User blog comment:Nayuta Ito/Ordinal Ruler/@comment-11227630-20180308062944/@comment-11227630-20180308094548

Even worse. If "with no ordinals in between" means there is no $$\gamma$$ such that $$\alpha<\gamma<\beta$$, then the only case will be $$\beta=\alpha+1$$, and the other 2 cases will fail.