A D11: Set $X$ is a

**countable set**if and only if \begin{equation} \exists \, E \subseteq \mathbb{N} : \text{Bij}(E \to X) \neq \emptyset \end{equation}

▼ | Set of symbols |

▼ | Alphabet |

▼ | Deduction system |

▼ | Theory |

▼ | Zermelo-Fraenkel set theory |

▼ | Set |

▼ | Binary cartesian set product |

▼ | Binary relation |

▼ | Map |

▼ | Bijective map |

▼ | Set of bijections |

Definition D16

Countable set

Formulation 0

Let $\mathbb{N}$ be the D225: Set of natural numbers.

A D11: Set $X$ is a**countable set** if and only if
\begin{equation}
\exists \, E \subseteq \mathbb{N} :
\text{Bij}(E \to X)
\neq \emptyset
\end{equation}

A D11: Set $X$ is a

Formulation 1

Let $\mathbb{N}$ be the D225: Set of natural numbers.

A D11: Set $X$ is a**countable set** if and only if there exists a D18: Map $f : E \to X$ such that

A D11: Set $X$ is a

(1) | $E \subseteq \mathbb{N}$ is a D78: Subset of $\mathbb{N}$ |

(2) | $f$ is a D468: Bijective map from $E$ to $X$ |

Children

▶ | D1881: Cocountable set |

▶ | D17: Finite set |

Results