User blog comment:Deedlit11/Ordinal Notations VII: Indescribables/@comment-25337554-20171217095001/@comment-30118230-20180111213144

There is a way to diagonalize over them,though. Several ways,actually:

1) Using the stage function. We can define I to be at stage 1,M to be at stage 2,K to be at stage 3,and so on. However,In order to define stage n or stage alpha,we need to find some common algorithm between these cardinal-collapsing functions.

2) Using Indescribables,which,as the title suggests,is exactly what Deedlit is going to do. The only question is how. The simplest way to do this is to define a "stage-function" S(n) so that S(n) or maybe S(n+1) corresponds to the first weakly $$\Pi^1_n$$ indescribable cardinal. Then S(alpha) can be extended to transfinite alpha,or even given a corresponding diagonalizer. And then we can define corresponding collapsing functions like \(\Psi_{S(n)}\) for example.

3) Using Taranovsky's ordinal notation.

Taranovsky has claimed that within $$C(C(\Omega_2 2+\text{____},0),0)$$$$C(\Omega_2 + C(\Omega_2,C(\Omega_2 2,0)),0)$$ corresponds to the Inaccessible cardinal within psi collapsing functions. He has also claimed that $$C(\Omega_2 + C(\Omega_2,C(\Omega_2 2,0))^2,0)$$ and $$C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^3,0)$$ correspond to the Mahlo and Kompact cardinals respectively. (again within their OCFs obviously)

So we can define somethin that works in a similar way as $$C(\Omega_2 + C(\Omega_2,C(\Omega_2 2,0))^\omega,0)$$