# Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as appropriately (= lexicographically) ordered rows in the cube $\omega^3$ (rows being sequences of cells parallel to the x-axis):

It “only” takes the ability to visualize higher-dimensional (hyper-)cubes to visualize arbitrary countable ordinals of the form $\omega^n$ as $\omega^{n-1}$ appropriately ordered copies of $\omega$.

But this ability comes to an end when thinking of the countable ordinal number $\omega^\omega$, since the $|\omega|$-dimensional hypercube $\omega^{|\omega|}$ consists of uncountably many cells and cannot represent a countable ordinal.1

While (presumably) the countable ordinal number $\omega^\omega$ cannot be visualized as a countable number of copies of $\omega$ (however carefully arranged) – can it be visualized in a similarly easy visual way, only “slightly” advanced? (For example, by replacing $\omega$ by some other (higher) ordinal number?)

If there were such a “similarly easy way”: Up to which limit would it lead? Up to $\epsilon_0$? And which “similarly easy way” would come beyond that limit?

1 Do $|\omega|$-dimensional vector spaces – e.g. Hilbert spaces – pose such serious problems?

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$\omega^\omega$ can be visualized, in what I think is a fairly nice way in a static 2D image featured on the wikipedia page for ordinal number:

Also, if you’re willing to allow dynamic visualizations, then Stephen Brooks’s transfinite number line goes well past $\epsilon_0$ (to $\Gamma_0$), as well as providing a more linear (if colorful) look at $\omega^\omega$:

It is wrong to think of $\omega^\omega$ as a hypercube of actually infinite dimension, which would correspond to the set of all infinite sequences of natural numbers, which is uncountable. One may think – if at all – of $\omega^\omega$ only as a hypercube of potentially infinite dimension, e.g. as $\bigcup_{n<\omega}\omega^n$, i.e. the functions $f:\omega \rightarrow \omega$ with finite support.

Especially: Other than $\omega^n$ – which is a well-ordered set of disjoint copies of $\omega$ – the ordinal $\omega^\omega$ cannot be thought of this way, even though the hypercube or the spiral representation might suggest it.