# numerable set

Let $X$ be a set. An *enumeration* on $X$ is a surjection from the set of natural numbers $\mathbb{N}$ to $X$.

A set $X$ is called *numerable* if there is a bijective^{} enumeration on $X$.

It is easy to show that $\mathbb{Z}$ and $\mathbb{Q}$ are numerable.

It is a standard fact that $\mathbb{R}$ is not numerable. For, if we suppose that the numbers [0,1] were countable^{}, we can arrange them in a list (given by the supposed bijection).

Representing them in a binary form, it is not hard to construct an element in [0,1], which is not in the list.

This contradiction^{} implies that [0,1]$\subset \mathbb{R}$ is not numerable.

Remark. If the enumeration $\mathbb{N}\to X$ is furthermore a computable function^{}, then we say that $X$ is *enumerable*. There exists numerable sets that are not enumerable.

Title | numerable set |

Canonical name | NumerableSet |

Date of creation | 2013-03-22 16:01:32 |

Last modified on | 2013-03-22 16:01:32 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 11 |

Author | juanman (12619) |

Entry type | Definition |

Classification | msc 97A80 |

Related topic | Calculus |

Related topic | TopicsOnCalculus |

Related topic | Denumerable |

Related topic | Countable |

Defines | enumeration |

Defines | enumerable |