User:Ynought/My ordinal notation

Ordinal notation
This function takes an ordinal \(\alpha\) and turns it into a different ordinal

Attempt 1
\(\Phi\alpha\) means either \(\beta_\alpha\) or \(\beta^\alpha\) where \(\beta\) is any ordinal or \(\alpha\times\beta\) or \(\alpha+\beta\) or it could be just \(\alpha\) or any combination of the cases listed above

\(\alpha[n]\) is the n-th term in the fundamental sequence of \(\alpha\)

\(\xi(0)[n]=n\)

\(\xi(\Phi\alpha+1)[n]=\xi(\Phi\alpha)+1\uparrow\uparrow n\)

\(\xi(\Phi\alpha)[n]=\xi(\Phi\alpha[n])\) if the previous case does not apply and the next case doesn't either.

\(\xi(\Phi\Omega)[n]=\text{sup}(\xi(\Phi\omega),\xi(\Phi\omega+\xi(\Phi\omega)),\xi(\Phi\omega+\xi(\Phi\omega+\xi(\Phi\omega)))...)\)

\(\xi(\Phi\Omega_n)=\xi(\Phi\Omega_{n-1}\uparrow\uparrow n)\uparrow\uparrow n\)

Attempt 2
\(\varsigma(0)[n]=n\)

If \(\alpha=\beta+1\):

\(\varsigma(\alpha)[n]=\varsigma(\alpha)\uparrow\uparrow n\)

Otherwise(when \(\alpha\neq\beta+1\)):

\(\varsigma(\alpha)[n]=\varsigma^n(\alpha[n])\)

\(f\) is some function:\(f^n(\alpha)=f(f^{n-1}(\alpha+1)+1)\) and \(f^0(\alpha)=f(\alpha+1)\)

Now to define some new sequences

\(\Omega_0[n]=\Omega[n]=\text{sup}(\varsigma(\omega),\varsigma(\omega+\varsigma(\omega)),...)\)

\(\Omega_k[n]=\Omega_{k-1}\uparrow\uparrow n\) and \(\Omega_\alpha[n]=\Omega_{\alpha[n]}\)

And finally \(\Psi(\alpha)=\Omega_\alpha\) then \(\tau[n]=\Psi^n(\Omega)\)

Attempting to define fundamental sequences
(i probaply don't need to define it for finite values) for:

for limit \(\beta\):\(\mu=\alpha^\beta)\) \(\mu[n]=\alpha^{\beta[n]}\)

for succesor \(\beta\):\(\mu=\alpha^\beta\) \(\mu[n]=\alpha^{\beta-1}n\)

for \(\alpha_1>\alpha_2...>\alpha_k\) and \(\beta_1>\beta_2...>\beta_k and every \(\beta\) is a limit : \(\mu=\beta_1^{\alpha_1}+\beta_2^{\alpha_2}...+\beta_k^{\alpha_k}\) then:

(\mu[n]=\beta_1^{\alpha_1}+\beta_2^{\alpha_2}...+\beta_k^{\alpha_k}[n]\)

Definition
\(f\) is the FGH

\(g\) is the SGH

when i say \(f_\alpha(n)\approx g_\alpha(n)\) i am saying that for that \(\alpha\) there exists an\(n\) for which holds \(g_\alpha(n+1)>f_\alpha(n))\)

\(\gamma(\beta)\) is the ordinal needed that \(g_\alpha(n)\approx f_\alpha(n)\) is the case for the \(\beta\)'th time

the way you get that for succesors is easy but if you come across a limit then :

\(\eta\) is the limit ordinal which you come across

so when \(\alpha=\eta+\delta\) then \(\gamma(\alpha)=\gamma(\eta[n]+\delta)\)

Formal definition?
\(f_\alpha(n)\approx g_\alpha(n)\) means \(\exists(n|f_\alpha(n)\leqslant g_\alpha(n+1))\)

for \(\alpha=\delta+1\):

\(\gamma(\alpha)[n]=\beta|\beta \text{ is the }\alpha\text{'th time that }f_\beta(n)\approx g_\beta(n)\)

for countable non 0 limit \(\alpha\):

\(\gamma(\alpha)[n]=\gamma(\alpha[n])\) here \(\alpha[n]\) denotes the n-th term in the fundamental sequnce leading up to \(\alpha\)

for countable limit \(\alpha\) + succesor

\(\gamma(\alpha+\eta)[n]=\gamma(\alpha[n]+\eta)\)

Beyond countables
if you come across a \(\Omega[n]\) anywhere then replace it with the n-th fixed point of \(\alpha\mapsto\gamma(\alpha)\)

and \(\Omega_0[n]=\Omega[n]\) and \(\Omega_{\beta+1}[n]\) is the n-th fixed point of \(\alpha\mapsto\gamma(\Omega_\beta^\alpha)\) and for countable limit alpha \(\Omega_\alpha[n]=\Omega_{\alpha[n]}\) and if you find Omegas in subscripts then treat them like usual.