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, \omega^{K+1}, \omega^{\omega^{K+1}}, \psi_{\Omega_{K+1}}(0), \Omega_{K+1}, M_{K+1}, \Xi(\alpha,K+1) for any \alpha.

So, we can make Mahlo-hierarchy above K using \Xi_{K_2}(0,\beta) = M_{K+\beta} and by the analogy for K_3, K_\omega, K_\Omega, K_K, K_{K_K}, K_\alpha. But my ideas doesn't stop here.

Note how we expressed \Omega_\alpha and M_\alpha in terms of larger functions?

\Omega_\alpha = \chi(0,\alpha)

M_\alpha = \chi_{\Xi(1,0)}(0,\alpha)


K_\alpha = \chi_{\Xi_{\sigma(1,0)}(0,0)}(0,\alpha)

Using \chi_\alpha(\beta,\lambda), where \alpha = \Xi_{\sigma(1,0)}(0,0), we can express the inaccessible hierarchy above K. Using \Xi_\alpha(\beta,\lambda), we express the Mahlo hierarchy above K. Finally, \sigma(\alpha,\beta) makes K-typed hierarchy above K. The diagonalizer of \sigma itself is supposed to be the next ordinal in the sequence \Omega, M, K, \cdots. Notating this as D, we can have D_2, D_\omega, D_\Omega, D_D, D_{D_D}, D_\alpha, and then we must have some new function to go further. Let's reconsider our strategy:

We can notate \chi_\alpha(\beta,\lambda) as \chi_\alpha(0,\beta,\lambda).

\Xi_\alpha(\beta,\lambda) = \chi_\alpha(1,\beta,\lambda)

\sigma_\alpha(\beta,\lambda) = \chi_\alpha(2,\beta,\lambda)

It other words, we turn the sequence of functions past binary \chi to the single ternary \chi function. Then let n-th term in the sequence \Omega, M, K, \cdots with index \alpha is represented by T(n,\alpha):

T(n,\alpha) = \chi_{\chi_{\cdots_{\chi_{\chi(n-1,1,0)}(n-2,0,0)}\cdots}(1,0,0)}(0,0,\alpha). Keep in mind, \chi_\alpha(0,\beta,\lambda) represents inaccessible hierarchy, \chi_\alpha(1,\beta,\lambda) represents Mahlo hierarchy, \chi_\alpha(2,\beta,\lambda) represents compact hierarchy, and so on.

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

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