Von Neumann Doğal Sayıları

The von Neumann natural numbers: a poster, size A1.

Posterin İngilizcesi, aşağıdadır // The poster's text in English is as follows:

In this picture can be seen the first six natural numbers, according to the definition of von Neumann. Here 0 is the empty set. Every circle in the picture is a non-empty set, and its elements are the contents of the circle. Sets with the same elements are the same as one another. Hence even though there are 32 zeros and 31 circles in the picture, in fact there are only six different sets. These sets are the first six natural numbers, namely 0, 1, 2, 3, 4, and 5. Moreover
5 = {0, 1, 2, 3, 4},
4 = {0, 1, 2, 3},
3 = {0, 1, 2},
2 = {0, 1},
1 = {0}.
The elements of the elements of a set need not be elements of that set. In fact, according to the Foundation Axiom, every nonempty set a has an element b such that a and b are disjoint from one another, that is a∩b=0.
If the elements of every element of a set are elements of the set, this set is transitive. The von Neumann natural numbers are transitive. Moreover, all elements of a von Neumann natural number are transitive. In fact, by definition every transitive set whose elements are transitive is an ordinal. If α is an ordinal, then the set α∪{α} is also an ordinal, and this ordinal is the successor of the ordinal α. If an ordinal is either empty or a successor, and moreover its every element is either empty or a successor, then this ordinal is a natural number.