The small countable ordinals

The ordinals begin with the following transfinite progression

$0,1,2,3,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega\cdot 2,\omega\cdot 2+1,\ldots,\omega\cdot 3,\ldots,\omega^2,\omega^2+1,\ldots,\omega^2+\omega,\ldots,\omega^2+\omega+1,\ldots,\omega^2+\omega\cdot 2,\ldots,\omega^3,\ldots,$

$\omega^\omega,\omega^\omega+1,\ldots,\omega^\omega+\omega,\ldots,\omega^\omega+\omega\cdot 2,\ldots,\omega^\omega\cdot 2,\ldots,\omega^{\omega^\omega},\ldots,\omega^{\omega^{\omega^\omega}},\ldots$

Counting to $\omega^2$

We explain here in detail how to count to $\omega^2$. This is something that anyone can learn to do, even young children.

The ordinals below $\epsilon_0$

We shall give here an account of the attractive finitary represenation of the ordinals below $\epsilon_0$.