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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.

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