User blog:Ikosarakt1/Ideas of going past psi(K)

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?

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

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

Then:

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

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, \Xi_\alpha(1,\beta,\lambda) represents Mahlo hierarchy, \Xi_\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.