Ordinal

In set theory, the ordinal numbers or ordinals are an extension of the natural numbers that describe the order types of well-ordered sets. A set \( S \) is well-ordered if each non-empty \( T \subseteq S \) has a least element.

Von Neumann definition

In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \).

Equivalence class definition

Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders.