Pictured above, we have the ordinal spiral, depicted using “matchstick notation.” The ordinal spiral is a beautiful and intuitive way to represent the structure of the ordinal numbers.
Quoting Wikipedia, “In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers are thus the ‘labels’ needed to arrange collections of objects in order.”
Note that “each turn of the spiral represents one power of omega.” Omega is, in turn, the first infinite ordinal. I’d rather not get into the technical details of how omega is defined. Instead, I’d like to encourage you to take a moment to appreciate it’s arithmetic and algebraic structure.
As a kid, I recall being endlessly obsessed by the idea of “being able to do math with infinity.” And there it is... stunning, isn’t it? Aren’t you tempted to factor out that omega in the middle of the matchstick, two turns in (from the middle-right)? Do it! Go ahead, you’re free now! Just be sure to thank my homie, Georg Cantor.
Mathematics is beautiful. <3
