In mathematics, ordinals extend the concept of counting beyond finite numbers and incorporate infinite sequences. From a graph theory perspective, you can think of ordinals as a hierarchy of "layers" or "levels" that represent different sizes and types of infinity.

In this blog, we define ordinals as a (maybe infinite) competitive graph $$$\Gamma$$$ (A directed graph is a competitive graph, if exactly one of $$$(u, v)$$$ and $$$(v, u)$$$ is in the edge set) without any loops.

Finite ordinals are natural numbers. For example, $$$0$$$ is the empty graph, and $$$1$$$ is the only graph with one vertex. Further example is as below: