Eretrandre Mathematics Forum

, as defined by Cantor, is the first uncountable ordinal, and contains all of the countable ordinals as elements. The countable order types are easy to visualize; they are just elements, corresponding with natural numbers, ordered in such a way that there are "infinite stretches", or "gaps", between sub-sequences of the elements. But it's not so easy to visualize an uncountable ordinal. Here then, is an idea I had which perhaps gives us a little glimpse into what the order type of looks like.

The Basic Idea

The idea is simple: instead of restricting ourselves to 1 dimension, where we get a picture that is very difficult to visualize much less understand, we use 2 dimensions instead, by 'wrapping' each stage around to fill 2D, and iterate that into a fractal-like structure.


The Infinite Image

Here's our first crack at it: first, we begin with a unit square. This represents 0, the first finite ordinal. Then we apply the successor function to get 1, which we represent by a rectangle to the right of the unit square, with the same height but half the width. Next, we append to the right of these two a third rectangle, still the same height, but with 1/4 the original width. We take the limit of the process by repeating this, ad infinitum, halving the width of each subsequent rectangle.

This gives us w, the first infinite limit ordinal. It fits in the 2x1 rectangular area (since the half-widths has a limit at 2).

This idea, of course, is not new. Many have used analogous schemes to visualize the transfinite ordinals. The difference comes in the next step. Instead of merely applying the successor function, we make the next step by copying the rectangles corresponding to w, halving their heights, and placing the copies above the originals. Now we have a 2x1.5 rectangular area representing the structure of 2*w. Then we take another copy of w, make its height 1/4 the original, and stack it on 2*w, to get a 2x1.75 rectangular area representing the structure of 3*w.

Now we take the limit, and end up with a 2x2 rectangular area, representing w*w.

Then, we take this 2x2 area, scale it by (1,1/2) so that its height remains the same, but with half its original width, and place this squished copy to the right of the original. So we get a 3x2 area representing . Repeat this and take the limit, and we get a 4x2 area representing .

Then we take the 4x2 area, scale it to half its height, and stack it on itself, and repeat this, halving the height each time, and take the limit, and we get a 4x4 area representing .

The pattern is now clear. At each step, we take the limit of pasting copies of the square area into halved areas, switching between the X and Y axes. Every two steps yields a square area, and so we can repeat this process.

Now, take the limit of this process, and we get an infinitely large square representing . Each rectangle (or square) represents a distinct order type in the ordinal.

The Finite Image

The picture above has the disadvantage of being infinitely large, so it still doesn't give us a full view of the completed set. Is it possible to create a completed view of the set? We shall try.

First, notice that every two steps of the above process yields a square 4 times larger than the original (twice the original's dimensions). This suggests that we can scale the picture after every two steps, so that it will still fit in a unit square area. We can repeat this process any even number of steps, to get a picture of the ordinal corresponding to an ever larger initial segment of . The picture remains within the unit square area.

Finally, we can take the limit as the number of steps go to infinity: we get a fractal-like image, which happens to be the same image we get if we take the limit of repeatedly substituting each rectangle in the 2x2 area for w*w with a suitably scaled version of itself. This final image gives us a view of the completed set . Right...?

Perhaps. Although this interpretation is very tempting, since it finally gives our mind a pictorial way to grasp the full structure of , which is uncountable, it suffers from a few fatal flaws:

1. Unlike the first, infinite, picture in the previous section, we can no longer identify individual elements in it. The reason for this is that not only have individual rectangles collapsed into dimensionless points, but entire square regions of the finite steps have collapsed into points. In the infinite picture, we still could, if given any subset ordinal, point out what its successor would be. But in this shrunk-down fractal picture, entire stretches of infinite numbers of successors have collapsed into points, and we can no longer distinguish between individual elements of the set, and we can no longer point out what the successor element of an initial segment should be.
2. This fractal has the property that the lower-left quadrant is identical to a smaller copy of itself. If we continue using the interpretation that a subset of the image should correspond with an ordinal which is an initial segment of the whole, we run into trouble: since the lower-left quadrant is identical in structure to the whole, the 'ordinal' represented by the whole must be a member of itself---a contradiction of the Axiom of Regularity (and indeed, the definition of an ordinal in the first place).
   

So, sadly to say, although this fractal picture is very compelling, in that it offers us a finite view of the completed set , it cannot possibly correspond with its actual structure. At the most, we can only say that it is an approximate visual aid, to help us grasp its full structure.

As a followup, I discuss the implications of the above attempts to visualize an uncountable ordinal.

Firstly, the two images tell us that the structure of is so complex that it can only be adequately represented by a fractal structure, or an infinite structure with fractal-like properties (the first, non-finite, picture).

Secondly, we must ask, why does the fractal picture have fatal flaws? The limit process we have used seems like it ought to accurately represent the completed , yet the resulting picture seems to require that it contain itself, contradicting the Axiom of Regularity. What went wrong? Why is it that while the first picture seems free of problems, just applying a scaling operation at each step causes problems to arise?

Thirdly, consider the implications this may have for the Continuum Hypothesis: In every finite step in the generation of the fractal picture, we have rectangles that correspond to individual elements in the set. Our construction ensures that these rectangles completely tile the unit square. As we take the limit of this process, each of these rectangles are reduced to points. Since the unit square is fully covered in the resulting fractal, that means the cardinality of the set of collapsed squares must be (at least) the cardinality of the continuum. This seems to imply the Continuum Hypothesis... if it were not for the fact that the resulting fractal can't possibly represent accurately, because it would also imply that it contains itself.

We can take this even further. Suppose we take the fractal picture of , and apply the same process to it as before: making a copy, halving its width, and concatenating it to the right of the original---this gives us . Similarly, applying this process along the vertical axis gives us . Taking the limit of this process, we arrive at .

The problem is that the picture thus arrived at is identical to the fractal picture for . In other words, we have no way to tell if the fractal actually represents , or any of the subsequent Alephs! Hence, the argument for the Continuum Hypothesis may be equally applied to any of these Alephs. Perhaps this is a little glimpse into why it is undecidable whether the cardinality of the continuum corresponds to any of the Alephs after the first.

Or perhaps this is an argument for equating ? Very well, but then we note that, at least according to the fractal, the structure contains itself, and therefore can't possibly correspond with a well-founded set. So perhaps, by taking the limit of the finitary steps, we have inadvertently stepped out of the universe of sets into the twilight zone between sets and classes? For, if we consider the fractal picture, every point in the picture corresponds to the limit of a structure which is isomorphic to the whole, so we may crudely understand that the unit square, although having only c points in it, has each of these c points correspond with an unspecified large number of rectangles from the finitary steps. But if this fractal is indicative of the structure of a well-ordered set, then the set must contain itself.

The disturbing thing about all of this, is that the limit that yielded this fractal involves only a countable number of steps. Even more disturbing is the question, if this fractal does not represent the actual structure of the uncountable ordinal, then what image would? If taking the limit of the finitary steps, which do correspond with the initial segments of the ordinal, gives us a picture which has gone "too far", then what exactly sits between the finitary steps and the limiting fractal which may correspond with the actual ordinal? How is it possible for there to be something between a limit and all possible finite steps leading up to the limit? Surely the limit must sit at most a countable number of steps beyond the finite steps; does this mean there are multiple distinct countable infinities (how can that be?!), with a lesser countable corresponding to an "intermediate" image that represents the actual ordinal?

Perhaps this is a reason to reconsider the Axiom of Regularity, and allow sets to contain themselves. The so-called New Foundations show that it is possible to have a consistent system that contains Quine Atoms, but not Russell's Paradox. In any case, these considerations give us a little glimpse into the pathological properties that infinite sets may have.

Actually, I discovered that the picture isn't quite this simple after all. I was under the impression that the equivalent of the Ackermann function on corresponded with , but in fact, I was wrong. The countable ordinals go far beyond that. In fact, they go far beyond the Church-Kleene ordinal, which is non-recursive. They even go beyond all describable ordinals, which means that there are actually countable ordinals which are so large as to be indescribable in finite space/time. So I'm just dreaming if I think that was anywhere remotely near being visualizable...