User blog comment:Nayuta Ito/Ordinal Ruler/@comment-11227630-20180309023417

Let $$\alpha[]=\alpha$$ if $$\alpha$$ is successor ordinal, or $$\alpha[0][]$$ if $$\alpha$$ is limit ordinal.

If the FS system fits $$\alpha+1<\beta[]$$, then we can't make ordinal ruler between $$\alpha$$ and $$\beta$$.

So, to make an ordinal ruler, there are some requirement for the FS system. Especially, in Monotonic FS systems, we can make ordinal ruler from $$\alpha[]$$ to $$\alpha$$.