Matchstick diagram

A matchstick diagram is a particular representation for ordinals. The idea is to lay down vertical lines left-to-right such that the diagram has an order type of the ordinal, where being left of a stick is equivalent to being lesser. Equivalently, enumerating a diagram will result in all ordinals less than the corresponding ordinal having their own stick. This second definition is simply another way of saying the first definition. Matchsticks may have unequal heights in order to make the diagram easier to read. Although matchstick diagrams can have infinitely many sticks, they are usually restricted to a finite region to make reading easier, as infinite area offers no special advantage.

Matchstick diagrams have the property that placing two diagrams side-by-side results in the sum of their corresponding ordinals. For example, placing the diagram for 1 before \(\omega\) results in the diagram \(1 + \omega = \omega\), and placing \(\omega\) before 1 results in \(\omega + 1 = \omega + 1 \). Another property of matchstick diagrams is that they can only represent countable ordinals.