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Cardinality of the continuum
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{{Short description|Cardinality of the set of real numbers}} In [[set theory]], the '''cardinality of the continuum''' is the [[cardinality]] or "size" of the [[Set (mathematics)|set]] of [[real numbers]] <math>\mathbb R</math>, sometimes called the [[Continuum (set theory)|continuum]]. It is an [[Infinite set|infinite]] [[cardinal number]] and is denoted by <math>\bold\mathfrak c</math> (lowercase [[Fraktur]] "'''c'''") or <math>\bold|\bold\mathbb R\bold|.</math><ref>{{Cite web | title=Transfinite number {{!}} mathematics | url=https://www.britannica.com/science/transfinite-number|access-date=2020-08-12 | website=Encyclopedia Britannica | language=en}}</ref> 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 [[aleph number#Aleph-nought|<math>\aleph_0</math>]], the cardinality of the continuum is {{block indent|<math>\mathfrak c = 2^{\aleph_0} > \aleph_0. </math>}} This was proven by [[Georg Cantor]] in his [[Cantor's first uncountability proof|uncountability proof]] of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his [[Cantor's diagonal argument|diagonal argument]] in 1891. Cantor defined cardinality in terms of [[bijective function]]s: 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, {{block indent|<math>|(a,b)| = |\mathbb R| = |\mathbb R^n|.</math>}} The smallest infinite cardinal number is <math>\aleph_0</math> ([[aleph number#Aleph-nought|aleph-null]]). The second smallest is <math>\aleph_1</math> ([[aleph number#Aleph-one|aleph-one]]). The [[continuum hypothesis]], which asserts that there are no sets whose cardinality is strictly between <math>\aleph_0</math> and {{nowrap|<math>\mathfrak c</math>}}, means that <math>\mathfrak c = \aleph_1</math>.<ref name=":0">{{Cite web| last=Weisstein| first=Eric W.| title=Continuum| url=https://mathworld.wolfram.com/Continuum.html | access-date=2020-08-12 | website=mathworld.wolfram.com | language=en}}</ref> The truth or falsity of this hypothesis is undecidable and [[Continuum hypothesis#Independence from ZFC|cannot be proven]] within the widely used [[Zermelo–Fraenkel set theory]] with axiom of choice (ZFC).
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