*Alternative meaning number of pitch classes in a set.*

In linguistics,

In mathematics,

He first established cardinality as an instrument to compare finite sets; i.e the sets {1,2,3} and {2,3,4} are not

Cantor invented the one-to-one correspondence, which easily showed that two finite sets had the same cardinality if there was a one-to-one correspondence between the members of the set. Using this one-to-one correspondence, he transferred the concept to infinite sets; i.e the set of natural numbers

Naming this cardinal number $aleph\_0$, aleph-null, Cantor proved that many subsets of

At this point, in 1874, there was a curiosity whether

But, later that year, Cantor succeeded in proving that there were higher-order cardinal numbers using the ingenious but simple Cantor's diagonal argument. This new cardinal number, called the "power of continuum", was termed

Cantor also developed a lot of the general theory of cardinal numbers; he proved that there is a transfinite cardinal number that is the smallest ($aleph\_0$, aleph-null) and that for every cardinal number, there is a next-larger cardinal ($aleph\_1,\; aleph\_2,\; aleph\_3,\; cdots$).

The later continuum hypothesis suggests that

More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions co-incide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e.g. 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c} which has 3 elements.However when dealing with infinite sets it is essential to distinguish between the two --- the two notionsare in fact different for infinite sets.Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the

The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.

A set

- 1 → a
- 2 → b
- 3 → c

We can then extend this to an equality-style relation.Two sets

The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It's possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicity write a segment of this mapping:

- 1 ↔ 2
- 2 ↔ 3
- 3 ↔ 4
- ...
- n ↔ n+1
- ...

When considering these large objects, we might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it doesn't; by considering the above example we can see that if some object "one greater than infinity" exists, then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, called ordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.

It is provable that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument;classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.

Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as

Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α. If the axiom of choice is not assumed and X does not have a well-ordering, the cardinality of X is defined to be the set of all sets which are equinumerous with X and have the least rank that a set equinumerous with X can have.

A set

If

- addition and multiplication of cardinal numbers is associative and commutative
- multiplication distributes over addition
- |
*X*|^{|Y| + |Z|}= |*X*|^{|Y|}× |*X*|^{|Z|} - |
*X*|^{|Y| × |Z|}= (|*X*|^{|Y|})^{|Z|} - (|
*X*| × |*Y*|)^{|Z|}= |*X*|^{|Z|}× |*Y*|^{|Z|}

The addition and multiplication of infinite cardinal numbers (assuming the axiom of choice) is easy: if

- |
*X*| + |*Y*| = |*X*| × |*Y*| = max{|*X*|, |*Y*|}.

On the other hand, 2

The continuum hypothesis (CH) states that there are no cardinals strictly between $aleph\_0$ and $2^\{aleph\_0\}$.The latter cardinal number is also often denoted by

- large cardinal.

- Hahn, Hans,
*Infinity*, Part IX, Chapter 2, Volume 3 of*The World of Mathematics*. New York: Simon and Schuster, 1956.

- http://web.archive.org/web/20010528181608/http://logweb.terrashare.com/text/logic/infini.txt
- http://www.ii.com/math/cardinals/