proof of Cantor’s theorem

The proof of this theorem is fairly using the following construction, which is central to Cantor’s diagonal argument.

Consider a function F:X𝒫(X) from a set X to its power setMathworldPlanetmath. Then we define the set ZX as follows:


Suppose that F is a bijection. Then there must exist an xX such that F(x)=Z. Then we have the following contradictionMathworldPlanetmathPlanetmath:


Hence, F cannot be a bijection between X and 𝒫(X).

Title proof of Cantor’s theorem
Canonical name ProofOfCantorsTheorem
Date of creation 2013-03-22 12:44:55
Last modified on 2013-03-22 12:44:55
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Proof
Classification msc 03E17
Classification msc 03E10