Ordinal number

Ordinal numbers (or ordinals) are numbers that show something's order, for example: 1st, 2nd, 3rd, 4th, 5th.

Suppose a person has four different T-shirts, and then lays them in front of the person, from left to right.

• At the far left, there is the red T-shirt.
• Right of that is the blue one.
• Then there is the yellow one.
• And finally, at the far right is an orange T-shirt.

If the person then starts counting the shirts from the left, he would first see the red shirt. So the red shirt is the first T-shirt. The blue shirt is the second T-shirt. The yellow shirt is the third one, and the orange T-shirt is the fourth one.

The first, second, third, and fourth in this case are ordinal numbers. They result from the fact that the person has many objects, and they give them an order (hence 'ordinal'). The person then simply counts those objects, and gives the ordinal numbers to them.

In set theory, ordinals are also ordinal numbers people use to order infinite sets. An example is the set [math]\displaystyle{ \omega_0 }[/math] (or [math]\displaystyle{ \omega }[/math] for short), which is the set containing all natural numbers (including 0). This is the smallest ordinal number that is infinite, and there are many more (such as [math]\displaystyle{ \omega }[/math] + 1).

Ordinal Number Media

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  Representation of the ordinal numbers up to ωω. One turn of the spiral corresponds to the mapping f(\alpha )=\omega (1+\alpha ). Since f has \omega ^{\omega } as least fixed point, larger ordinal numbers cannot be represented.

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  A graphical "matchstick" representation of the ordinal ω2. Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers.

Related pages

• Cardinal number
• Names of ordinal numbers in English
• Transfinite number