Editing Continuum hypothesis (section)

{{Short description|Proposition in mathematical logic}}
{{About|the hypothesis in set theory|the assumption in fluid mechanics|Continuum assumption|the album by Epoch of Unlight|The Continuum Hypothesis (album)}}
{{Use shortened footnotes|date=May 2021}}  

In [[mathematics]], specifically [[set theory]], the '''continuum hypothesis''' (abbreviated '''CH''') is a hypothesis about the possible sizes of [[infinite set]]s. It states:
{{Blockquote|There is no set whose [[cardinality]] is strictly between that of the [[integer]]s and the [[real number]]s.}}
Or equivalently:
{{Blockquote|Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers.}}

In [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]] (ZFC), this is equivalent to the following equation in [[aleph number]]s: <math>2^{\aleph_0}=\aleph_1</math>, or even shorter with [[beth number]]s: <math>\beth_1 = \aleph_1</math>.

The continuum hypothesis was advanced by [[Georg Cantor]] in 1878,{{r|Cantor1878}} and establishing its truth or falsehood is the first of [[Hilbert's problems|Hilbert's 23&nbsp;problems]] presented in 1900. The answer to this problem is [[independence (mathematical logic)|independent]] of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by [[Paul Cohen]], complementing earlier work by [[Kurt Gödel]] in 1940.{{r|Gödel1940}}

The name of the hypothesis comes from the term ''[[continuum (set theory)|the continuum]]'' for the real numbers.