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## Cardinal number
In mathematics, A natural 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. While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish between the two. This idea was developed by Georg Cantor. The position aspect leads to ordinal numbers, which were also discovered by Cantor, while the size aspect is generalized by the cardinal numbers described here.
Two sets
## Motivation
A set - 1 → 5
- 2 → 6
- 3 → 7
Y has cardinality greater than or equal to X. The advantage of this notion is that it can be extended to infinite sets.The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. Suppose you are a 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. In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a one-to-one mapping from the first to the second has been shown. This motivates the definition of an infinite set being any set which has a proper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}. 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 is easily 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. ## Formal definition
Formally, the order among cardinal numbers is defined as follows: |
A set
Note that without the axiom of choice there are sets which can not be well-ordered, and the definition of cardinal number given above does not work. It is still possible to define cardinal numbers (a mapping from sets to sets such that sets with the same
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 or Y is infinite and both are non-empty, then
- |
*X*| + |*Y*| = |*X*| × |*Y*| = max{|*X*|, |*Y*|}.
^{| X |} is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^{| X |} > | X | for any set X. This proves that there exists no largest cardinal. In fact, the class of cardinals is a proper class.
The continuum hypothesis (CH) states that there are no cardinals strictly between and .
The latter cardinal number is also often denoted by See also large cardinal. ## External links | |||||||

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