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)


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


From one to omega (above) and from one to w^w.


You may be able to use it for googology graph.

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