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

The Von Neumann definition of ordinals defines ordinals as objects in ZFC. Each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \).