User blog comment:Nayuta Ito/Ordinal Ruler/@comment-11227630-20180308062944

Consider such ordinal ruler using Wainer hierarchy and ratio 1:1. As a result, $$\omega^2$$ is at position both 1/2 and 5/8 - it goes wrong.
 * 1) Let $$\omega$$ at position 0 and $$\omega^\omega$$ at position 1.
 * 2) By case 2 ($$\alpha=\omega$$ and $$\beta=\omega^\omega$$), $$\omega^2$$ is at position 1/2.
 * 3) Again by case 2 ($$\alpha=\omega$$ and $$\beta=\omega^2$$), $$\omega2$$ is at position 1/4.
 * 4) Again by case 2 ($$\alpha=\omega2$$ and $$\beta=\omega^\omega$$), $$\omega^2$$ is at position 5/8.