Cardinal vs. ordinal vs. nominal numbers
The same numeral, such as “3,” can play different roles in everyday language. A number can be used to count (cardinal), to give a position in an order (ordinal), or as a label (nominal). For finite sets, cardinals and ordinals look the same, but for infinite sets they differ.
Definitions
A cardinal number is a number that tells how many.
The cardinal number of a set is its cardinality. Two sets have the same cardinal number if there is a bijection between them.
Examples
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The “3” in “3 apples”
The “3” in “3 apples” is a cardinal number because it tells how many apples there are.
Non-examples
- A phone number
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The “23” on a jersey
The “23” on a jersey is not a cardinal number: it doesn’t count items in a set.
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The “3rd” in “finished 3rd”
The “3rd” in “finished 3rd” is not a cardinal number, because it does not count the number of items in a collection.
Properties
- The cardinal numbers include the natural numbers and infinite cardinals, such as $\aleph_0$, the cardinality of $\nats$.
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.
An ordinal number is a number that tells position of an item in a list, such the first, second, and third.
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
-
The “3rd” in “finished 3rd”
The “3rd” in “finished 3rd” is an ordinal number because it tells a position in an order.
Non-examples
- A phone number
-
The “23” on a jersey
The “23” on a jersey is not an ordinal number: player 23 does not come after player 22.
-
The “3” in “3 apples”
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, $\aleph_0$.
A nominal number is a number used as a name or label, not to count or to order.
Examples
- A phone number
-
The “23” on a jersey
The “23” on a jersey is a nominal number because it is only a label.
Non-examples
-
The “3” in “3 apples”
The “3” in “3 apples” is not a nominal number, because it is not just a label.
-
The “3rd” in “finished 3rd”
The “3rd” in “finished 3rd” is not a nominal number, because its order matters.
Examples
| Example | cardinal number | ordinal number | nominal number | Why |
|---|---|---|---|---|
| The “3” in “3 apples” | ✓ | ✗ | ✗ | |
| The “3rd” in “finished 3rd” | ✗ | ✓ | ✗ | |
| The “23” on a jersey | ✗ | ✗ | ✓ | |
Missing example: omega_plus_1 | ||||