What's a (natural) number? What does it look like?

We represent numbers with symbols like 1, 2, 3, I, II, III. We are aware of objects that have a numeric property: A square has 4 sides.

But, what about the number itself? What is it? What does it look like?

In 1923, John von Neumann proposed a set-theoretic construction of natural numbers: We can start with the empty set, and then define the next number as the set containing all previous numbers:

0 = {} 1 = { {} } 2 = {0, 1} = { {}, {{}} }, 3 = {0, 1, 2} = { {}, {{}}, { {}, {{}} } } And so on.

"Number 7, Red", is inspired by von Neumann's construction. Circles are sets.