Cantor Normal Form
A remarkable ordinal notation system, as useful as the decimal number system, but for arbitrary ordinals and grounded in base ω rather than base ten.
Cantor proved a remarkable fact about ordinals and provided for us an ordinal notation system in which every ordinal admits a unique canonical representation by what we now call its Cantor normal form. The notation system is every bit as powerful and convenient as the familiar decimal number system is for representing our ordinary numbers, except that it works with arbitrary ordinals and uses base ω rather than base ten.
Welcome to this series of essays on the ordinals and ordinal arithmetic—you can find them in the ordinal-arithmetic tag. After building this foundation in the ordinals, we shall eventually return to my essay series on the surreal numbers, making use of our growing familiarity with the ordinals. You are welcome to join and follow along!
Let’s get into it, learning how to do ordinal arithmetic easily in the Cantor normal form. There are some surprising tricks that allow whole parts of the expressions to simplify away to nothing—they just disappear, leaving only the correct simplified expression behind.
