View source for Cardinality

''cardinal number, of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203801.png" />''

That property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203802.png" /> which is inherent in any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203803.png" /> equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203804.png" />. Here two sets are called equivalent (or equipotent or of the same cardinality) if it is possible to construct a one-to-one correspondence between them. Thus,  "defining by abstraction" , one may say that cardinality is that which is common to all equivalent sets. Since what is common to all finite sets equivalent to each other is the quantity, or number, of their elements, in its application to infinite sets the idea of cardinality is an analogue of the idea of quantity. The cardinality is a fundamental idea in set theory, due to G. Cantor. A set that is equivalent to the set of all natural numbers is called countable. The corresponding cardinality is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203805.png" /> (aleph null). The cardinality of sets equivalent to the set of real numbers is called the cardinality of the continuum and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203806.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203807.png" />. For example, the set of all algebraic numbers has countable cardinality, and the set of all closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203808.png" />-dimensional Euclidean space has the cardinality of the continuum. The Cantor–Bernshtein theorem: If for two sets, each is equivalent to a subset of the other, then the two sets are equivalent. In this case it is said that these sets have the same cardinality. If a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c0203809.png" /> is equivalent to a subset of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038010.png" />, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038011.png" /> is not equivalent to any subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038012.png" />, then it is said that the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038013.png" /> is greater than that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038014.png" />. Cantor's theorem: The cardinality of the set of all subsets of a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038015.png" /> is greater than the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020380/c02038016.png" />. This theorem allows one to construct a hierarchy of cardinalities (see [[Cardinal number|Cardinal number]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====
More information and references can be found in the article [[Cardinal number|Cardinal number]].