It is interesting to see how to go past psi(chi(Xi(K))). We can continue further with defining different functions on K: K*2, K*3, , , , , , for any .

So, we can make Mahlo-hierarchy above K using and by the analogy for . But my ideas doesn't stop here.

Note how we expressed and in terms of larger functions?

Then:

Using , where , we can express the inaccessible hierarchy above K. Using , we express the Mahlo hierarchy above K. Finally, makes K-typed hierarchy above K. The diagonalizer of itself is supposed to be the next ordinal in the sequence . Notating this as D, we can have , and then we must have some new function to go further. Let's reconsider our strategy:

We can notate as .

It other words, we turn the sequence of functions past binary to the single ternary function. Then let n-th term in the sequence with index is represented by :

. Keep in mind, represents inaccessible hierarchy, represents Mahlo hierarchy, represents compact hierarchy, and so on.

I know it can be a bit unclear, so wait for formal definition for all this like that.