# the set of all real transcendental numbers is uncountable

*Proof.*
Denote $\mathbb{T}$ and $\mathbb{A}$ be the set of real transcendental and real algebraic numbers^{} respectively. Suppose $\mathbb{T}$ is countable^{}. Then the union $\mathbb{T}\cup \mathbb{A}=\mathbb{R}$ is also countable, since $\mathbb{A}$ is also countable, which is a contradiction^{}. Therefore $\mathbb{T}$ must be uncountable. $\mathrm{\square}$

Title | the set of all real transcendental numbers is uncountable |
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Canonical name | TheSetOfAllRealTranscendentalNumbersIsUncountable |

Date of creation | 2013-03-22 16:08:05 |

Last modified on | 2013-03-22 16:08:05 |

Owner | gilbert_51126 (14238) |

Last modified by | gilbert_51126 (14238) |

Numerical id | 11 |

Author | gilbert_51126 (14238) |

Entry type | Theorem |

Classification | msc 03E10 |