At the end of this page, they describe a theory for larger cardinals. This page describes an indescribable cardinal ordinal collasping function.

We can use this notation: a  Π[a,b] indescribable cardinal means a Πab indescribable cardinal and a Σ[a,b] means a Σab indescribable cardinal. We can now invent a new function: E(a,b) means an [a,b] indescribable cardinal and the rank-into-rank cardinal (R) is the diagnolizer for a. E(R^a)= Π(1,0...0,0) with a+1 0's in my multi-argument indescribable array notation. ς(D,0)=E(R,0), but E(R,1)= the diagnolizer of ς(D[n],0). The zero can be inferred if it is only one entry., so S(a)= S(a,0). The rank-into rank cardinal can only go in the first entry.

I believe PNAN (I put a link to that page) has a level psi(psiE(eR+1)(0))

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