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definitionEdit

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 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.

Ordinalruler

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


use Edit

You may be able to use it for googology graph.


external linkEdit

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