User blog comment:Edwin Shade/Two Exceedingly Large Numbers/@comment-1605058-20171209210735

The problem is that one ordinal can be described in terms of \(\omega\) in many ways. For example, what ordinal is to \(\omega^2\) as \(\omega^2\) is to \(\omega\)?

On one hand, \(\omega^2\) is, well, \(\omega^2\), so one answer is \((\omega^2)^2=\omega^4\).

On the other hand, \(\omega^2\) is the limit of \(\omega1,\omega2,\omega3,\dots\), so a different answer is the limit of \(\omega^21,\omega^22,\omega^23,\dots\), i.e. \(\omega^3\).

On the third hand, we can describe \(\omega^2\) without refering to \(\omega\) at all, as the third ordinal (after \(\1,\omega\)) closed under addition.

On top of that there are of course the usual problem of "what fundamental sequences are you using?" for FGH, and for the latter number it is a nontrivial question as to whether a fixed point exists.