Ordinal number
Elementary
An ordinal number is a number that tells position of an item in a list, such the first, second, and third runner in a race.
High school
An ordinal number is a number that tells position of an item in a list, such the first, second, and third.
Graduate (set theory)
An ordinal number is the order type of a well-ordered set. The finite ordinals are the natural numbers, followed by infinite ordinals $\omega, \omega + 1, \omega + 2, \dots$
Examples
Examples
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The “3rd” in “finished 3rd”
Elementary
The “3rd” in “finished 3rd” is an ordinal number because it tells a position in an order.
Non-examples
Things that may be mistaken for examples.
- A phone number Elementary
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The “23” on a jersey
Elementary
The “23” on a jersey is not an ordinal number: player 23 does not come after player 22.
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The “3” in “3 apples”
Elementary
The “3” in “3 apples” is not an ordinal number, because it does not give a position in a list.
Properties
- For infinite sets, ordinals and cardinals differ: $\omega$ and $\omega + 1$ are different ordinals, but the corresponding sets have the same cardinality, $\aleph_0$.