Prime zeta function: Difference between revisions
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{{Portal:Mathematics/Feature article|img=|img-cap=|img-cred=|more=Continuum hypothesis|desc=The '''continuum hypothesis''' is a [[hypothesis]], advanced by [[Georg Cantor]], about the possible sizes of [[infinite set]]s. Cantor introduced the concept of [[cardinal number|cardinality]] to compare the sizes of infinite sets, and he showed that the set of [[integer]]s is strictly smaller than the set of [[real number]]s. The continuum hypothesis states the following: | |||
:There is no set whose size is strictly between that of the integers and that of the real numbers. | |||
Or mathematically speaking, noting that the [[Cardinal number|cardinality]] for the integers <math>|\mathbb{Z}|</math> is <math>\aleph_0</math> ("[[aleph number|aleph-null]]") and the [[cardinality of the real numbers]] <math>|\mathbb{R}|</math> is <math>2^{\aleph_0}</math>, the continuum hypothesis says | |||
:<math>\nexists \mathbb{A}: \aleph_0 < |\mathbb{A}| < 2^{\aleph_0}.</math> | |||
This is equivalent to: | |||
:<math>2^{\aleph_0} = \aleph_1</math> | |||
The real numbers have also been called [[Real line|''the continuum'']], hence the name.|class={{{class}}}}} | |||
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Revision as of 03:11, 24 October 2013
{{../Header}} Template:Portal:Mathematics/Feature article {{../Footer}}