In common usage, a cardinal number is a number used in counting (a counting number), such as 1, 2, 3, ....
In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method
of counting sets using it gives the same result. (This is
not true for the ordinal numbers.) In fact, the
cardinal numbers are obtained by collecting all ordinal
numbers which are obtainable by counting a given set. A set has 
(aleph-0) members if it
can be put into a one-to-one correspondence
with the finite ordinal numbers. The cardinality
of a set is also frequently referred to as the "power" of a set (Moore
1982, Dauben 1990, Suppes 1972).
In Georg Cantor's original notation, the symbol for a set 
annotated with a single overbar 
indicated 
stripped of any structure besides order, hence it represented
the order type of the set. A double overbar 
then indicated stripping the order from the set and thus
indicated the cardinal number of the set. However, in modern notation, the symbol
is used to denote the cardinal number
of set.
Cantor, the father of modern set theory, noticed that while the ordinal numbers 
, 
, ... were bigger than omega in the sense of order, they
were not bigger in the sense of equipollence. This
led him to study what would come to be called cardinal numbers. He called the ordinals
, 
, ... that are equipollent to the integers "the
second number class" (as opposed to the finite ordinals, which he called the
"first number class"). Cantor showed
1. The second number class is bigger than the first.
2. There is no class bigger than the first number class and smaller than the second.
3. The class of real numbers is bigger than the first number class.
One of the first serious mathematical definitions of cardinal was the one devised by Gottlob Frege and Bertrand Russell, who defined a cardinal number 
as the set of all sets equipollent
to 
.
(Moore 1982, p. 153; Suppes 1972, p. 109). Unfortunately, the objects produced
by this definition are not sets in the sense of Zermelo-Fraenkel
set theory, but rather "proper classes"
in the terminology of von Neumann.
Tarski (1924) proposed to instead define a cardinal number by stating that every set 
is associated with a cardinal number 
, and two sets 
and 
have the same cardinal number iff
they are equipollent (Moore 1982, pp. 52 and
214; Rubin 1967, p. 266; Suppes 1972, p. 111). The problem is that this
definition requires a special axiom to guarantee that cardinals exist.
A. P. Morse and Dana Scott defined cardinal number by letting 
be any set, then calling 
the set of all sets equipollent
to 
and of least possible rank (Rubin 1967, p. 270).
It is possible to associate cardinality with a specific set, but the process required either the axiom of foundation or the axiom of choice. However, these are two of the more controversial Zermelo-Fraenkel axioms. With the axiom of choice, the cardinals can be enumerated through the ordinals. In fact, the two can be put into one-to-one correspondence. The axiom of choice implies that every set can be well ordered and can therefore be associated with an ordinal number.
This leads to the definition of cardinal number for a set 
as the least ordinal
number 
such that 
and 
are equipollent. In this model, the cardinal numbers
are just the initial ordinals. This definition
obviously depends on the axiom of choice, because
if the axiom of choice is not true, then there
are sets that cannot be well ordered. Cantor believed that every set could be well
ordered and used this correspondence to define the 
s ("alephs"). For any ordinal
number 
,
.
An inaccessible cardinal cannot be expressed in terms of a smaller number of smaller cardinals.
