Numbers as Sets
Every child naturally learns how to count, but few non-mathematicians know that the "whole numbers" have been rigorously defined in terms of sets: each ordinal number is defined as the ordered set of all smaller ordinals. This Demonstration visually unpacks this recursive definition for the first few ordinals, letting you select an ordinal number and specifying how many levels deep the translation should be applied.
The definition of the ordinal numbers (whose finite members are the whole numbers) was originally due to Georg Cantor, whose groundbreaking treatment extended to transfinite numbers. The standard definition of the ordinals, due to John von Neumann, defines each ordinal as the well-ordered set of all smaller ordinals. Thus , since no previous ordinals have been defined, and then , , and in general, .
A succinct recursive version of this definition is , . This gives the same results as above; for example, .
It is interesting to note that this definition is not limited to finite numbers. Set theorists define the first transfinite number as , and continue far beyond this "smallest infinity".