User blog comment:Wythagoras/My Turing machines/@comment-6768393-20131012123348/@comment-5529393-20131013121727

I have to agree with LittlePeng9; I'm skeptical of any notation that purports to reach the ordinal of second order arithmetic. This includes Taranovsky's notation, although his notation is clearly quite strong. The exception is somebody like Rathjen proving it in a published paper.

Note that surpassing the notation for a weakly compact cardinal is not evidence for reaching second order arithmetic, as they are VASTLY far apart in complexity. I'm not sure that your notation surpasses a weakly compact cardinal though; it really is quite strong, and I'm not sure that I did a very good job explaining how strong it was.

Concerning reaching a $$Pi^1_n$$-indescribable cardinal, it would help if you defined the exact notation that uses such cardinals. There are "official" notations that use $$Pi^1_n$$-indescribable cardinals but they are incredibly complicated.