User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-25337554-20171208234807/@comment-32876686-20171209001357

Thus far I have not created a formalism of determining whether a given number corresponds to a valid ordinal, but it is to be assumed that it is fairly easy to figure out.

There may be an infinite number of ways to represent the same ordinal, but that is okay. My purpose in making these labeling systems was to demonstrate that there exists a way to label all ordinals less than a given ordinal $$\gamma$$ with a natural number, or simply put, that there exists a surjection, (which equates to a bijection in infinite sets), from the naturals to the countable ordinals.

After creating an enumeration scheme for a large enough countable ordinals I hope to stumble upon some general 'pattern' which will allow me to diagonal over the countable ordinals themselves, and assign each one a name, (if that can be done).