User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-78.22.170.27-20130619192628/@comment-5529393-20130624224425

Loader's ordinal was not expressly defined by Loader, we just use it to mean the smallest ordinal at which the FGH overtakes Loader's function D(n), which is a function that diagonalizes over the Calculus of Constructions. More information here: http://djm.cc/bignum-results.txt. I have said that Loader's ordinal surpasses all current recursive ordinal notations, and that it would be very difficult to beat.

If you are willing to evaluate ordinal notations, I would really be interested for someone knowlegeable to evaluate Taranovsky's notations: http://web.mit.edu/dmytro/www/other/OrdinalNotation.htm. Taranovsky does not seem to be a member of the proof-theoretic community, but he seems to know his stuff, so I wouldn't dismiss his claims out of hand. He claims to reach the proof-theoretic ordinal of second order arithmetic, which is an ambitious claim because I believe that surpasses any notation from Rathjen or Arai, while appearing simpler than their notations.

I would definitely be interested in anything you know about ordinal diagrams and Gordeev style notations. I would especially be interested in an explanation of Arai's stronger notations that go beyond \(\theta(\Omega_{\omega}, 0)\). Any help would be greatly appreciated.