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# Cardinality

The **cardinality** of a set is the number of elements of that set, represented by a cardinal number. Two sets have the same cardinality if and only if there is a bijection (one-to-one and onto correspondence) between the two sets. (While this is rather trivial for finite sets, the concept is essential for infinite sets.) The cardinality of a set ** Math error: Invalid format. ** is denoted in one of the following ways:

## Examples

## Cardinality of infinite sets

The cardinality of the set of natural numbers defines ** Math error: Invalid format. ** (pronounced aleph null or aleph nought, aleph being the first letter of the Hebrew alphabet), i.e. the cardinality of countably infinite sets (enumerable sets, denumerable sets), i.e. sets for which a one-to-one and onto correspondence with the set of natural numbers can be made

A set which has an injection into the natural numbers (i.e., either has a bijection to the natural numbers or is finite) is called *countable*. Georg Cantor proved that the the set of rational numbers are countable:

and it can be proved with a generalization of Cantor's idea that the set of algebraic numbers is also countable:

The cardinality of the real numbers, called the cardinality of the continuum and denoted or , is the cardinality of the power set of the natural numbers

The continuum hypothesis states that there are no cardinalities between ** Math error: Invalid format. ** and ** Math error: Invalid format. **, so that ** Math error: Invalid format. **.