The small countable ordinals

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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.

First, some terminology: the successor of an ordinal $\alpha$, sometimes written $S\alpha$, can be thought of as the ordinal coming directly after $\alpha$. Since it's "directly" after $\alpha$, there are no ordinals $\beta$ where $\alpha<\beta<S\alpha$.

We begin with the familiar natural numbers 0, 1, 2, 3, etc. After all of these is $\omega$, the least infinite ordinal. One way to think about what "infinite" means is that it's larger than all finite numbers. (There are other formulations such as Dedekind-infiniteness if we need to do this without reference to $\mathbb N$)

Then, we take the successor of $\omega$, $S\omega$, and get an ordinal equal to $\omega+1$. The $+$ here is a bit different from ordinary addition on natural numbers, that is often informally thought of as "putting two numbers together", and it's recursively defined for ordinals.

We continue with $SS\omega=\omega+2$, $SSS\omega=\omega+3$, etc., until we reach $\omega\times 2=\omega+\omega$. This ordinal is larger than $\omega+n$ for any natural number $n$. (Leaving out the noncommutativity part for this explanation?)

We can also add natural numbers to this ordinal, for example $\omega+\omega+5=SSSSS(\omega+\omega)$.

The ordinals below $\epsilon_0$

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

Kirby-Paris hydras