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