User blog comment:Tetramur/BEAF above SVO - comparison/@comment-5150073-20200219121934/@comment-35470197-20200220222036

In order to make "replace ω into Ω" valild, we need to fix a notation. The main idea here is perhaps to use a notation associated to transfinite hyperoperators. On the other hand, as Nama Naima pointed out, there is no agreed-upon definition of transfinite hyperoperator in googology. (I think that ω^^ω+1 = ε_0 is the most natural choice for mathematicians, but majority of googologists dislike it.)

I note that this approach does not cause a serious issue when we only consider ordinals below ω^^ω = ε_0, i.e. tetrational array level.

One candidate is to use a fixed definition of transfinite hyperoperator, as Nama Naima showed. For example, if we use a notation using Extended Buchholz's OCF expressed as transfinite composition, we can consider the following conversion: \begin{eqnarray*} & & \psi_v^{1+\omega^{\alpha}} := \psi_v \cirs \psi_{v+\alpha} & & BHO = \psi_0(\psi_2(0)) = \psi_0^{\omega \times \omega}(0) \\ & \mapsto & \psi_0^{\omega \times \Omega}(0) = psi_0(\psi_{\Omega}(0)) \end{eqnarray*} It is the ordinal corresponding to the limit of notations given as nested hydras such as HydHyd (in Japanese) or https://googology.wikia.org/wiki/User_blog:Wythagoras/Buchholz_hydra Wythadoras's hydra (the "analysis" by the creator looks wrong). I note that it is not a natural conversion, because we have α×Ω = Ω for any non-zero countable ordinal α.