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## definitionEdit

An ordinal ruler has these features:

• It has line (segments).
• 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 Edit

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$\beta$ with no graduation in between,
• 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)

### modified versionEdit

As many people make rulers with the procedure above, I found something problematic: 1. However big the ending ordinal is, if the first element of the fundamental sequence is

$\omega$ , The first quarter of the total ruler is left blank. 2. On the other hand, the latter half of the ruler is pretty dense.

To solve these problems. I suggest a new upgraded way:

Take an arbitrary ratio a:b.

• First, write two ordinals at the ends. By the definition, the left one must be smaller.
• For every line segment between$\alpha$ and$\beta$ with no graduation in between,
• 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 the point internally dividing in the ratio a:b.
• Else, write the smallest term of the fundamental sequence of$\beta$ which is bigger than$\alpha$.

If you take something like 2:1 or 3:1 as the ratio, it should solve the problems (but it will be harder to analyze the structure of it).

## examples Edit

The image is two of the ordinals made by above recipe with the ratio of 1:1.

## use Edit

You may be able to use it for googology graph.

There is an interactive ruler up to $\epsilon_{\epsilon_0}$.