User blog comment:Wythagoras/Strength of Hierarchical Nested Subscript Array Notation/@comment-7484840-20130730134356/@comment-10429372-20130731092010

I think I understand what you mean. Type 2 brackets use the universal bracket for nesting, so it will nest the whole array in the type n times, like Omega.

I've developed a powerful extension to HAN. You can use typed <> brackets to get psi_I(1)(0). After that, you can make brackets that diagonalize over angle brackets like angle brackets diagonalize over normal brackets. If you extend that far enough ( new brackets that diagonalize over that brackets, call level 4 brackets then get something like 1(1)2 brackets or beyond ), you can get something in the area of the Mahlo ordinals.

If you want, I can work this further out.