User blog:Boboris02/Analysis of Taranovsky's Ordinal Notation with "standard OCFs."

In this blog I will try to explain and familiarize people to Taranovsky's notation as well as make bounds for ordinals describable within his n=1 and n=2 systems. Note that I have some problems with understanding this notation fully myself,so if anyone reading this believes to understand it better than me,that please be sure to correct me for any mistake I make.

Definition
The actual definition of the notation is quite complicated,in my opinion. So I will try to break it down and simplify it.

Let's denote a binary relation of "\(\alpha\) is \(n\)-built from below by \(\beta\)" and a unary relation of "standart form" to ordinals.

\(\alpha\) is 0-built from below by \(\beta\) if \(\alpha<\beta\).

\(\alpha\) is (n+1)-built from below by \(\beta\) iff the standart representation of \(\beta\) does not use ordinals above \(\alpha\) unless it's an ordinal \(\gamma\) that is n-built from below by \(\beta\).

But what is standart representation?

Ordinals \(0,\Omega_n\) for \(0<n<\omega\) are axiomatically standart.

Ordinal \(C(\alpha,\beta)\) is in standart form iff: A "standart representation" of an ordinal is in the form \(C(\alpha,\beta)\) such that all of the ordinals used are \(0\) and \(\Omega_n\) respectively for an expression within the n-system.
 * Both \(\alpha\) and \(\beta\) are in standart form.
 * \(\alpha\) is in the form \(C(\gamma,\delta)\) where \(\beta\leq\delta\).
 * \(\beta\) is n-built from below by \(C(\alpha,\beta)\)

But what is an n-system?

The description of standart form above describes an infinite collection of functions,the concatenation of which is all of Taranovsky's C function.

An ordinal is in the scope of the n-system if it's representable in the form \(C(\Omega^{\alpha}_n \beta + \gamma ,\delta)\).

Every term within an n-system is valid if it has a standart representation within that said system.

With all of this said,let \(L(\alpha)\) be equal to the amount of \(C\)s in the standart representation of \(\alpha\) and let \(\alpha [n]\) denote the \(n\)th term of the fundamental sequence of \(\alpha\).

Then \(\alpha[n] = max\{\beta|\beta<\alpha\land L(\beta)\leq L(\alpha) + n\}\).

(I've removed the trinary lexicographic \(\{C,0,\Omega_n\}\) code because I don't feel like it's necessary here.)

Comparison of system n=1
\(C(0,\alpha)\) is simply the successor of (\alpha).

\(C(0,0)=1\)

\(C(0,C(0,0))=2\)

\(C(0,C(0,C(0,0)))=3\)

\(C(1,0)=\omega\)

\(C(1,1)=C(0,C(1,0))=\omega + 1\)

\(C(1,2)=C(0,C(0,C(1,0)))=\omega + 2\)

\(C(1,C(1,0))=\omega 2\)

\(C(1,C(1,1))=C(1,C(0,C(1,0)))=\omega 2 +1\)

\(C(1,C(1.C(1,1)))=\omega 3\)

\(C(2,0)=\omega^2\)

\(C(2,1)=C(0,C(2,0))=\omega^2 +1\)

\(C(2,2)=C(0,C(0,C(2,0)))=\omega^2 +2\)

\(C(2,C(1,0))=\omega^2 +\omega\)

\(C(2,C(2,0))=\omega^2 2\)

\(C(2,C(2,C(2,0)))=\omega^2 3\)

\(C(3,0)=\omega^3\)

Generally, \(\forall\alpha ,\beta < \Omega_1:C(\alpha,\beta) = \omega^\alpha + \beta\).

\(C(C(1,0),0) = C(\omega,0)=\omega^\omega\)

\(C(C(0,C(1,0)),0)=C(\omega +1,0) = \omega^{\omega+1}\)

\(C(C(1,C(1,0)),0)=\omega^{\omega 2}\)

\(C(C(2,0),0)=\omega^{\omega^2}\)

\(C(C(C(C(0,0),0),0),0)=C(C(C(1,0),0),0)=C(C(\omega ,0),0)=\omega^{\omega^\omega}\)

\(C(C(C(C(C(0,0),0),0),0),0)=\omega^{\omega^{\omega^\omega}}\)

\(C(\Omega ,0)=\varepsilon_0\)

\(C(C(\Omega ,0),C(\Omega ,0))=\varepsilon_0 2\)

\(C(C(\Omega ,0),C(C(\Omega ,0),C(\Omega ,0)))=\varepsilon_0 3\)

\(C(C(\Omega ,0)+1,C(\Omega ,0))=\varepsilon_0 \omega\)

\(C(C(\Omega ,0)+C(1,0),C(\Omega ,0))=\varepsilon_0 \omega^\omega\)

\(C(C(C(\Omega ,0),C(\Omega ,0)),C(\Omega ,0))=C(C(\Omega ,0)2,C(\Omega ,0)) = \varepsilon^{2}_0\)

\(C(C(C(\Omega ,0)+1,C(\Omega ,0)),C(\Omega ,0))=\varepsilon^{\omega}_0\)

\(C(C(C(C(\Omega ,0),C(\Omega ,0)),C(\Omega ,0)),C(\Omega ,0)) = C(C(C(\Omega ,0)2,C(\Omega ,0)),C(\Omega ,0))=\varepsilon^{\varepsilon_0}_0\)

\(C(C(C(C(C(\Omega ,0),C(\Omega ,0)),C(\Omega ,0)),C(\Omega ,0)),C(\Omega ,0)) = \varepsilon^{\varepsilon^{\varepsilon_0}_0}_0\)

\(C(\Omega ,C(\Omega ,0))=\varepsilon_1\)

\(C(\Omega ,C(\Omega ,C(\Omega ,0)))=\varepsilon_2\)

\(C(\Omega +1,0)=\varepsilon_\omega\)

\(C(\Omega +2,0)=\varepsilon_{\omega^2}\)

\(C(\Omega +C(1,0),0)=\varepsilon_{\omega^\omega}\)

\(C(\Omega +C(\Omega ,0),0)=\varepsilon_{\varepsilon_0}\)

\(C(\Omega +C(\Omega +C(\Omega ,0),0),0)=\varepsilon_{\varepsilon_{\varepsilon_0}}\)

\(C(\Omega 2,0)=\zeta_0\)

Generally,from \(C(\Omega ,0)\) on,the n=1 system is almost the same as the \(\vartheta\) function and somewhat to \(\psi\) function.

\(C(\Omega\omega ,0)=\varphi(\omega,0)\)

\(C(\Omega^2,0)=\Gamma_0\)

\(C(\Omega^\omega,0)=\psi(\Omega^{\Omega^\omega})\)

\(C(\Omega^\Omega,0)=\psi(\Omega^{\Omega^\Omega})\)

\(C(\Omega^{\Omega^\Omega},0)=\psi(\Omega^{\Omega^{\Omega^\Omega}})\)

So the limit of the n=1 system is the Bacchman-Howard ordinal.

Up to \(\psi(\psi_{I}(0))\)
\(C(C(\Omega_2 ,\Omega),0)=\psi(\varepsilon_{\Omega +1})\)

\(C(\Omega ,C(C(\Omega_2 ,\Omega),0))=\psi(\varepsilon_{\Omega +1}+1)\)

\(C(\Omega 2,C(C(\Omega_2 ,\Omega),0))=\psi(\varepsilon_{\Omega +1}+\Omega)\)

\(C(C(\Omega_2 ,\Omega),C(C(\Omega_2 ,\Omega),0))=\psi(\varepsilon_{\Omega +1}2)\)

\(C(C(\Omega_2 ,\Omega)+1,0)=\psi(\varepsilon_{\Omega +1}\omega)\)

\(C(C(\Omega_2 ,\Omega)+\Omega,0)=\psi(\varepsilon_{\Omega +1}\Omega)\)

\(C(C(\Omega_2 ,\Omega)+\Omega^\Omega,0)=\psi(\varepsilon_{\Omega +1}\Omega^\Omega)\)

\(C(C(\Omega_2 ,\Omega)+\Omega^{\Omega^\Omega},0) = \psi(\varepsilon_{\Omega +1}\Omega^{\Omega^\Omega})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega),\Omega),0)=\psi(\varepsilon^{2}_{\Omega +1})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega),\Omega)2,0)=\psi(\varepsilon^{3}_{\Omega +1})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega),\Omega)^2,0)=\psi(\varepsilon^{\varepsilon_{\Omega +1}}_{\Omega +1})\)

\(C(C(\Omega_2 ,\Omega)+C(C(C(C(C(\Omega_2 ,\Omega),\Omega),C(C(\Omega_2 ,\Omega),\Omega)),C(C(\Omega_2 ,\Omega),\Omega))\\,C(C(\Omega_2 ,\Omega),\Omega)),0) = C(C(\Omega_2,\Omega)+C(C(\Omega_2 ,\Omega),\Omega)^{C(C(\Omega_2 ,\Omega),\Omega)},0) \\= \psi(\varepsilon^{\varepsilon^{\varepsilon_{\Omega +1}}_{\Omega +1}}_{\Omega +1})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2,\Omega),C(C(\Omega_2,\Omega),\Omega)),0)=\psi(\varepsilon_{\Omega +2})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega)+1,\Omega),0)=\psi(\varepsilon_{\Omega +\omega})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega)+\Omega ,\Omega),0)=\psi(\varepsilon_{\Omega 2})\)

\(C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega)+C(C(\Omega_2 ,\Omega),\Omega),\Omega),0) = \psi(\varepsilon_{\varepsilon_{\Omega +1}})\)

\(C(C(C(\Omega_2 ,\Omega),C(\Omega_2 ,\Omega)),0)=C(C(\Omega_2 ,\Omega)2,0)=\psi(\psi_1(\Omega_2))\)

From here on it's a similar pattern:

\(C(\Omega_2 ,\Omega)\) corresponds to \(\Omega_2\)

\(C(C(C(\Omega_2,(\Omega_2 ,\Omega)),\Omega)\) corresponds to \(\varepsilon_{\Omega_2 +1}\)

\(C(\Omega_2,C(\Omega_2,\Omega))\) corresponds to \(\Omega_3\)

\(C(\Omega_2,C(\Omega_2,C(\Omega_2 ,\Omega)))\) corresponds to \(\Omega_4\)

\(C(C(\Omega_2 +1,0),0)=\psi(\Omega_\omega)\)

\(C(C(\Omega_2 +2,0),0)=\psi(\Omega_{\omega^2})\)

\(C(C(\Omega_2 +\Omega,0),0)=\psi(\Omega_\Omega)\)

\(C(C(\Omega_2+C(\Omega_2+\Omega ,0),0),0)=\psi(\Omega_{\Omega_\Omega})\)

\(C(C(\Omega_2 2,0),0)=\psi(\psi_I(0))\)

Up to the limit of "Standart OCFs"
[I'll do this later]