# cardinality of the continuum

The *cardinality of the continuum ^{}*, often denoted by $\mathrm{\pi \x9d\x94}$, is
the cardinality of the set $\mathrm{\beta \x84\x9d}$ of real numbers.
A set of cardinality $\mathrm{\pi \x9d\x94}$ is said to have

*continuum many*elements.

Cantorβs diagonal argument shows that $\mathrm{\pi \x9d\x94}$ is uncountable.
Furthermore, it can be shown that
$\mathrm{\beta \x84\x9d}$ is equinumerous with the power set^{} of $\mathrm{\beta \x84\x95}$, so $\mathrm{\pi \x9d\x94}={2}^{{\mathrm{\beta \x84\u0385}}_{0}}$.
It can also be shown that $\mathrm{\pi \x9d\x94}$ has uncountable cofinality.

It can also be shown that

$$\mathrm{\pi \x9d\x94}={\mathrm{\pi \x9d\x94}}^{{\mathrm{\beta \x84\u0385}}_{0}}={\mathrm{\beta \x84\u0385}}_{0}\beta \x81\u2019\mathrm{\pi \x9d\x94}=\mathrm{\pi \x9d\x94}\beta \x81\u2019\mathrm{\pi \x9d\x94}=\mathrm{\pi \x9d\x94}+\mathrm{\Xi \u038a}={\mathrm{\pi \x9d\x94}}^{n}$$ |

for all finite cardinals $n\beta \x89\u20af1$ and all cardinals $\mathrm{\Xi \u038a}\beta \x89\u20ac\mathrm{\pi \x9d\x94}$. See the article on cardinal arithmetic for some of the basic facts underlying these equalities.

There are many properties of $\mathrm{\pi \x9d\x94}$ 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 $\mathrm{\pi \x9d\x94}={\mathrm{\beta \x84\u0385}}_{n}$ is independent of ZFC.
(The case $n=1$ is the well-known
Continuum Hypothesis (http://planetmath.org/ContinuumHypothesis).)
The same is true for most other alephs,
although in some cases equality can be ruled out on the grounds of cofinality,
e.g., $\mathrm{\pi \x9d\x94}\beta \x89{\mathrm{\beta \x84\u0385}}_{\mathrm{{\rm O}\x89}}$.
In particular,
$\mathrm{\pi \x9d\x94}$ could be either ${\mathrm{\beta \x84\u0385}}_{1}$ or ${\mathrm{\beta \x84\u0385}}_{{\mathrm{{\rm O}\x89}}_{1}}$,
so it could be either a successor cardinal or a limit cardinal,
and either a regular cardinal or a singular cardinal.

Title | cardinality of the continuum |

Canonical name | CardinalityOfTheContinuum |

Date of creation | 2013-03-22 14:15:33 |

Last modified on | 2013-03-22 14:15:33 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 19 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E17 |

Classification | msc 03E10 |

Synonym | cardinal of the continuum |

Synonym | cardinal number of the continuum |

Related topic | CardinalNumber |

Related topic | CardinalArithmetic |

Defines | continuum many |