Cardinal vs Ordinal
In our day to day life, the use of numbers may take different forms in different situations. For example, when we count to figure out the size of a collection of objects, we count them as one, two, three, and so on. When we want to count something to get the sense of the position of the objects, we count them as first, second, third, and so on. In the first form of counting, numbers are said to be cardinal numbers. In the second form of counting, the numbers are considered as ordinal numbers. In this context, the concepts cardinal and ordinal are completely a matter of linguistics; cardinal and ordinal are adjectives.
However, the extension of the concept to sets in mathematics reveals a much deeper and broader perspective and cannot be treated in simple terms. In this article, we will try to understand the fundamental concepts of cardinal and ordinal numbers in mathematics.
Formal definitions of cardinal and ordinal numbers are provided in the set theory. The definitions are intricate and to understand them in perfect sense needs background knowledge in set theory. Therefore, we will turn towards a couple of examples, to understand the concepts heuristically.
Consider the two sets {1,3,6,4,5,2} and {bus, car, ferry, train, airplane, helicopter}. Each set lists a set of elements, and if we count the number of elements it is evident that each has the same number of elements, which is 6. Arriving at this conclusion we have taken the size of one set and compared to another using a number. Such a number is called a cardinal number. Therefore, we can say that a cardinal number is a number we can use to compare the size of the finite sets.
Again the first set of numbers can be arranged in ascending order considering the size of each element and comparing them. In the process of ordering, the numbers are considered as cardinals. Likewise, the set of all nonnegative integers can be ordered in a set; i.e {0,1,2,3,4,…..}. But in this case, the size of the set becomes infinite, and giving it in terms of ordinals is not possible. No matter how large a number you pick to give the size of the set, still there will be numbers left out of the set you pick and which are nonnegative integers.
Therefore, mathematicians define this infinite cardinal (which is the first) as Aleph-0, written as א (first letter in the Hebrew alphabet). Formally the ordinal number is the order type of a well ordered set. Therefore, the ordinal number of the finite sets can be given by cardinal numbers, but for infinite sets ordinal is given by transfinite numbers such as Aleph-0.
What is the difference between Cardinal and Ordinal Numbers?
• The cardinal number is a number that can be used to count, or to give the size of a finite ordered set. All cardinal numbers are ordinals.
• The ordinal numbers are numbers used to give the size of both finite and infinite ordered sets. The size of the finite ordered sets is given by usual Hindu-Arabic algebraic numerals, and the infinite set size is given by transfinite numbers.