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

Up to \(\omega\)

These ordinals in this category are the non-negative integers!

If \(\alpha\) is the integer, then \(\alpha[n] = \alpha\).

Up to \(\varepsilon_0\)

Let define \(\omega\) (the basic ordinal) as:

  • \(\omega[n] = n\)

And let define transfinite ordinal as the ordinal which is not equal into any integer.

Then we can define the additive properties of ordinals:

  • \(\alpha+\beta\) = \(\beta\)
    • If \(\beta > \alpha\) and both \(\alpha\) and \(\beta\) are any transfinite ordinals.
    • Or if \(\alpha\) is the integer and \(\beta\) is the transfinite ordinal.
  • \((\alpha+\beta)[n]\) = \(\alpha+(\beta[n])\) (If any 2 conditions on the first additive property didn't passed.)

Then we can extend to powers of \(\omega\):

  • \((\omega^{\alpha}*(\beta+1))[n] = \omega^{\alpha}*\beta+(\omega^{\alpha}[n])\) (\(\beta\) must be greater than 1)
  • \(\omega^{\alpha+1}[n] = \omega^{\alpha}*n\) (if \(\alpha + 1\) is a successor ordinal)
  • \(\omega^{\alpha}[n] = \omega^{\alpha[n]}\) (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 \(\omega^{1+\alpha}\); and \(\varphi(1,\alpha)\) equal to \(\varepsilon_{\alpha}\)

If \(\sigma\) is any either successor ordinal, limit ordinal, or integer; then \(\sigma+1\) is the successor ordinal.

  • \(\varphi(0,\beta)[0] = \omega^{1+\beta}\)
  • \(\varphi(\alpha+1,0)[0] = 0\)
  • \(\varphi(\alpha+1,\beta+1)[0] = \varphi(\alpha+1,\beta)+1\)
  • \(\varphi(\alpha+1,\beta+1)[n+1] = \varphi(\alpha+1,\varphi(\alpha+1,\beta+1)[n])\)
  • \(\varphi(\alpha,0)[n] = \varphi(\alpha[n],0)\)
  • \(\varphi(\alpha,\beta+1)[n] = \varphi(\alpha[n],\varphi(\alpha,\beta)+1)\)
  • \(\varphi(\alpha,\beta)[n] = \varphi(\alpha,\beta[n])\)

Up to \(\psi(\Omega^{\Omega^{\omega}})\)

let define \(A\) as all \(\alpha_{\beta}\) variables (where \(\beta\) can be integers and must be less than \(\gamma\)\); and define \(B\) as all \(\alpha_{\beta}\) variables (where \(\beta\) can be integers and must be greater than \(\gamma\)\).

Let \(\alpha_{\delta}\) will be last \(\alpha_{\beta}\) variable in any \(\delta+1\) arguments on \(\varphi()\). All first arguments of \(\alpha_{\beta}\) (only contains with 0) can be removed.

  • Definition coming soon.

Advanced

Let define \(\varphi(\alpha+1,n)\) is \(n+1\)-th term of \(\varphi(\alpha,\_)\) and \(\varphi(\alpha,n)\) is \(n+1\)-th term of \(\varphi(\beta,0)\).

Then \(\varphi(0,A) = \varphi(A)\) and \(\varphi(...,\alpha+1,0,...n)\) is \(n+1\)-th term of \(\varphi(...,\alpha,\_,...n)\)

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.

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