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 .
Read more about Cardinality Of The Continuum: Beth Numbers, The Continuum Hypothesis, Sets With Cardinality of The Continuum, Sets With Greater Cardinality
Famous quotes containing the word continuum:
“The further jazz moves away from the stark blue continuum and the collective realities of Afro-American and American life, the more it moves into academic concert-hall lifelessness, which can be replicated by any middle class showing off its music lessons.”
—Imamu Amiri Baraka (b. 1934)