Is there a graphical way to represent limit ordinals up to $\omega_1$?

Question

For some countable ordinals, I found that there are some, such as the graphical “matchstick” representation of the ordinal $\omega^2$ , and the spiral representation of the ordinal numbers up to $\omega^\omega$.

Additional caveats: Both use the intuitive trick of "compressing" the naturals (countable infinitely many) as they approach a limit ordinal. They compress them in a "similar" way as the peaks of the function $\sin(1/x)$ as $x$ approaches zero. So in some way, these representations make use of a higher cardinality set: the real line. But there are uncountable many limit ordinals, and they compress from countable many into uncountable many as we approach $\omega_1$. So, shouldn't we need a set of cardinality $\aleph_2$ to represent how a set of cardinality $\aleph_1$ can compress to reach a limit? Is there any graphical representation of a set of cardinality $\aleph_2$ anyways? (or are we limited by our intuition to cardinality $\aleph_1$, as physical 3-D space is of that same cardinality?

Answer 1

There are unfathomably large countable ordinals. And when I say, I mean with all seriousness. For example, a relatively simple one $\omega_1^{CK}$ which is the least ordinal that there is no $R\subseteq\Bbb N^2$ which is computable (in the sense that there is a Turing machine deciding whether or not an ordered pair is in $R$ or not), and the order type of $R$ (on its domain) is an ordinal, this alone is an ordinal so large that it is unfathomable to think how complicated it would have to look like.

In fact, even much much much smaller ordinals like $\varepsilon_0$ are so complicated to understand visually that it is nearly impossible to see them like that.

All this is augmented by the fact that was mentioned by tomasz and myself in a previous thread, there is no actual agreement between models of set theory (even those that have exactly the same ordinals) on which ordinal is $\omega_1$ or $\omega_2$. This unlike small ordinals like $\omega$ or $\omega^\omega$.

From all those point of views, $\omega_1$ is this far end of the universe, that you know is out there, but have no idea what lies beyond the horizon. Except that you know that most of the universe is beyond that horizon, regardless to your best efforts.

But in fact $\omega_1$ is only the edge of the countable universe. It is not even close to the end of the actual universe. So it's not even that.

My advice as to visually thinking about large ordinals is simply thinking about $\omega^\omega$, reiterated ad nauseum (or ad infinitum if you don't get nauseous), and somewhat "fade away" the edge, knowing that no matter how far you climb up the scale, you are left with the same end segment in the order. It's not a real visual representation, nor it attempts to be. It's just the way I see ordinals, and I find it helpful. If I want to put $\omega_1$ by $\omega_2$, then I imagine two lines one is much longer than the other, and somehow work with that.

You need to understand that our minds cannot really grasp an infinity visually. This is why we have so many "paradoxes" about infinite objects. They are beyond our wildest dreams. And if want to argue that it is easy to discern the real numbers from the integers, let me remind you that the rational numbers are also countable.

Answer 2

Then there would be a bijection between the elements of $\omega_1$ and the set $R$ of all such representations. Suppose every such ordinal has a finite representation. Since each representation is finite, there is a finite set $R_0$ with the empty representation, a finite set $R_1$ of representations of length 1, a finite set $R_2$ of representations of length 2, and so on. Then $R=\bigcup R_i$ is a countable union of finite sets and so is at most countable.

But $\omega_1$ is uncountable. So no such representations can exist.