User:Hyp cos/OCF vs Array Notation

I've compared Taranovsky's ordinal notation with my array notation, but most people don't understand both, so I'll compare ordinal collapsing functions (OCFs) with my array notation. I've compare \(\theta\) function up to \(\Pi_1^1\text{-TR}_0\), but further OCFs are defined in different ways, so I need to go back from the very beginning.

Bachmann's \(\psi\)

 * There's no comparisons with my array notation in this section.

Bachmann's \(\psi\) function was defined as follows, where \(\Omega\) is a "big" ordinal, say, the first uncountable cardinal. \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0,\Omega\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ & & \cup\{\omega^\gamma|\gamma\in C_n(\alpha,\beta)\} \\ & & \cup\{\psi(\gamma)|\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi(\alpha) &=& \min\{\beta<\Omega|C(\alpha,\beta)\cap\Omega\subseteq\beta\} \end{eqnarray*}

Generally, to get \(\psi(\alpha)\), we need to apply addition, omega-exponentiation and "previous" \(\psi\) to 0 and \(\Omega\) (i.e. set \(\beta=0\)), then choose a large enough \(\beta\) to cover all those results (i.e. fit \(C(\alpha,\beta)\cap\Omega\subseteq\beta\)). For \(\psi(\alpha+\Omega)\), if \(\alpha\in C(\alpha+\Omega,\beta)\), then we have \(\alpha+\Omega\), \(\psi(\alpha+\Omega)\), \(\alpha+\psi(\alpha+\Omega)\), \(\psi(\alpha+\psi(\alpha+\Omega))\), etc. in \(C(\alpha+\Omega,\beta)\). That suggests \(\psi(\alpha+\Omega)\) being the fixed point of \(\beta\mapsto\psi(\alpha+\beta)\). Similar things happen in \(\psi(\alpha_0+\omega^{\alpha_1+\omega^{\cdots^{\alpha_{k-1}+\omega^{\alpha_k+\Omega}}}})\), and thus \(\Omega\) appears analogous to the least 1-separator in EAN - the grave accent.

More detailed, \begin{eqnarray*} \psi(0) &=& \varepsilon_0 \\ \psi(1) &=& \varepsilon_1 \\ \psi(2) &=& \varepsilon_2 \\ \psi(\omega) &=& \varepsilon_\omega \\ \psi(\omega^\omega) &=& \varepsilon_{\omega^\omega} \\ \psi(\varepsilon_0) &=& \varepsilon_{\varepsilon_0} \\ \psi(\varepsilon_0+1) &=& \varepsilon_{\varepsilon_0+1} \\ \psi(\varepsilon_{\varepsilon_0}) &=& \varepsilon_{\varepsilon_{\varepsilon_0}} \\ \psi(\zeta_0) &=& \zeta_0 \\ \psi(\zeta_0+1) &=& \zeta_0 \\ \psi(\Omega) &=& \zeta_0 \\ \psi(\Omega+1) &=& \varepsilon_{\zeta_0+1} \\ \psi(\Omega+\omega) &=& \varepsilon_{\zeta_0+\omega} \\ \psi(\Omega+\varepsilon_0) &=& \varepsilon_{\zeta_0+\varepsilon_0} \\ \psi(\Omega+\psi(\Omega)) &=& \varepsilon_{\zeta_02} \\ \psi(\Omega+\psi(\Omega)+1) &=& \varepsilon_{\zeta_02+1} \\ \psi(\Omega+\psi(\Omega)2) &=& \varepsilon_{\zeta_03} \\ \psi(\Omega+\psi(\Omega+1)) &=& \varepsilon_{\varepsilon_{\zeta_0+1}} \\ \psi(\Omega+\psi(\Omega+\psi(\Omega))) &=& \varepsilon_{\varepsilon_{\zeta_02}} \\ \psi(\Omega+\psi(\Omega+\psi(\Omega+1))) &=& \varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}} \\ \psi(\Omega+\zeta_1) &=& \zeta_1 \\ \psi(\Omega+\zeta_1+1) &=& \zeta_1 \\ \psi(\Omega2) &=& \zeta_1 \\ \psi(\Omega2+1) &=& \varepsilon_{\zeta_1+1} \end{eqnarray*}

The definition of this \(\psi\) function seems similar to \(\vartheta\) function (just without the "\(\alpha\in C(\alpha,\beta)\)"), but the difference makes it follow a \(\psi\) way - not a \(\vartheta\) way. And the "\(\{\omega^\gamma|\gamma\in C_n(\alpha,\beta)\}\)" affects its strength.