are all possible ordered pairs , where is a member of and is a member of . For example, the Cartesian product of Power set of a set , denoted , is the set whose members are all of the possible subsets of . For example, the power set of is . Some basic sets of central importance are the set of natural numbers, the set of real numbers and the empty set—the unique set containing no elements. The empty set is also occasionally called the null set, though this name is ambiguous and can lead to several interpretations. Ontology A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number , known as its rank. The rank of a pure set is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set containing only the empty set is assigned rank 1. For each ordinal , the set is defined to consist of all pure sets with rank less than . The entire von Neumann universe is denoted . Formalized set theory Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a