User blog:Nayuta Ito/Ordinal Ruler

definition
An ordinal ruler has these features:
 * It has line (segments).
 * It has graduations.
 * It is graduated with ordinals.
 * The ordinals are in order. Smaller ones in the left and larger ones in the right.

(note that an ordinary ruler is an ordinal ruler)

how to make an ordinal ruler
There are many ways to make one, so I will write one of the simple ways.


 * First, write two ordinals at the ends. By the definition, the left one must be smaller.
 * For every line segment between $$\alpha$$ and $$\beta$$,
 * If $$\beta=\alpha+1$$, write nothing between.
 * If $$\alpha$$ is included in the fundamental sequence of $$\beta$$, write the next term of the fundamental sequence at halfway between them.
 * Else, write the smallest term of the fundamental sequence of $$\beta$$ which is bigger than $$\alpha$$.

(note that the graduations can be infinitely many)

examples
The image is two of the ordinals made by above recipe.



use
You may be able to use it for googology graph.