# Why aren't there $+\infty^{+\infty}$ real numbers?

I was reading this pop math piece on "the different sizes of Infinity." The article explains why the real numbers are uncountably infinite.

Taking a real number, my uneducated mathematical mind intuits that it could be considered as an infinitely-long word made up of letters drawn from an infinitely long alphabet (the rational numbers) in arbitrary combination (hence $+\infty$ to the power of $+\infty$ possible combinations). This would seem to suggest that the real numbers are countably infinite.

Of course, I know my reasoning must be wrong, but I do not have the mathematical background to find out why. Does anyone care to explain?

• You're looking for the Cantor diagonalization argument. en.wikipedia.org/wiki/Cantor's_diagonal_argument – user18862 Nov 12 '14 at 23:23
• All real numbers can be defined by infinite strings made from a finite alphabet. There are only $10$ digits but every real can be made from a string of them. – Jam Nov 12 '14 at 23:25
• In other words, you are asking us why $\aleph^\aleph>\aleph$ – Lucian Nov 12 '14 at 23:25
• @Lucian I think I am asking why |N^N| does not equal |R|. – user2398029 Nov 12 '14 at 23:27
• @louism $2^{\mathbb{N}}$ is the same size as $\mathbb{R}$. – Matt Samuel Nov 12 '14 at 23:28

The diagonal argument says that $\aleph_0<2^{\aleph_0}$ and $\aleph_0^{\aleph_0}=2^{\aleph_0}$ because $$2^{\aleph_0}\le\aleph_0^{\aleph_0}\le(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}$$