User blog comment:Edwin Shade/The OFF Function/@comment-1605058-20171201182048

For this to be well-defined, we need to fix some method of assigning to every ordinal a single expression for it involving \(\omega\)s. Without it, we would have to have for example \(10=[10,\omega]=[10,1+\omega]=1+10\). Less trivially, we could take on one hand \([10,\varepsilon_1]\) like you define it, while on the other hand we could use \([10,\varepsilon_1]=[10,(\omega\uparrow\uparrow\omega)\uparrow\uparrow\omega]=(10\uparrow\uparrow 10)\uparrow\uparrow 10\).