Editing Aleph number (section)

==Continuum hypothesis==
{{Main|Continuum hypothesis}}
{{See also|Beth number}}
The [[cardinality]] of the set of [[real number]]s ([[cardinality of the continuum]]) is 2<sup><math>\aleph_0</math></sup>. It cannot be determined from [[ZFC]] ([[Zermelo–Fraenkel set theory]] augmented with the [[axiom of choice]]) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity
:2<sup><math>\aleph_0</math></sup> = <math>\aleph_1</math>.<ref name=WolframCH>
{{cite web
 |url=http://mathworld.wolfram.com/ContinuumHypothesis.html
 |title=Continuum Hypothesis
 |last=Szudzik |first=Mattew
 |date=31 July 2018
 |website=Wolfram Mathworld
 |publisher=Wolfram Web Resources
 |access-date=15 August 2018
}}
</ref>

The CH states that there is no set whose cardinality is strictly between that of the natural numbers and the real numbers.<ref>
{{cite web
 |last=Weisstein |first=Eric W.
 |title=Continuum Hypothesis
 |url=https://mathworld.wolfram.com/ContinuumHypothesis.html
 |access-date=2020-08-12
 |website=mathworld.wolfram.com
 |language=en
}}
</ref> CH is independent of [[ZFC]]: It can be neither proven nor disproven within the context of that axiom system (provided that [[ZFC]] is [[consistency|consistent]]). That CH is consistent with [[ZFC]] was demonstrated by [[Kurt Gödel]] in 1940, when he showed that its negation is not a theorem of [[ZFC]]. That it is independent of [[ZFC]] was demonstrated by [[Paul Cohen]] in 1963, when he showed conversely that the CH itself is not a theorem of [[ZFC]] – by the (then-novel) method of [[Forcing (mathematics)|forcing]].<ref name=WolframCH/><ref>
{{cite arXiv
 |last=Chow |first=Timothy Y.
 |title=A beginner's guide to forcing
 |eprint=0712.1320
 |date=2007
|class=math.LO
 }}
</ref>