Counting Beyond Infinity

The most basic mathematical process is counting:
0, 1, 2, 3, 4, 5, 6, ...
In the 1870’s, the mathematician Georg Cantor invented (or found) how to count beyond infinity:
0, 1, 2, 3, ..., ω, ω+1, ...
The short film visualizes the transfinite numbers up to a number which is called ε₀.

There are many more transfinite numbers beyond ε₀. In any case, we can only visualize a tiny initial segment of all infinite numbers. It took half a century until John von Neumann gave the first rigorous definition of transfinite numbers inside axiomatic set theory. Still today, the concept of counting beyond infinity remains an intellectual challenge, even for professional mathematicians. It is fair to say that Cantor’s set theory marks the beginning of modern mathematics, where infinite objects coexist with finite ones. And the transfinite numbers form the spine of the universe of infinite sets in much the same way as the non-negative integers do in the realm of arithmetic and finite combinatorics. They are the mathematical key to understand infinity, and their richness and structure is a matter of research, debate, and fascination.

• MathFilm 2008 DVD