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 , , , , , and so on as the natural numbers. Here 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) is the empty set, but also .
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, , 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.