Axiom:Axiom of Abstraction

Given any property, there is a set which consists of all objects having that property.

From the definition of a set:

"A set is any aggregation of objects, called elements, which can be precisely defined in some way or other."

The looseness of the "in some way or other" leads to paradoxes (see Russell's Paradox). For a rigorous approach to set theory, it is necessary to specify exactly what the rules are by which one may build sets.

There are different ways of doing this, the best known perhaps being the Zermelo-Fraenkel axioms.