What is "cardinal number"

Number \Num"ber\ (n[u^]m"b[~e]r), n. [OE. nombre, F. nombre, L. numerus; akin to Gr. no`mos that which is dealt out, fr. ne`mein to deal out, distribute. See Numb, Nomad, and cf. Numerate, Numero, Numerous.]

1. That which admits of being counted or reckoned; a unit, or an aggregate of units; a numerable aggregate or collection of individuals; an assemblage made up of distinct things expressible by figures.

2. A collection of many individuals; a numerous assemblage; a multitude; many.

   Ladies are always of great use to the party they espouse, and never fail to win over numbers.
   --Addison.

3. A numeral; a word or character denoting a number; as, to put a number on a door.

4. Numerousness; multitude.

   Number itself importeth not much in armies where the people are of weak courage.
   --Bacon.

5. The state or quality of being numerable or countable.

   Of whom came nations, tribes, people, and kindreds out of number.
   --2 Esdras iii. 7.

6. Quantity, regarded as made up of an aggregate of separate things.

7. That which is regulated by count; poetic measure, as divisions of time or number of syllables; hence, poetry, verse; -- chiefly used in the plural.

   I lisped in numbers, for the numbers came.
   --Pope.

8. (Gram.) The distinction of objects, as one, or more than one (in some languages, as one, or two, or more than two), expressed (usually) by a difference in the form of a word; thus, the singular number and the plural number are the names of the forms of a word indicating the objects denoted or referred to by the word as one, or as more than one.

9. (Math.) The measure of the relation between quantities or things of the same kind; that abstract species of quantity which is capable of being expressed by figures; numerical value.

   Abstract number, Abundant number, Cardinal number, etc. See under Abstract, Abundant, etc.

   In numbers, in numbered parts; as, a book published in numbers.

1590s, "one, two, three," etc. as opposed to ordinal numbers "first, second, third," etc.; so called because they are the principal numbers and the ordinals depend on them (see cardinal (adj.)).

n. 1 A number used to denote quantity; a counting number. 2 (context mathematics English) A generalized kind of number used to denote the size of a set, including infinite sets. 3 (context grammar English) A word that expresses a countable quantity; a cardinal numeral.

n. the number of elements in a mathematical set; denotes a quantity but not the order [syn: cardinal]

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets.

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

There is a transfinite sequence of cardinal numbers:

$$0, 1, 2, 3, \ldots, n, \ldots ; \aleph_0, \aleph_1, \aleph_2, \ldots, \aleph_{\alpha}, \ldots.\$$
This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.

Cardinal

one

two

three

four

1

2

3

4

Ordinal

first

second

third

fourth

1st

2nd

3rd

4th

In linguistics, more precisely in traditional grammar, a cardinal number or cardinal numeral (or just cardinal) is a part of speech used to count, such as the English words one, two, three, but also compounds, e.g. three hundred and forty-two ( Commonwealth English) or three hundred forty-two ( American English). Cardinal numbers are classified as definite numerals and are related to ordinal numbers, such as first, second, third, etc.