What is an ordinal number? 



Ordinal numbers was introduced in 1883 by George cantor, it is the process of arranging natural numbers in orderly(increasing or decreasing order) manner. The arrangement is known as ordinality of numbers, this is mostly applied in set theory.
Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers provide the "labels" needed to arrange collections of objects in order.
An ordinal number is used to describe the order type of a well-ordered set. 
A well ordered set is a set with relation $<$ such that:
  1. Trichotomy law: for any elements $x$ and $y$, exactly one of this elements is true:  $x<y$, $y<x$  or $x=y$
  2. Transitive law: for any elements $x,y,z$ if $x<y$ and $y<z$ then $x<z$.
  3.  Well-foundedness: Every nonempty subset has a least element, that is, it has an element $x$ such that there is no other element $y$ in the subset where $y<x$.

Ordinal numbers are useful in ordering objects in a collection while  their counterpart cardinal numbers are used in quantifying the number of objects in a collection.
For example, any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal $42$ is the order type of the ordinals less than it, that is, the ordinals from $0$ (the smallest of all ordinals) to $41$ (the immediate predecessor of $42$), and it is generally identified as the set $\{0,1,2,…,41\}$.


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