Editing Set theory (section)

==Ontology==
{{Main|von Neumann universe}}
[[Image:Von Neumann Hierarchy.svg|thumb|right|300px|An initial segment of the von Neumann hierarchy]]

A set is [[pure set|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]] <math>\alpha</math>, known as its ''rank.'' The rank of a pure set <math>X</math> 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 {{math| <nowiki></nowiki> }} containing only the empty set is assigned rank 1. For each ordinal <math>\alpha</math>, the set <math>V_{\alpha}</math> is defined to consist of all pure sets with rank less than <math>\alpha</math>. The entire von Neumann universe is denoted&nbsp;<math>V</math>.