View source for Continuum, cardinality of the

The [[Cardinal number|cardinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257801.png" />, i.e. the cardinality of the set of all subsets of the natural numbers. The following sets have the cardinality of the continuum: 1) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257802.png" /> of all real numbers; 2) the set of all points in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257803.png" />; 3) the set of all irrational numbers in this interval; 4) the set of all points of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257805.png" /> is a positive integer; 5) the set of all transcendental numbers; and 6) the set of all continuous functions of a real variable. The cardinality of the continuum cannot be represented as a countable sum of smaller cardinal numbers. For any cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257806.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025780/c0257807.png" />,

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In particular,

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The [[Continuum hypothesis|continuum hypothesis]] states that the cardinality of the continuum is the first uncountable cardinal number, that is,

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,   A. Mostowski,   "Set theory" , North-Holland  (1968)</TD></TR></table>