Ordinal Number (Set Theory)
16 Sep 2016In set theory, an ordinal number (or ordinal) is a generalisation of the natural number that is used to describe a way to arrange a collection of objects in order - one after another. Put it in another way, ordinal numbers are the labels needed to arrange collections of objects in order.
For finite collections of objects, ordinal numbers are just counting numbers. On the other hand, a more generalised concept called well-ordering is introduced for infinite collections.
Every ordinal number is either zero, or a successor ordinal, or a limit ordinal. For example, 0, 1, 2, …, \omega, \omega + 1, \omega is a limit ordinal because, for any smaller ordinal, there is another ordinal larger than it, but still less than \omega. Also, 1, 2, … and \omega + 1 are successor ordinals.
Ordinals are distinct from cardinal numbers, which are useful for saying how many objects are in a collection. For finite sets, these two notions are not apparently different; however, different infinite ordinals may associate with the same cardinal.
Von Neumann Definition of Ordinals
The standard definition, suggested by John von Neumann, is: each ordinal is the well-ordered set of all smaller ordinals, in symbols x = [0, x). Formally,
A set S is an ordinal if and only if S is strictly well-ordered w.r.t. set membership and every element of S is also a subset of S.
Well-order
A well-order (or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is called a well-ordered set.
Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor. There may be elements besides the least element which have no predecessor. In a well-ordered set S, every subset T which has an upper bound has a least upper bound, i.e. the least element of the subset of all upper bounds of T in S. Furthermore, every well-ordered set is uniquely order isomorphic to a unique ordinal number.