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The continuum hypothesis says that =, i.e. is the smallest cardinal number bigger than , i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum ...
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor, who first stated and proved it at the end of the 19th century.
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three.
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers ... the set of sequences of integers (i.e. all functions ...
An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.
A set A is said to have cardinality smaller than or equal to the cardinality of a set B, if there exists a one-to-one function (an injection) from A into B. This is denoted |A| ≤ |B|. If A and B are not equinumerous, then the cardinality of A is said to be strictly smaller than the cardinality of B. This is denoted |A| < |B|.
A set has cardinality ℵ 0 if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are the set of natural numbers, irrespective of including or excluding zero, the set of all integers,
[a] Like the set of natural numbers, the set of integers is countably infinite. An integer may be regarded as a real number that can be written without a fractional component . For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2 , 5/4, and √ 2 are not.