# Zermelo-Fraenkel set theory

In this article, we will look at Zermelo-Fraenkel set theory, why it is necessary, and what it is. And as an example, we will see how it can be used to define the set of natural numbers.

# Naive set theory

Up until the start of the 20th century, a mathematical set was defined according to what we now call *naive set theory* (it wasn’t called that at the time, of course, because nobody had realised the problem with it). Loosely speaking, naive set theory says a set can contain any group of things that share a particular property, and that property can be described using natural language.

More formally, we say that for any property *P(x)* that is either true or false, we can form a set of all values of *x* for which *P(x)* is true. This is called the *axiom of unrestricted comprehension*, which is just a formal way of saying that a set can be constructed from any property *P*.

So for example, the set of all natural numbers less than 4 is the set {1, 2, 3}. The members don’t have to be numbers. We could define a set of all the vowels in the English language, which would be {A, E, I, O, U}.

A set can be infinite, for example, the set of the natural numbers, **N**, is infinite. The set of even numbers, **N2** would be {0, 2, 4, 6 … } (we are counting 0 as a natural number, very much a matter of opinion, but we include the empty set as a set, so it is useful to include 0 as a natural number). That set is infinite, but we can easily calculate the nth element, it is simply *2n*. So…