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I tried to define Hardy Hierarchy up to $H_{\psi(\chi(\Xi(K,0)))}(n)$, and I got the ruleset with 31 rules. It ignores the facts like $1+\omega = \omega$ for not making technical complexities, but it doesn't matter because ordinals can be treated just as notational symbols. Also, exponentiation isn't allowed: we must define it through $\psi$. So, the ruleset for which I was working for a few weeks:

The Ruleset

Rule 1:

$H_0(n) = n$

Rule 2:

$H_{\alpha+1}(n) = H_\alpha(n+1)$

Rule 3:

$H_\alpha(n) = H_{\alpha[n]}(n)$

Rule 4:

$(\alpha+\beta)[n] = \alpha+(\beta[n])$

Rule 5:

$\Omega_0 = 1$

Rule 6:

$\psi_{\Omega_{\alpha+1}}(0) = \Omega_\alpha$

Rule 7:

$\psi_{\Omega_{\alpha+1}}(\beta+1)[0] = 0$

Rule 8:

$\psi_{\Omega_{\alpha+1}}(\beta+1)[n+1] = \psi_{\Omega_{\alpha+1}}(\beta)+\psi_{\Omega_{\alpha+1}}(\beta+1)[n]$

Rule 9:

$\psi_\alpha(\beta)[n] = \psi_\alpha(\beta[n])$

Rule 10:

$\psi_\alpha(\beta+1)[0] = \psi_\alpha(\beta)+1$

Rule 11.

$\psi_\alpha(\beta+\lambda)[0] = \psi_\alpha(\beta)$

Rule 12.

$\psi_\alpha(\beta+\alpha)[n+1] = \psi_\alpha(\beta+\psi_\alpha(\beta+\alpha)[n])$

Rule 13. $(\lambda \geq \alpha)$

$\psi_\alpha(\beta+\lambda) = \psi_\alpha(\psi_\lambda(\beta+\lambda))$

Rule 14.

$\psi_{\chi_\alpha(\beta,\lambda)}(0)[0] = 1$

Rule 15.

$\psi_{\chi_\alpha(\beta,\lambda+1)}(0)[0] = \chi_\alpha(\beta,\lambda)+1$

Rule 16.

$\psi_{\chi_\alpha(\beta+1,\lambda)}(\delta)[n+1] = \chi_\alpha(\beta,\psi_{\chi_\alpha(\beta+1,\lambda)}(\delta)[n])$

Rule 17.

$\chi_\alpha(\beta,\lambda)[n] = \chi_\alpha(\beta,\lambda[n])$

Rule 18.

$\chi_\alpha(\beta,0)[n] = \chi_\alpha(\beta[n],0)$

Rule 19.

$\chi_\alpha(\beta,\lambda+1)[n] = \chi_\alpha(\beta[n],\chi_\alpha(\beta,\lambda)+1)$

Rule 20.

$\chi_\alpha(\beta+\alpha,0)[0] = \chi_\alpha(\beta,0)$

Rule 21.

$\chi_\alpha(\beta+\alpha,\lambda+1)[0] = \chi_\alpha(\beta+\alpha,\lambda)+1$

Rule 22.

$\chi_\alpha(\beta+\alpha,\lambda)[n+1] = \chi_\alpha(\beta+\chi_\alpha(\beta+\alpha,\lambda)[n],0)$

Rule 23. $(\lambda \geq \alpha)$

$\chi_\alpha(\beta+\lambda,\delta) = \chi_\alpha(\chi_\lambda(\beta+\lambda,\delta))$

Rule 24.

$M_0 = 0$

Rule 25.

$\chi_{M_{\alpha+1}}(0,\beta) = \Omega_{M_\alpha+\beta}$

Rule 26.

$\chi_{\Xi(\alpha+1,\beta+1)}(0,\lambda) = \Xi(\alpha,\Xi(\alpha+1,\beta)+\lambda)$

Rule 27.

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

Rule 28.

$\Xi(0,\alpha) = M_\alpha$

Rule 29.

$\Xi(\alpha,\beta)[n] = \Xi(\alpha,\beta[n])$

Rule 30.

$\Xi(\alpha,0)[n] = \Xi(\alpha[n],0)$

Rule 31.

$\Xi(\alpha,\beta+1)[n] = \Xi(\alpha[n],\Xi(\alpha,\beta)+1)$

These rules were written so that first rules handle smaller structures and extensions, and last otherwise.

Feel free to improve it (particularly in rule-conditions, I think they must be more formal.)

EDIT:

This ruleset is incorrect. The correct version is under construction.