In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers, sometimes called the continuum. It is an infinite cardinal number and is denoted by or (a lowercase fraktur script c).
The real numbers are more numerous than the natural numbers . Moreover, has the same number of elements as the power set of . Symbolically, if the cardinality of is denoted as, the cardinality of the continuum is
This was proven by Georg Cantor in his 1874 uncountability proof, part of his groundbreaking study of different infinities, and later more simply in his diagonal argument. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if and only if there exists a bijective function between them.
Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with This is also true for several other infinite sets, such as any n-dimensional Euclidean space (see Space filling curve). That is,
The smallest infinite cardinal number is (aleph-naught). The second smallest is (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between and implies that .
Other articles related to "cardinality of the continuum, the continuum, cardinality of the, cardinality":
... One of Cantor's most important results was that the cardinality of the continuum is greater than that of the natural numbers that is, there are more ... The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, (see Beth one) ...
... Sets with cardinality greater than include the set of all subsets of (i.e ... They all have cardinality (Beth two) ...
... One of Cantor's most important results was that the cardinality of the continuum is greater than that of the natural numbers (ℵ0) that is, there are more real numbers R than whole numbers N ... The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, (see Beth one) ... Cantor also showed that sets with cardinality strictly greater than exist (see his generalized diagonal argument and theorem) ...
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