The ordinals begin with the following transfinite progression $0,1,2,3,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega\cdot 2,\omega\cdot 2+1,\ldots,\omega\cdot 3,\ldots,\omega^2,\omega^2+1,\ldots,\omega^2+\omega,\ldots,\omega^2+\omega+1,\ldots,\omega^2+\omega\cdot 2,\ldots,\omega^3,\ldots,$ $\omega^\omega,\omega^\omega+1,\ldots,\omega^\omega+\omega,\ldots,\omega^\omega+\omega\cdot 2,\ldots,\omega^\omega\cdot 2,\ldots,\omega^{\omega^\omega},\ldots,\omega^{\omega^{\omega^\omega}},\ldots,\epsilon_{0}$ ## 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 [$\epsilon_0$](Epsilon_naught.md "Epsilon naught") We shall give here an account of the attractive finitary represenation of the ordinals below [$\epsilon_0$](Epsilon_naught.md "Epsilon naught").