set B was equal to {5, 4, 6}, every element of it would also be in A. The order of elements does not matter in this context and this would make the two sets equal to each other.

The great thing about set theory is that it does not only work with numbers or even strictly mathematical objects. We could just as well construct a set which is made of every kind of vegetable or of all cats with only three legs, although I’d hazard a guess that the mathematics of three-legged cats is probably not very interesting…