In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers <math>\mathbb R</math>, sometimes called the continuum. It is an infinite cardinal number and is denoted by <math>\bold\mathfrak c</math> (lowercase Fraktur "c") or <math>\bold|\bold\mathbb R\bold|.</math>
The real numbers <math>\mathbb R</math> are more numerous than the natural numbers <math>\mathbb N</math>. Moreover, <math>\mathbb R</math> has the same number of elements as the power set of <math>\mathbb N</math>. Symbolically, if the cardinality of <math>\mathbb N</math> is denoted as <math>\aleph_0</math>, the cardinality of the continuum is
This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. 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 <math>\mathbb R</math>, as well as with several other infinite sets, such as any n-dimensional Euclidean space <math>\mathbb R^n</math> (see space filling curve). That is,
The smallest infinite cardinal number is <math>\aleph_0</math> (aleph-null). The second smallest is <math>\aleph_1</math> (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between <math>\aleph_0</math> and , means that <math>\mathfrak c = \aleph_1</math>. This hypothesis is independent of the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC); that is, ZFC can neither prove that it is true nor that it is false.
Properties
Uncountability
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. That is, <math>{\mathfrak c}</math> is strictly greater than the cardinality of the natural numbers, <math>\aleph_0</math>:
In practice, this means that there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. For more information on this topic, see Cantor's first uncountability proof and Cantor's diagonal argument.
Cardinal equalities
A variation of 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. That is, <math>|A| < 2^{|A|}</math> (and so that the power set <math>\wp(\mathbb N)</math> of the natural numbers <math>\mathbb N</math> is uncountable). In fact, the cardinality of <math>\wp(\mathbb N)</math>, by definition <math>2^{\aleph_0}</math>, is equal to <math>{\mathfrak c}</math>. This can be shown by providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the Cantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same cardinality. In one direction, reals can be equated with Dedekind cuts, sets of rational numbers, That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality <math>{\mathfrak c}</math> = <math>\aleph_n</math> is independent of ZFC (case <math>n=1</math> being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out by König's theorem on the grounds of cofinality (e.g. <math>\mathfrak{c}\neq\aleph_\omega</math>). In particular, <math>\mathfrak{c}</math> could be either <math>\aleph_1</math> or <math>\aleph_{\omega_1}</math>, where <math>\omega_1</math> is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.
Sets with cardinality of the continuum
A great many sets studied in mathematics have cardinality equal to <math>{\mathfrak c}</math>. Some common examples are the following: