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# cardinality of the continuum

The *cardinality of the continuum*, often denoted by $\mathfrak{c}$, is
the cardinality of the set $\mathbb{R}$ of real numbers.
A set of cardinality $\mathfrak{c}$ is said to have *continuum many* elements.

Cantorβs diagonal argument shows that $\mathfrak{c}$ is uncountable. Furthermore, it can be shown that $\mathbb{R}$ is equinumerous with the power set of $\mathbb{N}$, so $\mathfrak{c}=2^{{\aleph_{0}}}$. It can also be shown that $\mathfrak{c}$ has uncountable cofinality.

It can also be shown that

$\mathfrak{c}=\mathfrak{c}^{{\aleph_{0}}}=\aleph_{0}\mathfrak{c}=\mathfrak{c}% \mathfrak{c}=\mathfrak{c}+\kappa=\mathfrak{c}^{n}$ |

for all finite cardinals $n\geq 1$ and all cardinals $\kappa\leq\mathfrak{c}$. See the article on cardinal arithmetic for some of the basic facts underlying these equalities.

There are many properties of $\mathfrak{c}$ that independent of ZFC, that is, they can neither be proved nor disproved in ZFC, assuming that ZF is consistent. For example, for every nonzero natural number $n$, the equality $\mathfrak{c}=\aleph_{n}$ is independent of ZFC. (The case $n=1$ is the well-known Continuum Hypothesis.) The same is true for most other alephs, although in some cases equality can be ruled out on the grounds of cofinality, e.g., $\mathfrak{c}\neq\aleph_{\omega}$. In particular, $\mathfrak{c}$ could be either $\aleph_{1}$ or $\aleph_{{\omega_{1}}}$, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.

## Mathematics Subject Classification

03E17*no label found*03E10

*no label found*

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