User:Hyp cos/OCF vs Array Notation p2

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Now the notations become complicated.

Collapsing higher Mahloness
Now we need another ordinal collapsing function to generate weakly Mahlo cardinals. Similar to "Using weakly Mahlos" section, we need another function after \(\psi\) and \(\chi\), and use a larger cardinal for collapsing - it should be a weakly Mahlo cardinal such that weakly Mahlo cardinals in it are stationary. That leads us to higher Mahloness: an ordinal \(\pi\) is \(\alpha\)-weakly Mahlo if it's an uncountable regular cardinal and the set of \(\gamma\)-weakly Mahlo cardinals in \(\pi\) is stationary for all \(\gamma<\alpha\). So 0-weakly Mahlo cardinals are just uncountable regular cardinals, and 1-weakly Mahlo cardinals are weakly Mahlo cardinals, and the "weakly Mahlo cardinal such that weakly Mahlo cardinals in it are stationary" is a 2-weakly Mahlo cardinal. And Mahloness can extend further. For example, an ordinal \(\pi\) is (1,0)-weakly Mahlo if it's \(\pi\)-weakly Mahlo. Generally, an ordinal \(\pi\) is \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo if it's \((\alpha_1,\alpha_2\cdots,\alpha_i,\gamma,\pi,\underbrace{0,0,\cdots0,0}_{n-i-2})\)-weakly Mahlo for all \(\gamma<\alpha_{i+1}\) and \(0\le i\le n-2\), and the set of \((\alpha_1,\alpha_2,\cdots\alpha_{n-1},\gamma)\)-weakly Mahlo cardinals in \(\pi\) is stationary for all \(\gamma<\alpha_n\).

Let \(M(\alpha_1,\alpha_2\cdots,\alpha_n;0)\) to be the first \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo cardinal, \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta+1)\) to be the next \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo cardinal after \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta)\), and \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta)=\sup\{M(\alpha_1,\alpha_2\cdots,\alpha_n;\gamma)|\gamma<\beta\}\) for limit ordinal \(\beta\). (Notice that there's a semicolon in this notation) So \(M(0;\alpha)=\Omega_{1+\alpha}\), \(M(1;\alpha)=M_{1+\alpha}\), and \(M(2;0)\) is the least "weakly Mahlo cardinal such that weakly Mahlo cardinals in it are stationary".

Let \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\) be the set of \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo cardinals below \(\sup\{M(1,\underbrace{0,0\cdots,0}_{n};0)|n<\omega\}\). So \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta)\) is the \(1+\beta\)-th ordinal in the closure of \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\).

\begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{M(\gamma_1,\gamma_2\cdots,\gamma_k;\delta)|\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi^{\gamma_1,\gamma_2\cdots,\gamma_k}(\delta)|\pi,\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\wedge\delta<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \chi_\pi^{\gamma_1,\gamma_2\cdots,\gamma_n}(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2\cdots,\gamma_n}\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\) (first uncountable cardinal), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(\chi_\pi(\alpha)\) is a shorthand for \(\chi_\pi^0(\alpha)\).

Compared with previous notations, this notation has changes on \(\chi\) and \(\psi\) - a \(\{\pi\}\) is unionized. Now the \(\pi\) isn't restricted to some \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\), so the \(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2\cdots,\gamma_n}\}\) may be empty for some value such as \(\pi=\Omega\), \(\alpha=0\). So we need to add a harmless \(\{\pi\}\) so that every \(\chi\) results some value. \(\psi\) is similar. And this \(\{\pi\}\) doesn't affect the values if the \(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2\cdots,\gamma_n}\}\) is not empty.

\(\psi(0)=1\), \(\psi_{\Omega_2}(0)=\Omega\omega\), \(\psi_{\Omega_3}(0)=\Omega_2\omega\), etc. those return the pattern of "Collapsing higher inaccessibility" section because we have \(\Omega_\gamma\) in \(C_{n+1}(\alpha,\beta)\) (even when \(\beta=0\)) again. Next, \(\psi_{\chi_M(0)}(0)=\psi_M(0)=\omega_*\) (the omega-fixed-point), where \(\chi_M(0)=I\), so this is consistent with \(\psi_I(0)=\omega_*\). \(\psi_{\Omega_{M+1}}(0)=M\omega\), \(\psi_{\Omega_{M+2}}(0)=\Omega_{M+1}\omega\), \(\psi_{M_2}(0)=\psi_{\chi_{M_2}(0)}(0)\) equals the next omega-fixed-point after \(M\), where \(\chi_{M_2}(0)\) equals the next weakly inaccessible after \(M\). \(\psi_{\chi_{M(2;0)}(0)}(0)=\psi_{\chi_{\chi^1_{M(2;0)}(0)}(0)}(0)=\psi_{\chi^1_{M(2;0)}(0)}(0)=\psi_{M(2;0)}(0)=M_*\) (the weakly Mahlo-fixed-point), where \(\chi_{M(2;0)}(0)=\chi_{\chi^1_{M(2;0)}(0)}(0)\) equals the least weakly inaccessible which is a limit of weakly Mahlos, and \(\chi^1_{M(2;0)}(0)=M(1,0)\) - the least weakly Mahlo which is a limit of weakly Mahlos. \(\chi^1_{M(2;1)}(0)=M(1,M(2;0)+1)\) - the next "weakly Mahlo which is a limit of weakly Mahlos" after \(M(2;0)\). \(\chi^1_{M(3;0)}(0)=\chi^1_{\chi^2_{M(3;0)}(0)}(0)\) is the least "weakly Mahlo which is a limit of 2-weakly Mahlos", where \(\chi^2_{M(3;0)}(0)\) is the least "2-weakly Mahlo which is a limit of 2-weakly Mahlos".

The superscript of \(\chi\) is useful at \(\chi^n_{M(\omega;0)}(0)\). \(\psi_{\chi^k_{M(\omega;0)}(0)}(0)=\psi_{M(\omega;0)}(0)=\sup\{M(n;0)|n<\omega\}\), \(\chi_{M(\omega;0)}(0)\) is the least "weakly inaccessible which is a limit of n-weakly Mahlos for all \(n<\omega\)", \(\chi^k_{M(\omega;0)}(0)\) is the least "k-weakly Mahlo which is a limit of n-weakly Mahlos for all \(n<\omega\)" (k > 0), whose supremum has cofinality \(\omega\) and is singular - that's \(\psi_{\chi^k_{M(\omega;0)}(1)}(1)=\psi_{M(\omega;0)}(1)\) instead of \(M(\omega;0)\).

\(\psi_{M(1,0;0)}(0)\) is the fixed point of \(\alpha\mapsto\psi_{M(\alpha;0)}(0)\). What's larger is \(\psi_{M(\psi_{M(1,0;0)}(0);0)}(1)=\sup\{\chi^\alpha_{M(\psi_{M(1,0;0)}(0);0)}(0)|\alpha<\psi_{M(1,0;0)}(0)\}\) (similar to \(\psi_{M(\omega;0)}(1)\)), then \(\chi_{M(\psi_{M(1,0;0)}(0);0)}(1)\), \(\chi^1_{M(\psi_{M(1,0;0)}(0);0)}(1)\), ..., \(\psi_{M(\psi_{M(1,0;0)}(0);0)}(2)\), \(M(\psi_{M(1,0;0)}(0);0)\), \(M(\psi_{M(1,0;0)}(0);1)\), \(M(\psi_{M(1,0;0)}(0)+1;0)\), until a next fixed point of \(\alpha\mapsto\psi_{M(\alpha;0)}(0)\) - \(\psi_{\chi_{M(1,0;0)}(0)}(1)\). What's larger is \(\chi_{M(1,0;0)}(0)\) - the fixed point of \(\alpha\mapsto\chi_{M(\alpha;0)}(0)\) (\(\chi^0\) is not so "normal" as \(\psi\), so it's "harder" to get a fixed point), then \(\chi^1_{M(1,0;0)}(0)\) - the fixed point of \(\alpha\mapsto\chi^1_{M(\alpha;0)}(0)\), etc. And \(\psi_{M(1,0;0)}(1)\) is the limit of \(\chi_{M(1,0;0)}(0)\), \(\chi^{\chi_{M(1,0;0)}(0)}_{M(1,0;0)}(0)\), \(\chi^{\chi^{\chi_{M(1,0;0)}(0)}_{M(1,0;0)}(0)}_{M(1,0;0)}(0)\), etc.

So the increase of \(\gamma\) in \(\chi^\gamma_\pi(\alpha)\) is also a way to get larger.

Here come comparisons between this OCF and pDAN. In the comparisons shown below, I'll use 2 kinds of statements. "A has recursion level \(\alpha\)" means s(n,n A 2) has growth rate \(\omega^{\omega^\alpha}\), while "A approximately corresponds to \(\alpha\)" means the separator A works similar to ordinal \(\alpha\) in other separators.
 * {1{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})\)
 * {1{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega})\)
 * {1 {1{1{1{1,,3,,}2}2,,2,,}2} 2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\psi_{M(2;0)}(M(2;0)^{M(2;0)}))\)
 * {1 {1{1{1{1,,3,,}2}1,2,,2,,}2} 2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}))\)
 * {1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\Omega_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})+1})\)
 * {1,,1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\chi_{M_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})+1}}(M(2;0)^{M(2;0)^\omega}))\)
 * {1,,1,,1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})+1})\)
 * {1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(M_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})+1}\)
 * {1{1,,2,,}1 {1{1{1{1,,3,,}2}1,2,,2,,}2} 2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})2})\)
 * {1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\psi_{\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})}(M(2;0)^{M(2;0)^\omega}+1))\)
 * {1{1,,2,,}1 {1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} 2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})}(M(2;0)^{M(2;0)^\omega}+1)\)
 * {1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega})\)
 * {1{1,,2,,}1,,3{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\psi_{\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega})}(M(2;0)^{M(2;0)^\omega}+1)}(M(2;0)^{M(2;0)^\omega}+2))\)
 * {1{1,,2,,}1,,3{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega})}(M(2;0)^{M(2;0)^\omega}+1)\)
 * {1{1,,2,,}1,,1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}))\)
 * {1{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega})\)
 * {1{1,,2,,}1,,1,,2{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+1))\)
 * {1{1,,2,,}1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0))\)
 * {1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} 2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))\)
 * {1{1,,2,,}1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))\)
 * {1,,1,,1,,2{1,,2,,}1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)+M_{\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))+1})\)
 * {1{1,,2,,}2{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(M_{\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))+1}\)
 * {1{1,,2,,}1,,2{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))}(M(2;0)^{M(2;0)^\omega}+\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))+1)\)
 * {1{1,,2,,}1{1,,2,,}2,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))}(M(2;0)^{M(2;0)^\omega}+M(2;0)+1)}(M(2;0)^{M(2;0)^\omega}+M(2;0)+1)\)
 * {1{1,,2,,}1{1,,2,,}1,,3{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))}(M(2;0)^{M(2;0)^\omega}+M(2;0)+1)\)
 * {1{1,,2,,}1{1,,2,,}1,,1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)+\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)))\)
 * {1{1,,2,,}1{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0))\)
 * {1{1,,2,,}1,,1,,2{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)+\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)+1))\)
 * {1{1,,2,,}1{1,,2,,}1,,1,,2{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)2)\)
 * {1{1,,2,,}1{1,,2,,}1{1,,2,,}1,,1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)^2)\)
 * {1{2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)\)
 * {1{2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)\)
 * {1{1,,2,,}2{2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(M_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)+1}\)
 * {1{1,,2,,}1{1,,2,,}2{2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)+1)\)
 * {1{1,,2,,}1{1,,2,,}1{1,,2,,}2{2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)+1)\)
 * {1{2,,2,,}3{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega+1)\)
 * {1{2,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)\)
 * {1{2,,2,,}1,,3{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega+1)\)
 * {1{2,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega)\)
 * {1{2,,2,,}1,,1,,2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega+\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^\omega))\)
 * {1{2,,2,,}1{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{\omega+1})\)
 * {1{3,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{\omega^2})\)
 * {1{1{1,,3,,}2}2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)})\)
 * {1{1{1,,3,,}2}2,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)})\)
 * {1{1{1,,3,,}2}2,,2,,}3{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)})}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)}+1)\)
 * {1{1{1,,3,,}2}2,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)})\)
 * {1{1{1,,3,,}2}2,,2,,}1,,3{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi_{\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)})}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)}+1)\)
 * {1{1{1,,3,,}2}2,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)}+1)\)
 * {1{1{1,,3,,}2}3,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}+M(2;0)^{M(2;0)^2})\)
 * {1{1{1{1,,3,,}2}1,2,,2,,}3} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}2)\)
 * {1{1{1{1,,3,,}2}1,2,,2,,}3} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega}2)\)
 * {1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}3} approximately corresponds to \(M_{\psi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}2)+1}\)
 * {1{1,,2,,}1,,2{1{1{1,,3,,}2}1,2,,2,,}3} approximately corresponds to \(\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega}2)\)
 * {1{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}1,2,,2,,}3} approximately corresponds to \(\chi^1_{M(2;0)}(M(2;0)^{M(2;0)^\omega}2)\)
 * {1{1{1{1,,3,,}2}1,2,,2,,}1,,2} has recursion level \(\psi(M(2;0)^{M(2;0)^\omega+1})\)
 * {1{1{1{1,,3,,}2}1,2,,2,,}1,,2} approximately corresponds to \(\chi_{M(2;0)}(M(2;0)^{M(2;0)^\omega+1})\)
 * {1{1{1{1,,3,,}2}2,2,,2,,}2} approximately corresponds to \(\psi_{M(2;0)}(M(2;0)^{M(2;0)^{\omega+1}})\)
 * {1{1{1{1,,3,,}2}2,2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^{\omega+1}})\)
 * {1{1{1{1,,3,,}2}1{1{1,,3,,}2}2,,2,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^{M(2;0)}})\)
 * {1{1{1,,3,,}2}2{1,,3,,}2} has recursion level \(\psi(M(2;0)^{M(2;0)^{M(2;0)^{M(2;0)}}})\)
 * {1{1,,3,,}2} approximately corresponds to \(M(2;0)\)
 * {1,,2{1,,3,,}2} approximately corresponds to \(\Omega_{M(2;0)+1}\)
 * {1{1,,2,,}2{1,,3,,}2} approximately corresponds to \(M_{M(2;0)+1}\)
 * {1{1,,2,,}1 {1{1,,2,,}1,,2{1,,3,,}2} 2{1,,3,,}2} approximately corresponds to \(\psi_{M(2;1)}(0)\)
 * {1{1,,2,,}1,,2{1,,3,,}2} approximately corresponds to \(\chi_{M(2;1)}(0)\)
 * {1{1,,2,,}1,,3{1,,3,,}2} approximately corresponds to \(\chi_{\chi^1_{M(2;1)}(0)}(1)\)
 * {1{1,,2,,}1{1,,2,,}2{1,,3,,}2} approximately corresponds to \(\chi^1_{M(2;1)}(0)\)
 * {1{1,,2,,}1{1,,2,,}3{1,,3,,}2} approximately corresponds to \(\chi^1_{M(2;1)}(1)\)
 * {1{1,,2,,}1{1,,2,,}1{1,,2,,}2{1,,3,,}2} approximately corresponds to \(\chi^1_{M(2;1)}(M(2;1))\)
 * {1{1,,2,,}1{1,,2,,}1,,2{1,,3,,}2} has recursion level \(\psi(M(2;1))\)
 * {1{1,,2,,}1{1,,2,,}2{1,,2,,}2{1,,3,,}2} approximately corresponds to \(\chi^1_{M(2;1)}(M(2;1)+1)\)
 * {1{1,,2,,}1{1,,2,,}1{1,,2,,}3{1,,3,,}2} approximately corresponds to \(\chi^1_{M(2;1)}(M(2;1)2)\)
 * {1{1{1{1,,3,,}3}2,,2,,}2{1,,3,,}2} approximately corresponds to \(\chi^1_{M(2;1)}(M(2;1)^{M(2;1)})\)
 * {1{1,,3,,}3} approximately corresponds to \(M(2;1)\)
 * {1{1,,2,,}1{1,,2,,}1,,2{1,,3,,}3} has recursion level \(\psi(M(2;2))\)
 * {1{1,,3,,}1,2} has recursion level \(\psi(M(2;\omega))\)
 * {1{1,,3,,}1,2} approximately corresponds to \(M(2;\omega)\)
 * {1{1,,3,,}2,2} approximately corresponds to \(M(2;\omega+1)\)
 * {1{1,,3,,}1{1{1,,3,,}2}2} approximately corresponds to \(M(2;M(2;0))\)
 * {1{1,,3,,}1,,2} has recursion level \(\psi(\psi_{M(3;0)}(0))\)
 * {1{1,,3,,}1{1{1,,3,,}1,,2}2} approximately corresponds to \(\psi_{M(3;0)}(0)\)
 * {1{1,,3,,}2{1{1,,3,,}1,,2}2} approximately corresponds to \(M(2;\psi_{M(3;0)}(0)+1)\)
 * {1{1,,3,,}1{1{1,,3,,}1,,2}3} approximately corresponds to \(\psi_{\chi_{M(3;0)}(0)}(1)\)
 * {1{1,,3,,}1,,2} approximately corresponds to \(\chi_{M(3;0)}(0)\)
 * {1{1,,3,,}2,,2} approximately corresponds to \(M(2;\chi_{M(3;0)}(0)+1)\)
 * {1{1,,3,,}1,,3} approximately corresponds to \(\chi_{\chi^1_{M(3;0)}(0)}(1)\)
 * {1{1,,3,,}1,,1,,2} approximately corresponds to \(\chi_{\chi^1_{M(3;0)}(0)}(\chi^1_{M(3;0)}(0))\)
 * {1{1,,3,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{M(3;0)}(0)\)
 * {1{1,,3,,}2{1,,2,,}2} approximately corresponds to \(M(2;\chi^1_{M(3;0)}(0)+1)\)
 * {1{1,,3,,}1,,2{1,,2,,}2} approximately corresponds to \(\chi_{\chi^1_{\chi^2_{M(3;0)}(0)}(1)}(1)\)
 * {1{1,,3,,}1{1,,2,,}3} approximately corresponds to \(\chi^1_{\chi^2_{M(3;0)}(0)}(1)\)
 * {1{1,,3,,}1{1,,2,,}1,,2} has recursion level \(\psi(\chi^2_{M(3;0)}(0))\)
 * {1{1,,3,,}1{1,,2,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{\chi^2_{M(3;0)}(0)}(\chi^2_{M(3;0)}(0))\)
 * {1{1,,3,,}1{1,,2,,}2{1,,2,,}2} approximately corresponds to \(\chi^1_{\chi^2_{M(3;0)}(0)}(\chi^2_{M(3;0)}(0)+1)\)
 * {1{1,,3,,}1{2,,2,,}2} approximately corresponds to \(\psi_{\chi^2_{M(3;0)}(0)}(\chi^2_{M(3;0)}(0)^\omega)\)
 * {1{1,,3,,}1{1{1{1,,3,,}1{1,,3,,}2}2,,2,,}2} approximately corresponds to \(\psi_{\chi^2_{M(3;0)}(0)}(\chi^2_{M(3;0)}(0)^{\chi^2_{M(3;0)}(0)})\)
 * {1{1,,3,,}1{1,,3,,}2} has recursion level \(\psi(\chi^2_{M(3;0)}(0)^{\chi^2_{M(3;0)}(0)})\)
 * {1{1,,3,,}1{1,,3,,}2} approximately corresponds to \(\chi^2_{M(3;0)}(0)\)
 * {1{1,,3,,}2{1,,3,,}2} approximately corresponds to \(M(2;\chi^2_{M(3;0)}(0)+1)\)
 * {1{1,,3,,}1,,2{1,,3,,}2} approximately corresponds to \(\chi_{\chi^2_{M(3;0)}(1)}(1)\)
 * {1{1,,3,,}1{1,,2,,}2{1,,3,,}2} approximately corresponds to \(\chi^1_{\chi^2_{M(3;0)}(1)}(1)\)
 * {1{1,,3,,}1{1,,3,,}3} approximately corresponds to \(\chi^2_{M(3;0)}(1)\)
 * {1{1,,3,,}1{1,,3,,}1,,2} has recursion level \(\psi(M(3;0))\)
 * {1{1,,3,,}1{1,,3,,}1,,2} approximately corresponds to \(\chi_{M(3;0)}(M(3;0))\)
 * {1{1,,3,,}1{1,,3,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{M(3;0)}(M(3;0))\)
 * {1{1,,3,,}1{1,,3,,}1{1,,3,,}2} approximately corresponds to \(\chi^2_{M(3;0)}(M(3;0))\)
 * {1{1{1{1,,4,,}2}2,,3,,}2} approximately corresponds to \(\chi^2_{M(3;0)}(M(3;0)^{M(3;0)})\)
 * {1{1,,4,,}2} has recursion level \(\psi(M(3;0)^{M(3;0)})\)
 * {1{1,,4,,}2} approximately corresponds to \(M(3;0)\)
 * {1{1,,4,,}3} approximately corresponds to \(M(3;1)\)
 * {1{1,,4,,}1{1{1,,4,,}2}2} approximately corresponds to \(M(3;M(3;0))\)
 * {1{1,,4,,}1,,2} approximately corresponds to \(\chi_{M(4;0)}(0)\)
 * {1{1,,4,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{M(4;0)}(0)\)
 * {1{1,,4,,}1{1,,3,,}2} approximately corresponds to \(\chi^2_{M(4;0)}(0)\)
 * {1{1,,4,,}1{1,,4,,}2} approximately corresponds to \(\chi^3_{M(4;0)}(0)\)
 * {1{1,,4,,}1{1,,4,,}3} approximately corresponds to \(\chi^3_{M(4;0)}(1)\)
 * {1{1,,4,,}1{1,,4,,}1,,2} has recursion level \(\psi(M(4;0))\)
 * {1{1,,4,,}1{1,,4,,}1{1,,4,,}2} approximately corresponds to \(\chi^3_{M(4;0)}(M(4;0))\)
 * {1{1{1{1,,5,,}2}2,,4,,}2} approximately corresponds to \(\chi^3_{M(4;0)}(M(4;0)^{M(4;0)})\)
 * {1{1,,5,,}2} approximately corresponds to \(M(4;0)\)
 * {1{1,,6,,}2} approximately corresponds to \(M(5;0)\)
 * {1{1,,1,2,,}2} has recursion level \(\psi(\psi_{M(\omega;0)}(0))\)
 * {1{1,,1,2,,}2} approximately corresponds to \(\psi_{M(\omega;0)}(0)\)
 * {1,,2{1,,1,2,,}2} approximately corresponds to \(\Omega_{\psi_{M(\omega;0)}(0)+1}\)
 * {1{1,,2,,}2{1,,1,2,,}2} approximately corresponds to \(M_{\psi_{M(\omega;0)}(0)+1}\)
 * {1{1,,3,,}2{1,,1,2,,}2} approximately corresponds to \(M(2;\psi_{M(\omega;0)}(0)+1)\)
 * {1{1,,4,,}2{1,,1,2,,}2} approximately corresponds to \(M(3;\psi_{M(\omega;0)}(0)+1)\)
 * {1{1,,1,2,,}3} approximately corresponds to \(\psi_{\chi_{M(\omega;0)}(0)}(1)\)
 * {1{1,,1,2,,}1,,2} has recursion level \(\psi(\chi_{M(\omega;0)}(0))\)
 * {1{1,,1,2,,}1,,2} approximately corresponds to \(\chi_{M(\omega;0)}(0)\)
 * {1{1,,1,2,,}1,,3} approximately corresponds to \(\chi_{\chi^1_{M(\omega;0)}(0)}(1)\)
 * {1{1,,1,2,,}1,,1,,2} approximately corresponds to \(\chi_{\chi^1_{M(\omega;0)}(0)}(\chi^1_{M(\omega;0)}(0))\)
 * {1{1,,1,2,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{M(\omega;0)}(0)\)
 * {1{1,,1,2,,}1{1,,3,,}2} approximately corresponds to \(\chi^2_{M(\omega;0)}(0)\)
 * {1{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(\psi_{M(\omega;0)}(1)\)
 * {1{1,,2,,}2{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(M_{\psi_{M(\omega;0)}(1)+1}\)
 * {1{1,,3,,}2{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(M(2;\psi_{M(\omega;0)}(1)+1)\)
 * {1{1,,1,2,,}2{1,,1,2,,}2} approximately corresponds to \(\psi_{\chi_{M(\omega;0)}(1)}(2)\)
 * {1{1,,1,2,,}1,,2{1,,1,2,,}2} approximately corresponds to \(\chi_{M(\omega;0)}(1)\)
 * {1{1,,1,2,,}1{1,,2,,}2{1,,1,2,,}2} approximately corresponds to \(\chi^1_{M(\omega;0)}(1)\)
 * {1{1,,1,2,,}1{1,,3,,}2{1,,1,2,,}2} approximately corresponds to \(\chi^2_{M(\omega;0)}(1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}3} approximately corresponds to \(\psi_{M(\omega;0)}(2)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,2} has recursion level \(\psi(M(\omega;0))\)
 * {1{1,,1,2,,}1{1,,1,2,,}1 {1{1,,1,2,,}1{1,,1,2,,}1,,2} 2} approximately corresponds to \(\psi_{M(\omega;0)}(M(\omega;0))\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,2} approximately corresponds to \(M(\omega;0)\)
 * {1{1,,1,2,,}2{1,,1,2,,}1,,2} approximately corresponds to \(\psi_{M(\omega;1)}(0)\)
 * {1{1,,1,2,,}1{1,,1,2,,}2,,2} approximately corresponds to \(\psi_{M(\omega;1)}(1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}3,,2} approximately corresponds to \(\psi_{M(\omega;1)}(2)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,3} approximately corresponds to \(M(\omega;1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,1,,2} has recursion level \(\psi(\psi_{M(\omega+1;0)}(0))\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,1,,2} approximately corresponds to \(\chi_{M(\omega+1;0)}(0)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{M(\omega+1;0)}(0)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1)\)
 * {1{1,,2,,}2{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(M_{\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1)+1}\)
 * {1{1,,1,2,,}2{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(\psi_{M(\omega;\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1))}(1)\)
 * {1{1,,1,2,,}1,,2{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(\chi_{M(\omega;\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1))}(1)\)
 * {1{1,,1,2,,}1{1,,2,,}2{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(\chi^1_{M(\omega;\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1))}(1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}2{1,,1,2,,}2} approximately corresponds to \(\psi_{M(\omega;\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1))}(2)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,2{1,,1,2,,}2} approximately corresponds to \(M(\omega;\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1))\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,3{1,,1,2,,}2} approximately corresponds to \(M(\omega;\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(1)+1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,1,,2{1,,1,2,,}2} approximately corresponds to \(\chi_{\chi^\omega_{M(\omega+1;0)}(0)}(1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,2,,}2{1,,1,2,,}2} approximately corresponds to \(\chi^1_{\chi^\omega_{M(\omega+1;0)}(0)}(1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}3} approximately corresponds to \(\psi_{\chi^\omega_{M(\omega+1;0)}(0)}(2)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}1,,2} approximately corresponds to \(\chi^\omega_{M(\omega+1;0)}(0)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}1,,3} approximately corresponds to \(\chi^\omega_{M(\omega+1;0)}(1)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}1,,1,,2} has recursion level \(\psi(M(\omega+1;0))\)
 * {1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}1{1,,1,2,,}2} approximately corresponds to \(\chi^\omega_{M(\omega+1;0)}(M(\omega+1;0))\)
 * {1{1{1{1,,2,2,,}2}2,,1,2,,}2} approximately corresponds to \(\chi^\omega_{M(\omega+1;0)}(M(\omega+1;0)^{M(\omega+1;0)})\)
 * {1{1,,2,2,,}2} approximately corresponds to \(M(\omega+1;0)\)
 * {1{1,,1,3,,}2} approximately corresponds to \(\psi_{M(\omega2;0)}(0)\)
 * {1{1,,1,3,,}1{1,,1,3,,}1,,2} approximately corresponds to \(M(\omega2;0)\)
 * {1{1,,1{1{1,,2,,}2}2,,}1{1,,1{1{1,,2,,}2}2,,}1,,2} approximately corresponds to \(M(M;0)\)
 * {1{1,,1,,2,,}2} has recursion level \(\psi(\psi_{M(1,0;0)}(0))\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{M(1,0;0)}(0)=\psi_{M(\psi_{M(1,0;0)}(0);0)}(0)\)
 * {1{1,,1,2,,}1{1,,1,2,,}1,,2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(M(\omega;\psi_{M(1,0;0)}(0))\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_{M(\psi_{M(1,0;0)}(0);0)}(0)}(1)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1,,2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_{M(\psi_{M(1,0;0)}(0);0)}(0)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1{1,,2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi^1_{M(\psi_{M(1,0;0)}(0);0)}(0)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{M(\psi_{M(1,0;0)}(0);0)}(1)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1,,2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(M(\psi_{M(1,0;0)}(0);0)\)
 * {1{1,,2 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(M(\psi_{M(1,0;0)}(0)+1;0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}3} approximately corresponds to \(\psi_{\chi_{M(1,0;0)}(0)}(1)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\chi_{M(1,0;0)}(0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,2,,}2} approximately corresponds to \(\chi^1_{M(1,0;0)}(0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2} approximately corresponds to \(\chi^{\psi_{M(1,0;0)}(0)}_{M(1,0;0)}(0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{M(1,0;0)}(1)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_{M(1,0;0)}(1)}(2)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}3} approximately corresponds to \(\psi_{M(1,0;0)}(2)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(M(1,0;0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}1,,2} has recursion level \(\psi(M(1,0;0))\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}2{1,,1{1{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\psi_{M(1,0;1)}(0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}1,,3} approximately corresponds to \(M(1,0;1)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1 {1,,1{1{1,,1,,2,,}2}2,,}1 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi^{1,0}_{M(1,1;0)}(0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1 {1,,1{1{1,,1,,2,,}2}2,,}1 {1,,1{1{1,,1,,2,,}2}2,,}1 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi^{1,0}_{M(1,1;0)}(M(1,1;0))\)
 * {1{1,,2{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(M(1,1;0)\)
 * {1{1,,1{1{1,,1,,2,,}2}3,,}2} has recursion level \(\psi(\psi_{M(2,0;0)}(0))\)
 * {1{1,,1{1{1,,1,,2,,}2}3,,}2} approximately corresponds to \(\psi_{M(2,0;0)}(0)\)
 * {1{1,,1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2,,}2} has recursion level \(\psi(\psi_{M(1,0,0;0)}(0))\)
 * {1{1,,1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2,,}2} has recursion level \(\psi(\psi_{M(1,0,0,0;0)}(0))\)

Using a weakly compact
Now we need another ordinal collapsing function to diagonalize over \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\), and use a "large" ordinal as a diagonalizer - a weakly compact cardinal. Weakly compact cardinals have many equivalent definitions. The weakly compact cardinal is regular, weakly inaccessible, weakly Mahlo, \(\alpha\)-weakly Mahlo, \((\alpha,\beta)\)-weakly Mahlo, \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo, and so on. It's large enough for the new type of collapsing function.

Let \(K\) denote the weakly compact cardinal, \(\Omega_0=0\) and \(\Omega_\alpha\) is the \(\alpha\)-th uncountable cardinal. Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0,K\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\xi,\gamma)|\pi,\xi,\gamma\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ A(\alpha) &=& \{\beta<K|C(\alpha,\beta)\cap K\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A(\xi)\text{ is stationary in }\beta\} \\ \chi_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\).

Here, the \(\Omega_\gamma\) is added to the generation of \(C(\alpha,\beta)\) to prevent the notation stopping at \(K\omega\). Now both the \(\xi\) and the \(\alpha\) in \(\chi_\pi(\xi,\alpha)\) are in the induction, resulting a more complicated system. To go through it, we first look at \(\chi_\pi(0,0)\), then \(\chi_\pi(0,1)\), \(\chi_\pi(1,0)\) and \(\chi_\pi(1,1)\), then \(\chi_\pi(\xi,\alpha)\) for \(\xi,\alpha\le2\), and so on.

\(\psi_\pi(0)=1\) for \(\pi\le K\). For \(\pi>K\), we have \(K\),\(K2\), \(K3\), etc. so \(\psi_{\Omega_{K+1}}(0)=K\omega\), \(\psi_{\Omega_{K+2}}(0)=\Omega_{K+1}\omega\), \(\psi_{\Omega_{K+\omega+1}}(0)=\Omega_{K+\omega}\omega\), etc. \(A(0)\) is the set of uncountable regular cardinals \(\pi<K\) that ordinals \(<\pi\) cannot reach it by addition and \(\Omega_\gamma\), so \(A(0)\) is the set of weakly inaccessible cardinals below \(K\). Then \(\chi_M(0,0)=\chi_K(0,0)=I\).

\(\psi_I(1)\) equals the omega-fixed-point. \(\psi_{I_2}(1)=\psi_K(1)\) equals the next omega-fixed-point after \(I\). \(\chi_M(0,1)=\chi_K(0,1)=I_2\). Note that for \(\pi\le K\), \(\psi_K(\alpha)\) is always the largest one among \(\psi_\pi(\alpha)\), and so do \(\psi_K(\xi,\alpha)\). \(A(1)\) contains ordinal \(\beta\) such that \(A(0)\) is stationary in \(\beta\) (so \(\beta\) is weakly Mahlo) and ordinals \(<\beta\) cannot reach it by addition, \(\Omega_\gamma\), \(\chi_\pi(0,0)\) and \(\psi_\pi(0)\) (this condition seems useless currently), so \(A(1)\) is the set of weakly Mahlo cardinals below \(K\). Then \(\chi_K(1,0)=\chi_K(1,1)=M\) (note that \(C(1,\beta)\) doesn't contain \(\chi_\pi(1,0)\)).

Next, \(\psi_K(2)\) equals the next omega-fixed-point after \(M\). \(\chi_K(0,2)=I_{M+1}\) and \(\chi_K(1,2)=M_2\). \(\chi_M(0,2)=I_3\). \(A(2)\) is the set of 2-weakly Mahlos below \(K\), so \(\chi_K(2,0)=M(2;0)\), followed by \(\chi_K(2,1)=\chi_K(2,2)=M(2;0)\). This "initial platform" also happens on \(\chi_\pi(\xi,0)\) for other \(\pi\). For \(\alpha\le\xi\), \(C(\alpha,\beta)\) don't contain \(\chi_\pi(\xi,0)\).

\(A(\omega)\) is the set of \(\omega\)-weakly Mahlos below \(K\), so \(\chi_K(\omega,0)=M(\omega;0)\) (not \(\sup\{\chi_K(n,0)|n<\omega\}\), which is smaller and equal to \(\psi_K(\omega)\)).

Let \(M_@\) be the "\(\psi_{M(1,0;0)}(0)\) in previous section: Collapsing higher Mahloness" or \(\sup\{M,M(M;0),M(M(M;0);0),\cdots\}\), which has cofinality \(\omega\). \(\psi_K(M_@)=M_@\), \(\chi_K(M_@,0)=M(M_@;0)\), and \(\psi_K(M_@+1)=M_@\) gets stuck. Consider \(A(M_@+1)\). An ordinal \(\beta\) in it should be \(\gamma+1\)-weakly Mahlo for \(\gamma\in C(M_@+1,\beta)\cap(M_@+1)\). Also \(C(M_@+1,\beta)\cap K\subseteq\beta\), resulting \(\beta\) to be at least \(M_@\). It should be weakly inaccessible, so we need to set at least \(\beta=I_{M_@+1}\); but then (for \(\gamma=M_@\)) it should be \(M_@+1\)-weakly Mahlo. As a result, \(A(M_@+1)\) is the set of \(M_@+1\)-weakly Mahlos below \(K\) - the \(A(\alpha)\) doesn't get stuck here.

\(\chi_K(0,I(2,M_@+1)+1)=I(2,M_@+1)\), \(\chi_K(1,M(1,M_@+1)+1)=M(1,M_@+1)\), etc. higher Mahloness also get stuck gradually.

Next important step is \(M(1,0;0)\). \(\chi_K(M(1,0;0),0)=M(1,0;0)\), and it's first fixed point of \(\alpha\mapsto\chi_K(\alpha,0)\). In \(C(M(1,0;0)+1,\beta)\) we have 0, \(\chi_K(0,0)=I\), \(\chi_K(I,0)=M(I;0)\), \(\chi_K(\chi_K(I,0),0)=M(M(I;0);0)\), etc. For an ordinal \(\beta\) in \(A(M(1,0;0)+1)\), we need it \(\gamma+1\)-weakly Mahlo for \(\gamma\in C(M(1,0;0)+1,\beta)\cap(M(1,0;0)+1)\), so \(I\)-weakly Mahlo, \(M(I;0)\)-weakly Mahlo, and so on. That makes the least \(\beta=M(1,0;0)\) in \(A(M(1,0;0)+1)\), the same as \(\min A(M(1,0;0))\), and it's not \(M(1,0;0)+1\)-weakly Mahlo. But for next ordinal in \(A(M(1,0;0)+1)\), it's at least \(M(1,0;0)+1\), then it's require to be \(M(1,0;0)+1\)-weakly Mahlo. So \(A(M(1,0;0)+1)\) contains \(M(1,0;0)\), and \(M(1,0;0)+1\)-weakly Mahlos below \(K\). Further, the \(M(1,0;0)\) will remain in \(A(\alpha)\) for \(M(1,0;0)\le\alpha\le K\). \(\chi_K(M(1,0;0),M(1,0;0)+1)=M(1,0;0)\). \(\chi_K(M(1,0;0)+1,0)=M(1,0;0)\), \(\chi_K(M(1,0;0)+1,1)=M(1,0;0)\), ..., \(\chi_K(M(1,0;0)+1,M(1,0;0))=M(1,0;0)\), and \(\chi_K(M(1,0;0)+1,M(1,0;0)+1)=M(1,0;0)\). For \(\alpha\le K\), \(C(\alpha,M(1,0;0))\) always cannot reach \(M(1,0;0)\), and \(M(1,0;0)\in A(\alpha)\), so all the \(\psi_K(\alpha)\) and \(\chi_K(\xi,\alpha)\) get stuck here.

Next, \(A(M(1,0;0)+2)\) contains \(M(1,0;0)\), and \(M(1,0;0)+2\)-weakly Mahlos below \(K\). That continues up to \(M(1,0;1)\), and next \(A(M(1,0;1)+1)\) contains \(M(1,0;0)\), \(M(1,0;1)\), and \(M(1,0;1)+1\)-weakly Mahlos below \(K\). Continue upward, when \(\alpha\) increase, there'll be more ordinals remaining in \(A(\alpha)\), until \(K\). \(A(K)\) contains all the remains: \(M(1,0;0)\), \(M(1,0;1)\), \(M(1,0;2)\), etc. - that's exactly (1,0)-weakly Mahlo cardinals below \(K\).

\(\psi_K(K+1)\) equals next omega-fixed-point after \(M(1,0;0)\), \(\chi_K(0,K+1)=I_{M(1,0;0)+1}\), \(\chi_K(1,K+1)=M_{M(1,0;0)+1}\), ..., \(\chi_K(M(1,0;0),K+1)=M(M(1,0;0);1)\) - the 2nd \(M(1,0;0)\)-weakly Mahlo, \(\chi_K(M(1,0;0)+1,K+1)=M(M(1,0;0)+1;0)\) - first \(M(1,0;0)+1\)-weakly Mahlo, \(\chi_K(M(1,0;0)+2,K+1)=M(M(1,0;0)+2;0)\), ..., \(\chi_K(M(1,0;1),K+1)=M(1,0;1)\), \(\chi_K(M(1,0;1)+1,K+1)=M(1,0;1)\) (which still needs one more step to get unstuck), ..., \(\chi_K(K,K+1)=M(1,0;1)\). In \(C(K+1,\beta)\cap(K+1)\) we have \(K\), so ordinals in \(A(K)\) are stationary in ordinals in \(A(K+1)\). Then \(A(K+1)\) is the set of (1,1)-weakly Mahlos below \(K\), and \(\chi_K(K+1,0)=\chi_K(K+1,K+1)=M(1,1;0)\).

Next, \(\chi_K(0,K+2)=I_{M(1,1;0)+1}\), \(\chi_K(1,K+2)=M_{M(1,1;0)+1}\), ..., \(\chi_K(M(1,0;0),K+2)=M(M(1,0;0);M(1,1;0)+1)\), \(\chi_K(M(1,0;0)+1,K+2)=M(M(1,0;0)+1;M(1,1;0)+1)\), ..., \(\chi_K(M(1,1;0),K+2)=M(M(1,1;0);1)\), \(\chi_K(M(1,1;0)+1,K+2)=M(M(1,1;0)+1;0)\), ..., \(\chi_K(M(1,0;M(1,1;0)+1),K+2)=M(1,0;M(1,1;0)+1)\), \(\chi_K(M(1,0;M(1,1;0)+1)+1,K+2)=M(1,0;M(1,1;0)+1)\), ..., \(\chi_K(K,K+2)=M(1,0;M(1,1;0)+1)\), \(\chi_K(K+1,K+2)=M(1,1;1)\), \(A(K+2)\) is the set of (1,2)-weakly Mahlos below \(K\), and \(\chi_K(K+2,0)=\chi_K(K+2,K+2)=M(1,2;0)\).

Here come comparisons between this OCF and pDAN. In the comparisons shown below, I'll use 2 kinds of statements. "A has recursion level \(\alpha\)" means s(n,n A 2) has growth rate \(\omega^{\omega^\alpha}\), while "A approximately corresponds to \(\alpha\)" means the separator A works similar to ordinal \(\alpha\) in other separators.
 * {1{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(K)=M_@\)
 * {1,,2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\Omega_{\psi_K(K)+1}\)
 * {1,,1,,2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(0,\psi_K(K))\)
 * {1,,1,,3{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(0,\psi_K(K)+1)\)
 * {1,,1,,1,,2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(0,K)\)
 * {1{1,,2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(1,\psi_K(K))\)
 * {1{1,,2,,}1{1,,2,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(1,K)\)
 * {1{1,,3,,}2{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(2,\psi_K(K))\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_K(\psi_K(K),0)}(K+1)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1,,2 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_{\chi_K(\psi_K(K),0)}(0,K+1)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1{1,,2,,}2 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_{\chi_K(\psi_K(K),0)}(1,K+1)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1 {1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_K(\psi_K(K),0)}(K+2)\)
 * {1{1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1 {1,,1 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}1,,2 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(\psi_K(K),0)\)
 * {1{1,,2 {1{1,,1{1{1,,1,,2,,}2}2,,}2} 2,,}2 {1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(\psi_K(K)+1,0)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}3} approximately corresponds to \(\psi_{\chi_K(K,0)}(K+1)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\chi_{\chi_K(K,0)}(0,K+1)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,2,,}2} approximately corresponds to \(\chi_{\chi_K(K,0)}(1,K+1)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_K(K,0)}(K+2)\)
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\chi_K(K,0)=M(1,0;0)\)
 * {1{1,,2{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(K+1,0)\)
 * {1{1,,1{1{1,,1,,2,,}2}3,,}2} approximately corresponds to \(\psi_K(K2)\)
 * {1{1,,1{1{1,,1,,2,,}2}3,,}1{1,,1{1{1,,1,,2,,}2}3,,}1,,2} approximately corresponds to \(\chi_K(K2,0)\)
 * {1{1,,1{1{1,,1,,2,,}2}1,2,,}2} approximately corresponds to \(\psi_K(K\omega)\)
 * {1{1,,1{1{1,,1,,2,,}2}1,2,,}3} approximately corresponds to \(\psi_{\chi_K(K\omega,0)}(K\omega+1)\)
 * {1{1,,1{1{1,,1,,2,,}2}1,2,,}1,,2} approximately corresponds to \(\chi_{\chi_K(K\omega,0)}(0,K\omega+1)\)
 * {1{1,,1{1{1,,1,,2,,}2}1,2,,}1{1,,2,,}2} approximately corresponds to \(\chi_{\chi_K(K\omega,0)}(1,K\omega+1)\)
 * {1{1,,1{1{1,,1,,2,,}2}1,2,,}1{1,,1{1{1,,1,,2,,}2}1,2,,}2} approximately corresponds to \(\psi_{\chi_K(K\omega,0)}(K\omega+2)\)
 * {1{1,,1{1{1,,1,,2,,}2}1,2,,}1{1,,1{1{1,,1,,2,,}2}1,2,,}1,,2} approximately corresponds to \(\chi_K(K\omega,0)\)
 * {1{1,,2{1{1,,1,,2,,}2}1,2,,}2} approximately corresponds to \(\chi_K(K\omega+1,0)\)
 * {1{1,,1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(K^2)\)
 * {1{1,,1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2,,}1 {1,,1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2,,} 1,,2} approximately corresponds to \(\chi_K(K^2,0)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}2} has recursion level \(\psi(K^\omega)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(K^\omega)\)
 * {1,,1,,2{1,,1{2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(0,K^\omega)\)
 * {1{1,,2,,}2{1,,1{2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(1,K^\omega)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}3} has recursion level \(\psi(K^\omega+\psi_{\chi_K(K^\omega,0)}(K^\omega+1))\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}3} approximately corresponds to \(\psi_{\chi_K(K^\omega,0)}(K^\omega+1)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\chi_{\chi_K(K^\omega,0)}(0,K^\omega+1)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,2,,}2} approximately corresponds to \(\chi_{\chi_K(K^\omega,0)}(1,K^\omega+1)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_K(K^\omega,0)}(K^\omega+2)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\chi_K(K^\omega,0)\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}1,,2} has recursion level \(\psi(K^\omega+\chi_K(K^\omega,0))\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}1,,3} has recursion level \(\psi(K^\omega+\chi_K(K^\omega,K^\omega+1))=\psi(K^\omega+\chi_{\chi_K(K^\omega+1,0)}(K^\omega,K^\omega+1))\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}1,,1 {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}1,,2} 2} has recursion level \(\psi(K^\omega+\chi_{\chi_K(K^\omega+1,0)}(K^\omega,K^\omega+\chi_K(K^\omega,0)))\)
 * {1{1,,1{2{1,,1,,2,,}2}2,,}1{1,,1{2{1,,1,,2,,}2}2,,}1,,1,,2} has recursion level \(\psi(K^\omega+\chi_K(K^\omega+1,0))\)
 * {1{1,,2{2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(K^\omega+1,0)\)
 * {1{1,,1{2{1,,1,,2,,}2}3,,}2} approximately corresponds to \(\psi_K(K^\omega2)\)
 * {1{1,,1{2{1,,1,,2,,}2}1{2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(K^{\omega2})\)
 * {1{1,,1{3{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(K^{\omega^2})\)
 * {1{1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(K^K)\)
 * {1{1{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level \(\psi(K^K)\)
 * {1{1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}3} approximately corresponds to \(\psi_{\chi_K(K^K,0)}(K^K+1)\)
 * {1{1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}3} has recursion level \(\psi(K^K+\psi_{\chi_K(K^K,0)}(K^K+1))\)
 * {1{1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}1 {1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_{\chi_K(K^K,0)}(K^K+2)\)
 * {1{1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}1 {1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}1,,2} approximately corresponds to \(\chi_K(K^K,0)\)
 * {1{1,,2{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\chi_K(K^K+1,0)\)
 * {1{1,,1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}3,,}2} has recursion level \(\psi(K^K2)\)
 * {1{1,,1{2{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,}2} has recursion level \(\psi(K^{K\omega})\)
 * {1{1{1,,1,,2,,}2}3{1,,1,,2,,}2} has recursion level \(\psi(K^{K^2})\)
 * {1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level \(\psi(K^{K^K})\)
 * {1{1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level \(\psi(K^{K^{K^K}})\)
 * {1,,2{1,,1,,2,,}2} has recursion level \(\psi(\Omega_{K+1})\)
 * {1{1,,1{1,,2{1,,1,,2,,}2}2,,}2} approximately corresponds to \(\psi_K(\Omega_{K+1})\)
 * {1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} approximately corresponds to \(\psi_{\Omega_{K+1}}(\Omega_{K+1})\)
 * {1,,2{1,,1,,2,,}2} approximately corresponds to \(\Omega_{K+1}\)
 * {1,,3{1,,1,,2,,}2} has recursion level \(\psi(\Omega_{K+2})\)
 * {1,,1{1{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level \(\psi(\Omega_{K2})\)
 * {1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level \(\psi(\Omega_{\Omega_{K+1}})\)
 * {1,,1{1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level \(\psi(\Omega_{\Omega_{\Omega_{K+1}}})\)

Further guess
Ordinal collapsing functions beyond that are too complicated for me to understand. So I only have some guess.

Here are previous correspondence: Here's one guess, fitting Taranovsky's correspondence: Here's another guess, not fitting Taranovsky's correspondence:
 * {1,,1,,2} - first recursively inaccessible, or first weakly inaccessible
 * s(n,n{1,,2,,2}2) - the growth rate limit in KP + "there exists a recursively inaccessible"
 * {1,,1,,3} - 2nd recursively inaccessible, or 2nd weakly inaccessible
 * {1,,1,,1,,2} - first recursively 2-inaccessible, or first weakly 2-inaccessible
 * {1{2,,}1,,2} - first recursively \(\omega\)-inaccessible, or first weakly \(\omega\)-inaccessible
 * {1{1{1{1,,2,,}2}2,,}1,,2} - first recursively (1,0)-inaccessible, or first weakly (1,0)-inaccessible
 * {1{1,,2,,}2} - first recursively Mahlo, or first weakly Mahlo
 * s(n,n{1,,2{1,,2,,}2}2) - the growth rate limit in KP + "there exists a recursively Mahlo"
 * {1{1,,2,,}3} - 2nd recursively Mahlo, or 2nd weakly Mahlo
 * {1{1,,2,,}1,,2} - first recursively inaccessible limit of recursively Mahlos
 * {1{1,,2,,}1,,1,,2} - first recursively inaccessible limit recursively inaccessible limit of recursively Mahlos
 * {1{1,,2,,}1{1,,2,,}2} - first recursively Mahlo limit of recursively Mahlos
 * {1{1,,2,,}1{1,,2,,}1{1,,2,,}2} - first recursively Mahlo limit of recursively Mahlo limit of recursively Mahlos
 * {1{1,,3,,}2} - first recursively 2-Mahlo, or first weakly 2-Mahlo (weakly Mahlo in which weakly Mahlos are stationary)
 * {1{1,,3,,}1{1,,3,,}2} - first recursively 2-Mahlo limit of recursively 2-Mahlos
 * {1{1,,4,,}2} - first recursively 3-Mahlo, or first weakly 3-Mahlo
 * {1{1,,1,2,,}1{1,,1,2,,}1,,2} - first recursively \(\omega\)-Mahlo, or first weakly \(\omega\)-Mahlo
 * {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}1,,2} - first recursively (1,0)-Mahlo, or first weakly (1,0)-Mahlo
 * {1{1,,1,,2,,}2} - first \(\Pi_3\)-reflecting, or first weakly compact
 * s(n,n{1,,2{1,,1,,2,,}2}2) - the growth rate limit in KP + \(\Pi_3\)-reflection
 * {1{1,,1,,2,,}3} - 2nd \(\Pi_3\)-reflecting
 * {1{1,,1,,2,,}1,,2} - first recursively inaccessible limit of \(\Pi_3\)-reflectings
 * {1{1,,1,,2,,}1,,1,,2} - first recursively inaccessible limit of recursively inaccessible limit of \(\Pi_3\)-reflectings
 * {1{1,,1,,2,,}1{1,,2,,}2} - first recursively Mahlo limit of \(\Pi_3\)-reflectings
 * {1{1,,1,,2,,}1{1,,3,,}2} - first recursively 2-Mahlo limit of \(\Pi_3\)-reflectings
 * {1{1,,1,,2,,}1{1,,1,,2,,}2} - first \(\Pi_3\)-reflecting limit of \(\Pi_3\)-reflectings
 * {1{1,,2,,2,,}2} - first \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * {1{1,,3,,2,,}2} - first \(\Pi_2\)-reflecting onto \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * {1{1,,1,,3,,}2} - first \(\Pi_3\)-reflecting onto \(\Pi_3\)-reflectings
 * {1{1,,1,,1,,2,,}2} - first \(\Pi_4\)-reflecting, or first \(\Pi^1_2\)-indescribable
 * s(n,n{1,,2{1,,1,,1,,2,,}2}2) - the growth rate limit in KP + \(\Pi_4\)-reflection
 * {1{1,,1,,1,,2,,}1,,2} - first recursively inaccessible limit of \(\Pi_4\)-reflectings
 * {1{1,,1,,1,,2,,}1{1,,2,,}2} - first recursively Mahlo limit of \(\Pi_4\)-reflectings
 * {1{1,,1,,1,,2,,}1{1,,1,,2,,}2} - first \(\Pi_3\)-reflecting limit of \(\Pi_4\)-reflectings
 * {1{1,,1,,1,,2,,}1{1,,1,,1,,2,,}2} - first \(\Pi_4\)-reflecting limit of \(\Pi_4\)-reflectings
 * {1{1,,2,,1,,2,,}2} - first \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
 * {1{1,,1,,2,,2,,}2} - first \(\Pi_3\)-reflecting onto \(\Pi_4\)-reflectings
 * {1{1,,1,,1,,3,,}2} - first \(\Pi_4\)-reflecting onto \(\Pi_4\)-reflectings
 * {1{1,,1,,1,,1,,2,,}2} - first \(\Pi_5\)-reflecting, or first \(\Pi^1_3\)-indescribable
 * s(n,n{1,,2{1,,1,,1,,1,,2,,}2}2) - the growth rate limit in KP + \(\Pi_5\)-reflection
 * {1{1,,1,,1,,1,,1,,2,,}2} - first \(\Pi_6\)-reflecting, or first \(\Pi^1_4\)-indescribable
 * s(n,n{1,,2{1,,1,,1,,1,,1,,2,,}2}2) - the growth rate limit in KP + \(\Pi_6\)-reflection
 * s(x,x{1{1{2,,}2,,}2}2) - the growth rate limit in KP + \(\Pi_n\)-reflection
 * {1{1,,1,,2,,}3} - 2nd weakly compact
 * {1{1,,1,,2,,}1,,2} - first weakly inaccessible limit of weakly compacts
 * {1{1,,1,,2,,}1,,1,,2} - first weakly inaccessible limit of weakly inaccessible limit of weakly compacts
 * {1{1,,1,,2,,}1{1,,2,,}2} - first weakly Mahlo limit of weakly compacts
 * {1{1,,1,,2,,}1{1,,3,,}2} - first weakly 2-Mahlo limit of weakly compacts
 * {1{1,,1,,2,,}1{1,,1,,2,,}2} - first weakly compact limit of weakly compacts
 * {1{1,,2,,2,,}2} - first weakly Mahlo in which weakly compacts are stationary
 * {1{1,,3,,2,,}2} - first weakly Mahlo in which weakly Mahlos in which weakly compacts are stationary are stationary
 * {1{1,,1,,3,,}2} - first weakly compact in which weakly compacts are stationary
 * {1{1,,1,,4,,}2} - first weakly compact in which weakly compacts in which weakly compacts are stationary are stationary
 * {1{1,,1,,1,,2,,}2} - first weakly compact with "weakly compact condition" over weakly compacts
 * {1{1,,1,,1,,1,,2,,}2} - first weakly compact with "weakly compact condition" over weakly compacts with "weakly compact condition" over weakly compacts
 * {1{1{1,,2,,}2,,}2} - first \(\Pi_4\)-reflecting, or first \(\Pi^1_2\)-indescribable
 * s(n,n{1,,2{1{1,,2,,}2,,}2}2) - the growth rate limit in KP + \(\Pi_4\)-reflection
 * {1{1{1,,2,,}2,,}1,,2} - first weakly inaccessible limit of \(\Pi^1_2\)-indescribables
 * {1{1{1,,2,,}2,,}1{1,,2,,}2} - first weakly Mahlo limit of \(\Pi^1_2\)-indescribables
 * {1{1{1,,2,,}2,,}1{1,,1,,2,,}2} - first weakly compact limit of \(\Pi^1_2\)-indescribables
 * {1{1{1,,2,,}2,,}1{1{1,,2,,}2,,}2} - first \(\Pi^1_2\)-indescribable limit of \(\Pi^1_2\)-indescribables
 * {1{1,,2{1,,2,,}2,,}2} - first weakly Mahlo in which \(\Pi^1_2\)-indescribables are stationary
 * {1{1,,1,,2{1,,2,,}2,,}2} - first weakly compact in which \(\Pi^1_2\)-indescribables are stationary
 * {1{1,,1{1,,2,,}3,,}2} - first \(\Pi^1_2\)-indescribable in which \(\Pi^1_2\)-indescribables are stationary
 * {1{1,,1{1,,2,,}1,,2,,}2} - first weakly compact with "weakly compact condition" over \(\Pi^1_2\)-indescribables
 * {1{1,,1{1,,2,,}1{1,,2,,}2,,}2} - first \(\Pi^1_2\)-indescribable with "weakly compact condition" over \(\Pi^1_2\)-indescribables
 * {1{1,,1{1,,3,,}2,,}2} - first \(\Pi^1_2\)-indescribable with "\(\Pi^1_2\)-indescribable condition" over \(\Pi^1_2\)-indescribables
 * {1{1,,1{1,,1,,2,,}2,,}2} - first \(\Pi_5\)-reflecting, or first \(\Pi^1_3\)-indescribable
 * s(n,n{1,,2{1{1,,1,,2,,}2,,}2}2) - the growth rate limit in KP + \(\Pi_5\)-reflection
 * {1{1,,1{1{1,,2,,}2,,}2,,}2} - first \(\Pi_6\)-reflecting, or first \(\Pi^1_4\)-indescribable
 * s(n,n{1,,2{1{1{1,,2,,}2,,}2,,}2}2) - the growth rate limit in KP + \(\Pi_6\)-reflection
 * s(x,x{1{1`,,2,,}2}2) - the growth rate limit in KP + \(\Pi_n\)-reflection

Using weakly compacts
Let \(K_0=0\), \(K_{\alpha+1}\) be the next weakly compact cardinal after \(K_\alpha\), and \(K_\alpha=\sup\{K_\beta|\beta<\alpha\}\) for limit ordinal \(\alpha\). Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{K_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\xi,\gamma)|\pi,\xi,\gamma\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ A_\pi(\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A_\pi(\xi)\text{ is stationary in }\beta\} \\ \chi_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A_\pi(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\) (first uncountable cardinal), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(K\) is a shorthand for \(K_1\).

Inaccessibility over weakly compacts
To express ordinals beyond the "weakly compact-fixed-point" \(K_*\), we need a "weakly inaccessible cardinal which is a limit of weakly compact cardinals", followed by higher inaccessibility. A "weakly Mahlo cardinal which is a limit of weakly compact cardinals" can collapse and express all those inaccessibility. Then higher Mahloness can be expressed by a "weakly compact cardinal which is a limit of weakly compact cardinals" - denoted by \(K(1,0)\).

Further, let \(K(\alpha_1,\alpha_2,\cdots,\alpha_n,\beta,0,0,\cdots,0,0)\) be the first weakly compact cardinal \(\pi\) such that \(\pi=K(\alpha_1,\alpha_2,\cdots,\alpha_n,\delta,\pi,0,\cdots,0,0)\) for all \(\delta<\beta\), \(K(\alpha_1,\alpha_2,\cdots,\alpha_n,\beta,0,0,\cdots,0,\gamma+1)\) be the next weakly compact cardinal \(\pi\) such that \(\pi=K(\alpha_1,\alpha_2,\cdots,\alpha_n,\delta,\pi,0,\cdots,0,0)\) for all \(\delta<\beta\) after \(K(\alpha_1,\alpha_2,\cdots,\alpha_n,\beta,0,0,\cdots,0,\gamma)\), and \(K(\alpha_1,\alpha_2,\cdots,\alpha_n,\beta)=\sup\{K(\alpha_1,\alpha_2,\cdots,\alpha_n,\delta)|\delta<\beta\}\) for limit ordinal \(\beta\). This definition is similar to \(I(\alpha_1,\alpha_2,\cdots,\alpha_n,\beta)\), except "uncountable regular cardinal" is substituted into "weakly compact cardinal", so it's called "inaccessibility over weakly compacts". \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{K(\gamma_1,\gamma_2,\cdots,\gamma_k,\delta)|\gamma_1,\gamma_2,\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\xi,\gamma)|\pi,\xi,\gamma\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ A_\pi(\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A_\pi(\xi)\text{ is stationary in }\beta\} \\ \chi_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A_\pi(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\) (first uncountable cardinal), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(K\) is a shorthand for \(K(0)\).

Mahloness over weakly compacts
The collapsing function using an "ordinal such that the set of weakly compact cardinals in it are stationary" can express all the \(K(\alpha_1,\alpha_2,\cdots,\alpha_n,\beta)\). There are still ordinals such that the set of the previous kind of ordinals in it are stationary. And all these Mahloness can be expressed by the collapsing of a weakly compact cardinal such that the set of weakly compact cardinals in it are stationary.

Now Mahloness over weakly compacts comes in. An ordinal \(\pi\) is "\(\alpha\)-Mahlo over weakly compacts" if it's weakly compact and the set of "\(\gamma\)-Mahlo over weakly compacts" ordinals in \(\pi\) are stationary for all \(\gamma<\alpha\). (So 0-Mahlo over weakly compacts are just weakly compacts) Next, an ordinal \(\pi\) is "(1,0)-Mahlo over weakly compacts" if it's "\(\pi\)-Mahlo over weakly compacts". Generally, ordinal \(\pi\) is "\((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-Mahlo over weakly compacts" if it's "\((\alpha_1,\alpha_2,\cdots,\alpha_i,\beta,\pi,\underbrace{0,\cdots,0}_{n-i-2})\)-Mahlo over weakly compacts" for all \(\beta<\alpha_{i+1}\) and \(0\le i\le n-2\), and the set of "\((\alpha_1,\alpha_2,\cdots,\alpha_{n-1},\beta)\)-Mahlo over weakly compacts" in it are stationary for all \(\beta<\alpha_n\).

Let \(K(\alpha_1,\alpha_2,\cdots,\alpha_n;0)\) be the first "\((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-Mahlo over weakly compacts" ordinal, \(K(\alpha_1,\alpha_2,\cdots,\alpha_n;\beta+1)\) be the next "\((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-Mahlo over weakly compacts" ordinal after \(K(\alpha_1,\alpha_2,\cdots,\alpha_n;\beta)\), and \(K(\alpha_1,\alpha_2,\cdots,\alpha_n;\beta)=\sup\{K(\alpha_1,\alpha_2,\cdots,\alpha_n;\delta)|\delta<\beta\}\) for limit ordinal \(\beta\). And let \(M^{\alpha_1,\alpha_2,\cdots,\alpha_n}\) be the set of "\((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-Mahlo over weakly compacts" ordinals \(<\sup\{K(1,\underbrace{0,\cdots,0,0}_n)|n<\omega\}\). \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{K(\gamma_1,\gamma_2,\cdots,\gamma_k;\delta)|\gamma_1,\gamma_2,\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{X^{\gamma_1,\gamma_2,\cdots,\gamma_k}_\pi(\delta)|\gamma_1,\gamma_2,\cdots,\gamma_k,\pi,\delta\in C_n(\alpha,\beta)\wedge\delta<\alpha\} \\ &\cup& \{\chi_\pi(\xi,\gamma)|\pi,\xi,\gamma\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ X^{\gamma_1,\gamma_2,\cdots,\gamma_n}_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in M^{\gamma_1,\gamma_2,\cdots,\gamma_n}\}\cup\{\pi\}) \\ A_\pi(\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A_\pi(\xi)\text{ is stationary in }\beta\} \\ \chi_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A_\pi(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega\) is first uncountable cardinal, \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(K\) is a shorthand for \(K(0;0)\).

Collapsing higher weak compactness
The weakly compact cardinal is a "focus point" with many definitions, but not all of them can extend to higher "weak compactness" that suits the use of ordinal collapsing functions. Here, we use the definition of \(\Pi_1^1\)-indescribability, and extend it to higher weak compactness and \((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-weakly compact cardinals.

Ordinal \(\pi\) is \(\Pi_1^1\)-indescribable if for every \(\Pi_1\) first-order unary formula \(\phi\), \(\forall S\in V_{\pi+1}(V_{\pi+1}\models\phi(S)\rightarrow\exists\alpha\in\pi(V_{\alpha+1}\models\phi(S\cap V_\alpha)))\); \(\pi\) is \(\Pi_1^1\)-indescribable over set \(A\) if for every \(\Pi_1\) first-order unary formula \(\phi\), \(\forall S\in V_{\pi+1}(V_{\pi+1}\models\phi(S)\rightarrow\exists\alpha\in A\cap\pi(V_{\alpha+1}\models\phi(S\cap V_\alpha)))\).

Next, ordinal \(\pi\) is \(\alpha\)-weakly compact if it's \(\Pi_1^1\)-indescribable over \(\gamma\)-weakly compact ordinals for all \(\gamma<\alpha\). So 0-weakly compact ordinals are just ordinals, and 1-weakly compact ordinals are normal weakly compact cardinals; the collapsing of the least 2-weakly compact ordinals can express all the \(X^{\gamma_1,\gamma_2,\cdots,\gamma_n}_\pi(\alpha)\) in "Mahloness over weakly compacts". Generally, ordinal \(\pi\) is \((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-weakly compact if it's \((\alpha_1,\alpha_2,\cdots,\alpha_i,\gamma,\pi,\underbrace{0,\cdots,0}_{n-i-2})\)-weakly compact for all \(\gamma<\alpha_{i+1}\) and \(0\le i\le n-2\), and it's \(\Pi_1^1\)-indescribable over the set of \((\alpha_1,\alpha_2,\cdots,\alpha_{n-1},\gamma)\)-weakly compact ordinals for all \(\gamma<\alpha_n\).

Let \(B^0\) be the set of uncountable regular cardinals below \(\alpha\), \(B^{\alpha_1,\alpha_2,\cdots,\alpha_n}\) be the set of \((\alpha_1,\alpha_2,\cdots,\alpha_n)\)-weakly compact ordinals below \(\alpha=\sup\{\min B^{1,\underbrace{0,0\cdots,0}_n}|n<\omega\}\). Let \(K[\alpha_1,\alpha_2,\cdots,\alpha_n,0]\) be the least ordinal in \(B^{\alpha_1,\alpha_2,\cdots,\alpha_n}\), \(K[\alpha_1,\alpha_2,\cdots,\alpha_n,\beta+1]\) be the next ordinal in \(B^{\alpha_1,\alpha_2,\cdots,\alpha_n}\) after \(K[\alpha_1,\alpha_2,\cdots,\alpha_n,\beta]\), and \(K[\alpha_1,\alpha_2,\cdots,\alpha_n,\beta]=\sup\{K[\alpha_1,\alpha_2,\cdots,\alpha_n,\gamma]|\gamma<\beta\}\) for limit ordinal \(\beta\). \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{K[\gamma_1,\gamma_2,\cdots,\gamma_k,\delta]|\gamma_1,\gamma_2,\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Psi^{\gamma_1,\gamma_2,\cdots,\gamma_k}_\pi(\xi,\delta)|\gamma_1,\gamma_2,\cdots,\gamma_k,\pi,\xi,\delta\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\delta<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ A^{\gamma_1,\gamma_2,\cdots,\gamma_n}_\pi(\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in B^{\gamma_1,\gamma_2,\cdots,\gamma_n} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A^{\gamma_1,\gamma_2,\cdots,\gamma_n}_\pi(\xi)\text{ is stationary in }\beta\} \\ \Psi^{\gamma_1,\gamma_2,\cdots,\gamma_n}_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2,\cdots,\gamma_n}_\pi(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega=K[0,0]\) is first uncountable cardinal, \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(K\) is a shorthand for \(K[1,0]\).

Using \(\Pi_2^1\)-indescribables
A \(\Pi_m^n\)-indescribable cardinal is an ordinal \(\pi\) such that for every \(\Pi_m\) first-order unary formula \(\phi\), \(\forall S\in V_{\pi+1}(V_{\pi+n}\models\phi(S)\rightarrow\exists\alpha\in\pi(V_{\alpha+n}\models\phi(S\cap V_\alpha)))\).

Let \(\kappa_0=0\), \(\kappa_{\alpha+1}\) be the next \(\Pi_2^1\)-indescribable cardinal after \(\kappa_\alpha\), and \(\kappa_\alpha=\sup\{\kappa_\beta|\beta<\alpha\}\) for limit ordinal \(\alpha\). Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\kappa_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\Psi_\pi(\nu,\xi,\gamma)|\pi,\nu,\xi,\gamma\in C_n(\alpha,\beta)\wedge\nu,\xi,\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ B_\pi(\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\nu\in C(\alpha,\beta)\cap\alpha)\beta\text{ is }\Pi_1^1\text{-indescribable over }B_\pi(\nu)\} \\ A_\pi(\nu,\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in B_\pi(\nu) \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A_\pi(\nu,\xi)\text{ is stationary in }\beta\} \\ \Psi_\pi(\nu,\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A_\pi(\nu,\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*} And \(\Omega\) is first uncountable cardinal, \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(\kappa\) is a shorthand for \(\kappa_1\).

Using \(\Pi_n^1\)-indescribables
Now define indescribability over sets. Ordinal \(\pi\) is \(\Pi_m^n\)-indescribable over set \(A\) (denoted \(\pi\in\Pi_m^n[A]\)) if for every \(\Pi_m\) first-order unary formula \(\phi\), \(\forall S\in V_{\pi+1}(V_{\pi+n}\models\phi(S)\rightarrow\exists\alpha\in A\cap\pi(V_{\alpha+n}\models\phi(S\cap V_\alpha)))\).

Let \(\varpi_n\) be the least \(\Pi_n^1\)-indescribable cardinal. Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\}\cup\{\varpi_n|1\le n<\omega\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Phi_\pi(\gamma_1,\gamma_2,\cdots,\gamma_k)|\pi,\gamma_1,\gamma_2,\cdots,\gamma_k\in C_n(\alpha,\beta)\wedge\gamma_1,\gamma_2,\cdots,\gamma_k<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ P^m_\pi(\alpha) &=& \{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)\beta\in\Pi_m^1[P^m_\pi(\xi)]\} \\ P^m_\pi(\alpha_1,\alpha_2,\cdots,\alpha_n) &=& \{\beta<\pi|C(\alpha_n,\beta)\cap\pi\subseteq\beta\wedge\beta\in P^{m+1}_\pi(\alpha_1,\alpha_2,\cdots,\alpha_{n-1}) \\ & & \wedge(\forall\xi\in C(\alpha_n,\beta)\cap\alpha_n)\beta\in\Pi_m^1[P^m_\pi(\alpha_1,\alpha_2,\cdots,\alpha_{n-1},\xi)]\} \\ \Phi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \\ \Phi_\pi(\alpha_1,\alpha_2,\cdots,\alpha_n) &=& \min(\{\beta<\pi|C(\alpha_n,\beta)\cap\pi\subseteq\beta\wedge\beta\in P^1_\pi(\alpha_1,\alpha_2,\cdots,\alpha_{n-1})\}\cup\{\pi\}) \end{eqnarray*} And \(\Phi(\alpha)\) is a shorthand for \(\Phi_{\varpi_1}(\alpha)\).