Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or 'size' of the set of real numbers R {displaystyle mathbb {R} } , sometimes called the continuum. It is an infinite cardinal number and is denoted by | R | {displaystyle |mathbb {R} |} or c {displaystyle {mathfrak {c}}} (a lowercase fraktur script 'c').For instance, for all a , b ∈ R {displaystyle a,bin mathbb {R} } such that a < b {displaystyle a<b} we can define the bijection In set theory, the cardinality of the continuum is the cardinality or 'size' of the set of real numbers R {displaystyle mathbb {R} } , sometimes called the continuum. It is an infinite cardinal number and is denoted by | R | {displaystyle |mathbb {R} |} or c {displaystyle {mathfrak {c}}} (a lowercase fraktur script 'c'). The real numbers R {displaystyle mathbb {R} } are more numerous than the natural numbers N {displaystyle mathbb {N} } . Moreover, R {displaystyle mathbb {R} } has the same number of elements as the power set of N {displaystyle mathbb {N} } . Symbolically, if the cardinality of N {displaystyle mathbb {N} } is denoted as ℵ 0 {displaystyle aleph _{0}} , 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; the inequality was later stated 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 R . {displaystyle mathbb {R} .} This is also true for several other infinite sets, such as any n-dimensional Euclidean space R n {displaystyle mathbb {R} ^{n}} (see space filling curve). That is, The smallest infinite cardinal number is ℵ 0 {displaystyle aleph _{0}} (aleph-null). The second smallest is ℵ 1 {displaystyle aleph _{1}} (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between ℵ 0 {displaystyle aleph _{0}} and c {displaystyle {mathfrak {c}}} , implies that c = ℵ 1 {displaystyle {mathfrak {c}}=aleph _{1}} . Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite; i.e. c {displaystyle {mathfrak {c}}} is strictly greater than the cardinality of the natural numbers, ℵ 0 {displaystyle aleph _{0}} : In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. See Cantor's first uncountability proof and Cantor's diagonal argument. A variation on Cantor's diagonal argument can be used to prove Cantor's theorem which states that the cardinality of any set is strictly less than that of its power set, i.e. | A | < 2 | A | {displaystyle |A|<2^{|A|}} , and so the power set ℘ ( N ) {displaystyle wp (mathbb {N} )} of the natural numbers N {displaystyle mathbb {N} } is uncountable. In fact, it can be shown that the cardinality of ℘ ( N ) {displaystyle wp (mathbb {N} )} is equal to c {displaystyle {mathfrak {c}}} : By the Cantor–Bernstein–Schroeder theorem we conclude that