User blog comment:Wythagoras/My Turing machines/@comment-6768393-20131012123348/@comment-10429372-20131013123215

You can see the rules for dollar function here.

About strength of weakly compact ordinal notation, I think it works like this:

The first level diagonalisators are $$\Omega$$, $$I$$, $$M$$,$$\Xi(3,0)$$, $$\Xi(4,0)$$, ...

Then K is a second level diagonalisator.

And I think that dollar function can diagonalize over nth level diagonalisators.