See the next article, “Self-similarity,” for an animation of the image here.
I have long been fascinated by von Neumann’s definition of the natural numbers (and more generally the ordinals). In developing axioms for set theory, Zermelo used the sets 0, {0}, {{0}}, {{{0}}}, {{{{0}}}}, and so on as the natural numbers. Here 0 is the empty set. Zermelo’s method works, but is not so elegant as von Neumann’s later proposal to consider each natural number as the set of all natural numbers that are less than it is, so that (again) 0 is the empty set, but also n + 1 = {0, 1, …, n}.
The von Neumann ordinals are like fractals. Fractals are self-similar; that is, in a description quoted by Wikipedia, they are “the same from near as from far”. All of the von Neumann natural numbers look the same from afar—from far enough away that their smallest element, 0, cannot be seen. I worked out a visual representation of this in 2010. Today I put this representation on a poster for display in my department in Istanbul:
Searching Google images under “von Neumann natural numbers” or “von Neumann ordinals” yields nothing like this. The closest I find is the following image of ω—that is, the set of all natural numbers—, from a blog called Nature Loves Math:
But this picture does not show that every natural number is the set of all of its predecessors.
My own picture does not show all of ω.
My poster in size A1, with English translation of the Turkish text, is among my departmental web pages.
My image has been used (with my permission) to illustrate an article in a French popular science magazine: Roger Mansuy, “L’ensemble vide, créateur d’infinis,” La Récherche, juillet-août 2020.
Polytropy 
5 Comments
Here is a way to represent finite ordinals with a fractal: Imagine that the relation $x \epsilon y$ is interpreted as “y is contained in x”. Then the following nested circles correspond to finite ordinals with any natural number n being represented by the circles of radius 1/2^n: http://www.wolframalpha.com/input/?i=plot%7Bx%5E2%2By%5E2%3D1%2C+%28x%2B1%2F2%29%5E2%2By%5E2%3D1%2F4%2C%28x%2B3%2F4%29%5E2%2By%5E2%3D1%2F16%2C+%28x-1%2F4%29%5E2%2By%5E2%3D1%2F16%2C+%28x%2B7%2F8%29%5E2%2By%5E2%3D1%2F64%2C+%28x%2B3%2F8%29%5E2%2By%5E2%3D1%2F64%2C+%28x-1%2F8%29%5E2%2By%5E2%3D1%2F64%2C+%28x-5%2F8%29%5E2%2By%5E2%3D1%2F64%7D (Wolfram Alpha did not let me draw all the circles to represent 4).
In other words, Burak, I think your picture is as follows. Being a simple closed curve, a circle divides the plane into two regions, one having finite area, one not. If the circle represents a set, then we normally think of the elements of the set as lying within the region that has finite area. But we could just as well think of the elements of the set as lying in the other region. In one sense of the word, each of the two regions determined by the circle is *finite*, because it has a boundary that keeps some points away from it.
Under this conception, the circle representing zero or the empty set will have nothing *outside* it, because the outside holds the elements of the set. The circle representing one will lie inside the circle representing zero. But then to represent two, we need two circles: one to lie “inside” Circle One, thus “containing” it; and one lying inside Circle Zero (thus containing it), but *not* inside Circle One. These two new circles *together* represent Two; and so on.
Your picture will give us representations of all of the natural numbers, within a finite region (namely the original circle, representing zero). This is a good feature. Unfortunately I do not see how the set omega of all of these natural numbers can itself be represented; do you?
By the way, I think the figures we are discussing are not really fractals. I consider the list of criteria in the Wikipedia article http://en.wikipedia.org/wiki/Fractal :
1) self-similarity,
2) “Fine or detailed structure at arbitrarily small scales”,
3) “Irregularity locally and globally that is not easily described in traditional Euclidean geometric language”,
4) “Simple and “perhaps recursive” definitions”.
I don’t think the third criterion is really met here. In any case, the figures seem to have Hausdorff dimension one.
I did the same thing but had no real idea what I was doing and only recently found out they were referred to as the Von Neumann ordinals, after which I did a google images search and found your page. Here’s my representation; I made this in 2014 after first starting to think about it in 2012 or 2013
Yery nice; thanks!
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