Cardinality of the Continuum

In set theory, the cardinality of the continuum is the cardinality, or size, of the set of real numbers, sometimes called simply the continuum, and it is an infinite cardinal number usually denoted by a lowercase Fraktur letter c. The real numbers are more numerous than the natural numbers, and in fact the set of real numbers has exactly the same number of elements as the power set of the natural numbers, a fact proven by Georg Cantor in his uncountability proof of 1874 and restated more simply in his diagonal argument of 1891; Cantor defined cardinality in terms of bijective functions, so that two sets are considered to have the same cardinality exactly when a bijection exists between them. Between any two real numbers, no matter how close together, there are infinitely many further real numbers, and Cantor showed that any such open interval contains exactly as many points as the entire real line, or indeed as any n-dimensional Euclidean space; the continuum hypothesis, which asks whether any cardinality lies strictly between that of the natural numbers and that of the continuum, was later shown to be independent of the standard Zermelo-Fraenkel axioms with the axiom of choice, meaning those axioms can neither prove nor disprove it.