# Zermelo-Fraenkel set theory

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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…

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