the set of all real transcendental numbers is uncountable
Proof.
Denote 𝕋 and 𝔸 be the set of real transcendental and real algebraic numbers respectively. Suppose 𝕋 is countable
. Then the union 𝕋∪𝔸=ℝ is also countable, since 𝔸 is also countable, which is a contradiction
. Therefore 𝕋 must be uncountable. □
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 |