User blog:Googleaarex/Fundamental sequences for my ordinal notation

Today, I will define fundamental sequences for my ordinal notation.

Up to \(\omega\)
These ordinals in this category are the non-negative integers!

Up to \(\varepsilon_0\)
Let define \(\omega\) (the basic ordinal) as:
 * \(\omega[n] = n\)

Then we can extend to powers of \(\omega\):
 * \((\omega^{\alpha}\*\beta+\gamma)[n] = \omega^{\alpha}\*\beta+(\gamma)[n]\)
 * \((\omega^{\alpha+1}\*(\beta+1))[n] = \omega^{\alpha+1}\*\beta+\omega^{\alpha}\*n\) (if \(\alpha + 1\) is a successor ordinal)
 * \((\omega^{\alpha}\*(\beta+1))[n] = \omega^{\alpha}\*\beta+\omega^{\alpha[n]}\) (if \(\alpha\) is a limit ordinal)

But what's successor ordinal and limit ordinal? A successor ordinal is the ordinal that other ordinal added by 1; and the limit ordinal is the ordinal that NOT a successor ordinal.

Up to \(\zeta_0\)

 * \(\varepsilon_{0}[0] = 1\)
 * \(\varepsilon_{\alpha+1}[0] = \varepsilon_{\alpha}+1\) (if \(\alpha + 1\) is a successor ordinal)
 * \(\varepsilon_{\alpha+1}[n+1] = \omega^{\varepsilon_{\alpha + 1}[n]}\) (if \(\alpha + 1\) is a successor ordinal)
 * \(\varepsilon_{\alpha}[n] = \varepsilon_{\alpha[n]}\) (if \(\alpha\) is a limit ordinal)

Really easy.

Up to \(\varphi(1,0,0)\)
\(\varphi(0,\alpha)\) is equal to \(\omega^{\alpha}\); and \(\varphi(1,\alpha)\) equal to \(\varepsilon_{\alpha}\)

Definition coming soon.

Up to \(\psi(\Omega^{\Omega^{\omega}})\)
Coming soon.

Up to \(\psi(\varepsilon_{\Omega+1})\)
Coming soon.

Up to \(\psi(\Omega_2)\)
Coming soon.

Up to \(\psi(\psi_{\chi(0)}(0))\)
Coming soon.

Up to \(\psi(\psi_{\chi(M)}(0))\)
Coming soon.

Up to \(\psi(\psi_{\Xi(0)}(0))\)
Coming soon.

Up to ???
Coming soon.