Diagonalization

Diagonalization is a technique first used by George Cantor, a German mathematician. He used it to show that the cardinality of the real numbers is not equal to the cardinality of the natural numbers, thereby demonstrating the real numbers are not countable.

Proof of the non-countability of real numbers

There exists a map $f:[0,1]\rightarrow \mathbb {R}$ (in fact all infinitly supported probability distribution does this). Therefor there are as many number in $[0,1]$ as $\mathbb {R}$.

We will now use proof by contradiction to show that the numbers in $[0,1]$ are uncountable.

Assume that that they are countable then we can list them as such,

$0.a_{11}a_{12}a_{13}a_{14}a_{15}\dots$

$0.a_{21}a_{22}a_{23}a_{24}a_{25}\dots$

$0.a_{31}a_{32}a_{33}a_{34}a_{35}\dots$

$0.a_{41}a_{42}a_{43}a_{44}a_{45}\dots$

$\vdots$

Where $a_{ij}\in \{0,1,2,3,4,5,6,7,8,9\}$

Construct the number,

$a=0.a_{1}a_{2}a_{3}a_{4}\dots$, where

$a_{i}=1$ when $a_{ii}\neq 1$ and $a_{i}=2$ when $a_{ii}=1$.

Therefor $a$ is not in the list, so we have a contradition and our assumption is false, the number $[0,1]$ are not countable. Therefor $\mathbb {R}$ is uncountable.

References

Komolgorov, Introduction to Real Analysis. (You can find it inalmost any book store).