Set theory articulates all of mathematics according to one very simple relationship: grouping.

In set theory, there are only four fundamentals: existence, containment, equality, and negation. Something can exist, it can be contained in something else that exists, two things can be the same, and statements can be negated.

The rest is then defined by a series of axioms, called the Zermelo-Frankel (ZF) axioms.

For instance, the axiom of empty set says that there exists a set such that there doesn't exist anything contained in that set-i.e., there is such a thing as a set which is empty.

The axiom of comprehension says that sets are equal if and only if the things they contain are equal.

And so on, until there are enough axioms to encompass all of mathematics.

Set theory can even account for such things as mappings and ordered pairs with these simple axioms; an ordered pair (a,b) is a set of the form {a,{a,b}}, and a mapping is a set of ordered pairs. Gradually these concepts can be extended out into functions, formulas, cardinality, ordinal numberings, and so on.

In short, set theory is one of the best known ways to *axiomatize* mathematics; that is, to define explicitly what assumptions are going in before we can do any mathematical work. There are other ways, such as Peano arithmetic, and for all practical purposes the results are the same; but the nicest feature of set theory is that it is so elegantly simple and yet so powerful in extension.

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