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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.

Fundamental sequences: Let \(\iota\) be an initial ordinal, \(\lambda\) be the limit of all ordinals the OCF uses. Define the following for every OCF:

  • \(L(\alpha)=\min\{n<\omega|\alpha\in C_n(\lambda,\iota)\}\)
  • \(\alpha[n]=\max\{\beta<\alpha|L(\beta)\le L(\alpha)+n\}\)

where

  • \((\lambda,\iota)=(\varepsilon_{\Omega+1},0)\) in "Bachmann's \(\psi\)"
  • \((\lambda,\iota)=(\sup\{\Omega,\Omega_\Omega,\Omega_{\Omega_\Omega},\cdots\},2)\) in "Collapsing higher cardinality"
  • \((\lambda,\iota)=(\sup\{I+1,\Omega_{I+1},\Omega_{\Omega_{I+1}},\cdots\},0)\) in "Using a weakly inaccessible"
  • \((\lambda,\iota)=(\sup\{I,I_I,I_{I_I},\cdots\},2)\) in "Using weakly inaccessibles"
  • \((\lambda,\iota)=(\sup\{I(1,\underbrace{0,0\cdots,0}_n)|n<\omega\},0)\) in "Collapsing higher inaccessibility" (only works up to \(\psi(I(1,0,0\cdots,0))\))
  • \((\lambda,\iota)=(\sup\{M,M_M,M_{M_M},\cdots\},2)\) in "Using weakly Mahlos"
  • \((\lambda,\iota)=(\sup\{M(1,\underbrace{0,0\cdots,0}_n)|n<\omega\},0)\) in "Inaccessibility over weakly Mahlos" (only works up to \(\psi(M(1,0,0\cdots,0))\))

Bachmann's \(\psi\)

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} \\ \psi(\Omega2+\psi(\Omega)) &=& \varepsilon_{\zeta_1+\zeta_0} \\ \psi(\Omega2+\psi(\Omega)+1) &=& \varepsilon_{\zeta_1+\zeta_0+1} \\ \psi(\Omega2+\psi(\Omega2)) &=& \varepsilon_{\zeta_12} \\ \psi(\Omega2+\psi(\Omega2)+1) &=& \varepsilon_{\zeta_12+1} \\ \psi(\Omega2+\psi(\Omega2+1)) &=& \varepsilon_{\varepsilon_{\zeta_1+1}} \\ \psi(\Omega2+\zeta_2) &=& \zeta_2 \\ \psi(\Omega3) &=& \zeta_2 \\ \psi(\Omega3+1) &=& \varepsilon_{\zeta_2+1} \\ \psi(\Omega4) &=& \zeta_3 \\ \psi(\omega^{\Omega+1}) &=& \zeta_\omega \\ \psi(\omega^{\Omega+\varepsilon_0}) &=& \zeta_{\varepsilon_0} \\ \psi(\omega^{\Omega+\psi(\Omega)}) &=& \zeta_{\zeta_0} \\ \psi(\omega^{\Omega+\psi(\Omega2)}) &=& \zeta_{\zeta_1} \\ \psi(\omega^{\Omega+\psi(\omega^{\Omega+1})}) &=& \zeta_{\zeta_\omega} \\ \psi(\omega^{\Omega+\varphi(3,0)}) &=& \varphi(3,0) \\ \psi(\omega^{\Omega2}) &=& \varphi(3,0) \\ \psi(\omega^{\Omega2}+1) &=& \varepsilon_{\varphi(3,0)+1} \\ \psi(\omega^{\Omega2}+\psi(\omega^{\Omega2})) &=& \varepsilon_{\varphi(3,0)2} \\ \psi(\omega^{\Omega2}+\Omega) &=& \zeta_{\varphi(3,0)+1} \\ \psi(\omega^{\Omega2}+\omega^{\Omega+1}) &=& \zeta_{\varphi(3,0)+\omega} \\ \psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2})}) &=& \zeta_{\varphi(3,0)2} \\ \psi(\omega^{\Omega2}2) &=& \varphi(3,1) \\ \psi(\omega^{\Omega2+1}) &=& \varphi(3,\omega) \\ \psi(\omega^{\Omega3}) &=& \varphi(4,0) \\ \psi(\omega^{\Omega4}) &=& \varphi(5,0) \\ \psi(\omega^{\omega^{\Omega+1}}) &=& \varphi(\omega,0) \\ \psi(\omega^{\omega^{\Omega+1}}+1) &=& \varepsilon_{\varphi(\omega,0)+1} \\ \psi(\omega^{\omega^{\Omega+1}}+\Omega) &=& \zeta_{\varphi(\omega,0)+1} \\ \psi(\omega^{\omega^{\Omega+1}}+\omega^{\Omega2}) &=& \varphi(3,\varphi(\omega,0)+1) \\ \psi(\omega^{\omega^{\Omega+1}}2) &=& \varphi(\omega,1) \\ \psi(\omega^{\omega^{\Omega+1}+\Omega}) &=& \varphi(\omega+1,0) \\ \psi(\omega^{\omega^{\Omega+1}2}) &=& \varphi(\omega2,0) \\ \psi(\omega^{\omega^{\Omega+2}}) &=& \varphi(\omega^2,0) \\ \psi(\omega^{\omega^{\Omega+\varepsilon_0}}) &=& \varphi(\varepsilon_0,0) \\ \psi(\omega^{\omega^{\Omega+\Gamma_0}}) &=& \Gamma_0 \\ \psi(\omega^{\omega^{\Omega2}}) &=& \Gamma_0 \\ \psi(\omega^{\omega^{\Omega2}}+1) &=& \varepsilon_{\Gamma_0+1} \\ \psi(\omega^{\omega^{\Omega2}}+\Omega) &=& \zeta_{\Gamma_0+1} \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+1}}) &=& \varphi(\omega,\Gamma_0+1) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\Gamma_0}}) &=& \varphi(\Gamma_0,1) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\Gamma_0}}2) &=& \varphi(\Gamma_0,2) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\Gamma_0}+1}) &=& \varphi(\Gamma_0,\omega) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\Gamma_0}+\Omega}) &=& \varphi(\Gamma_0+1,0) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\Gamma_0}2}) &=& \varphi(\Gamma_02,0) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\Gamma_0+1}}) &=& \varphi(\omega^{\Gamma_0+1},0) \\ \psi(\omega^{\omega^{\Omega2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega2}}+1)}}) &=& \varphi(\varepsilon_{\Gamma_0+1},0) \\ \psi(\omega^{\omega^{\Omega2}}2) &=& \Gamma_1 \\ \psi(\omega^{\omega^{\Omega2}+1}) &=& \Gamma_\omega \\ \psi(\omega^{\omega^{\Omega2}+\Omega}) &=& \varphi(1,1,0) \\ \psi(\omega^{\omega^{\Omega2}+\Omega2}) &=& \varphi(1,2,0) \\ \psi(\omega^{\omega^{\Omega2}+\omega^{\Omega+1}}) &=& \varphi(1,\omega,0) \\ \psi(\omega^{\omega^{\Omega2}2}) &=& \varphi(2,0,0) \\ \psi(\omega^{\omega^{\Omega2}2}2) &=& \varphi(2,0,1) \\ \psi(\omega^{\omega^{\Omega2}2+\Omega}) &=& \varphi(2,1,0) \\ \psi(\omega^{\omega^{\Omega2}3}) &=& \varphi(3,0,0) \\ \psi(\omega^{\omega^{\Omega2+1}}) &=& \varphi(\omega,0,0) \\ \psi(\omega^{\omega^{\Omega3}}) &=& \varphi(1,0,0,0) \\ \psi(\omega^{\omega^{\Omega4}}) &=& \varphi(1,0,0,0,0) \\ \psi(\omega^{\omega^{\omega^{\Omega+1}}}) &=& \text{SVO} \\ \psi(\omega^{\omega^{\omega^{\Omega2}}}) &=& \text{LVO} \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.

Collapsing higher cardinality

This notation is a step toward User:Deedlit11's ordinal notations, where \(\Omega_0=0\), \(\Omega_\alpha\) is the \(\alpha\)-th uncountable cardinal, and \(\pi\) represents uncountable regular cardinals (at this point, uncountable regular cardinals can be written as \(\Omega_{\alpha+1}\)). \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ 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_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\).

Compared with Bachmann's \(\psi\), this notation drops \(\omega^\gamma\), which doesn't modify the strength at this point; it adds \(\Omega_\gamma\), to get higher cardinalities; it drops \(\Omega\) in \(C_0(\alpha,\beta)\) because it already has \(\Omega_\gamma\). To go through this notation, the \(\alpha\) in \(\psi_\pi(\alpha)\) is the main index but the \(\pi\) is not. We can first go through \(\psi_\pi(0)\), then \(\psi_\pi(1)\), \(\psi_\pi(2)\), etc.

\(C(0,0)=\{0\}\), so we can set \(\beta=1\), and thus \(\psi_\pi(0)=1\).

\(C(1,0)\) contains all natural number n, \(\Omega_n\), \(\Omega_nn\), \(\Omega_{\Omega_n}\), and so on. Then \(\psi_\pi(1)\) depends on the \(\pi\). \(\psi(1)=\omega\), \(\psi_{\Omega_2}(1)=\Omega\omega\), \(\psi_{\Omega_3}(1)=\Omega_2\omega\), etc. For the term \(\psi_{\Omega_{\omega+1}}(1)\), although \(\Omega_\omega\notin C(1,0)\), when we set \(\beta=\Omega_\omega\), then \(\Omega_\omega\in C(1,\Omega_\omega)\), and we need \(\beta=\Omega_\omega\omega\) to get \(\psi_{\Omega_{\omega+1}}(1)\).

Next, \(\psi(2)=\omega^2\), \(\psi_{\Omega_2}(2)=\Omega\omega^2\), \(\psi_{\Omega_{\omega+1}}(2)=\Omega_\omega\omega^2\), etc.

\(\psi(\varepsilon_0+1)=\varepsilon_0=\psi(\varepsilon_0)\); however, for \(\psi_{\Omega_2}(\varepsilon_0+1)\) we cannot set \(\beta=\varepsilon_0\) because \(\Omega\in C(\varepsilon_0+1,\varepsilon_0)\), and further \(\Omega\varepsilon_0\in C(\varepsilon_0+1,\Omega)\), and finally \(\psi_{\Omega_2}(\varepsilon_0+1)=\Omega\varepsilon_0\omega\). At this point, \(\psi_\Omega\) gets stuck but \(\psi_\pi\) with higher cardinalities don't.

Continue upwards, we have \(\psi_{\Omega_2}(\Omega)=\Omega^2\), \(\psi_{\Omega_2}(\Omega2)=\Omega^3\), \(\psi_{\Omega_2}(\psi_{\Omega_2}(1))=\psi_{\Omega_2}(\Omega\omega)=\Omega^\omega\), \(\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(1)))=\psi_{\Omega_2}(\Omega^\omega)=\Omega^{\Omega^\omega}\), \(\psi_{\Omega_2}(\varepsilon_{\Omega+1})=\varepsilon_{\Omega+1}\) (and \(\psi(\varepsilon_{\Omega+1})=\text{BHO}\)).

Next, \(\psi_{\Omega_2}(\varepsilon_{\Omega+1}+1)=\varepsilon_{\Omega+1}\) - both \(\psi_\Omega\) and \(\psi_{\Omega_2}\) get stuck, but \(\psi_\pi\) with higher cardinalities (e.g. \(\psi_{\Omega_3}\)) don't. Up to \(\alpha=\Omega_2\), we still have \(\psi(\Omega_2)=\text{BHO}\) and \(\psi_{\Omega_2}(\Omega_2)=\varepsilon_{\Omega+1}\). Next, BHO and \(\varepsilon_{\Omega+1}\in C(\Omega_2+1,0)\), so they get unstuck - \(\psi(\Omega_2+1)=\text{BHO}\times\omega\) and \(\psi_{\Omega_2}(\Omega_2+1)=\varepsilon_{\Omega+1}\omega\).

\(\psi(\Omega_2+\varepsilon_{\Omega+1}+1)\) doesn't get stuck because \(C(\Omega_2+1,0)\) already contains \(\varepsilon_{\Omega+1}\). Every time an \(\Omega\) is added in or out of \(\psi_{\Omega_2}()\), the outside \(\psi_\Omega\) will get stuck, but the inside \(\psi_{\Omega_2}\) doesn't. \(\psi_{\Omega_2}\) gets stuck again at \(\psi_{\Omega_2}(\Omega_2+\varepsilon_{\Omega+2})=\varepsilon_{\Omega+2}=\psi_{\Omega_2}(\Omega_22)\), while \(\psi(\Omega_2+\varepsilon_{\Omega+2})=\psi(\Omega_22)\) also gets stuck.

And so on... At \(\psi_{\Omega_3}(\varepsilon_{\Omega_2+1})=\varepsilon_{\Omega_2+1}=\psi_{\Omega_3}(\Omega_3)\), \(\psi_\Omega\), \(\psi_{\Omega_2}\) and \(\psi_{\Omega_3}\) get stuck...

\(\psi(\Omega_\omega)=\sup_{n<\omega}\{\psi(\Omega_n)\}\) and \(\psi(\Omega_\omega2)=\sup_{n<\omega}\{\psi(\Omega_\omega+\Omega_n)\}\).

Here're some comparisons between ordinals and recursion levels of array notation separators. Separator A has recursion level \(\alpha\) if s(n,n A 2) has growth rate \(\omega^{\omega^\alpha}\).

Ordinal Array notation separator
\(\psi(\Omega_2)\) {1{1``2`}2} or {1,,3}
\(\psi(\Omega_2)+1\) {2{1``2`}2}
\(\psi(\Omega_2)2\) {1{1{1``2`}2}2{1``2`}2}
\(\psi(\Omega_2+1)\) {1{1{1``2`}2}1,2{1``2`}2}
\(\psi(\Omega_2+\psi(\Omega_2))\) {1{1{1``2`}2}1{1{1``2`}2}2{1``2`}2}
\(\psi(\Omega_2+\psi(\Omega_2+1))\) {1{2{1``2`}2}2{1``2`}2}
\(\psi(\Omega_2+\psi(\Omega_2+\psi(\Omega_2)))\) {1{1{1{1``2`}2}2{1``2`}2}2{1``2`}2}
\(\psi(\Omega_2+\psi(\Omega_2+\psi(\Omega_2+\psi(\Omega_2))))\) {1{1{1{1``2`}2}1{1{1``2`}2}2{1``2`}2}2{1``2`}2}
\(\psi(\Omega_2+\Omega)\) {1`2{1``2`}2}
\(\psi(\Omega_2+\Omega+1)\) {1{1`2{1``2`}2}1,2`2{1``2`}2}
\(\psi(\Omega_2+\Omega2)\) {1`3{1``2`}2}
\(\psi(\Omega_2+\Omega^2)\) {1`1`2{1``2`}2}
\(\psi(\Omega_2+\Omega^\Omega)\) {1{1`2`}1{1``2`}2}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2))\) {1{1``2`}3}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2)+\Omega)\) {1`2{1``2`}3}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2)2)\) {1{1``2`}4}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+1))\) {1{1``2`}1,2}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+\Omega))\) {1{1``2`}1`2}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(\Omega_2)))\) {1{1``2`}1{1``2`}2}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(\Omega_2+1)))\) {1{2``2`}2}
\(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(\Omega_2))))\) {1{1{1``2`}2``2`}2}
\(\psi(\Omega_22)\) {1{1``3`}2}
\(\psi(\Omega_22+\Omega)\) {1`2{1``3`}2}
\(\psi(\Omega_22+\psi_{\Omega_2}(\Omega_2))\) {1{1``2`}2{1``3`}2}
\(\psi(\Omega_22+\psi_{\Omega_2}(\Omega_22))\) {1{1``3`}3}
\(\psi(\Omega_23)\) {1{1``4`}2}
\(\psi(\Omega_2\omega)\) {1{1``1,2`}2}
\(\psi(\Omega_2\Omega)\) {1{1``1`2`}2}
\(\psi(\Omega_2\psi_{\Omega_2}(\Omega_2))\) {1{1``1{1``2`}2`}2}
\(\psi(\Omega_2\psi_{\Omega_2}(\Omega_2+1))\) {1{1``1{1``2`}1,2`}2}
\(\psi(\Omega_2\psi_{\Omega_2}(\Omega_22))\) {1{1``1{1``3`}2`}2}
\(\psi(\Omega_2\psi_{\Omega_2}(\Omega_2\omega))\) {1{1``1{1``1,2`}2`}2}
\(\psi(\Omega_2\psi_{\Omega_2}(\Omega_2\psi_{\Omega_2}(\Omega_2)))\) {1{1``1{1``1{1``2`}2`}2`}2}
\(\psi(\Omega_2^2)\) {1{1``1``2`}2}
\(\psi(\Omega_2^2+\Omega)\) {1`2{1``1``2`}2}
\(\psi(\Omega_2^2+\psi_{\Omega_2}(\Omega_2^2))\) {1{1``1``2`}3}
\(\psi(\Omega_2^2+\psi_{\Omega_2}(\Omega_2^2+\psi_{\Omega_2}(\Omega_2^2)))\) {1{1``1``2`}1{1``1``2`}2}
\(\psi(\Omega_2^2+\psi_{\Omega_2}(\Omega_2^2+\psi_{\Omega_2}(\Omega_2^2+\psi_{\Omega_2}(\Omega_2^2))))\) {1{1{1``1``2`}2``1``2`}2}
\(\psi(\Omega_2^2+\Omega_2)\) {1{1``2``2`}2}
\(\psi(\Omega_2^2+\Omega_22)\) {1{1``3``2`}2}
\(\psi(\Omega_2^2+\Omega_2\Omega)\) {1{1``1`2``2`}2}
\(\psi(\Omega_2^2+\Omega_2\psi_{\Omega_2}(\Omega_2^2))\) {1{1``1{1``1``2`}2``2`}2}
\(\psi(\Omega_2^22)\) {1{1``1``3`}2}
\(\psi(\Omega_2^23)\) {1{1``1``4`}2}
\(\psi(\Omega_2^2\Omega)\) {1{1``1``1`2`}2}
\(\psi(\Omega_2^2\psi_{\Omega_2}(\Omega_2^2))\) {1{1``1``1{1``1``2`}2`}2}
\(\psi(\Omega_2^3)\) {1{1``1``1``2`}2}
\(\psi(\Omega_2^\omega)\) {1{1{2``}2`}2}
\(\psi(\Omega_2^\omega+\Omega_2)\) {1{1``2{2``}2`}2}
\(\psi(\Omega_2^\omega+\Omega_22)\) {1{1``3{2``}2`}2}
\(\psi(\Omega_2^\omega+\Omega_2\omega)\) {1{1``1,2{2``}2`}2}
\(\psi(\Omega_2^\omega+\Omega_2\Omega)\) {1{1``1`2{2``}2`}2}
\(\psi(\Omega_2^\omega+\Omega_2^2)\) {1{1``1``2{2``}2`}2}
\(\psi(\Omega_2^\omega+\Omega_2^3)\) {1{1``1``1``2{2``}2`}2}
\(\psi(\Omega_2^\omega2)\) {1{1{2``}3`}2}
\(\psi(\Omega_2^\omega\omega)\) {1{1{2``}1,2`}2}
\(\psi(\Omega_2^\omega\Omega)\) {1{1{2``}1`2`}2}
\(\psi(\Omega_2^{\omega+1})\) {1{1{2``}1``2`}2}
\(\psi(\Omega_2^{\omega+2})\) {1{1{2``}1``1``2`}2}
\(\psi(\Omega_2^{\omega2})\) {1{1{2``}1{2``}2`}2}
\(\psi(\Omega_2^{\omega^2})\) {1{1{3``}2`}2}
\(\psi(\Omega_2^{\omega^\omega})\) {1{1{1,2``}2`}2}
\(\psi(\Omega_2^{\Omega})\) {1{1{1`2``}2`}2}
\(\psi(\Omega_2^{\Omega_2})\) {1{1{1``2``}2`}2}
\(\psi(\Omega_2^{\Omega_2\omega})\) {1{1{2``2``}2`}2}
\(\psi(\Omega_2^{\Omega_2^2})\) {1{1{1``3``}2`}2}
\(\psi(\Omega_2^{\Omega_2^\omega})\) {1{1{1``1,2``}2`}2}
\(\psi(\Omega_2^{\Omega_2^\Omega})\) {1{1{1``1`2``}2`}2}
\(\psi(\Omega_2^{\Omega_2^{\Omega_2}})\) {1{1{1``1``2``}2`}2}
\(\psi(\Omega_2^{\Omega_2^{\Omega_2^{\Omega_2}}})\) {1{1{1{1``2``}2``}2`}2}
\(\psi(\varepsilon_{\Omega_2+1})=\psi(\Omega_3)\) {1{1{1```2``}2`}2} or {1,,4}
\(\psi(\Omega_3+1)\) {1{1{1{1```2``}2`}2}1,2{1{1```2``}2`}2}
\(\psi(\Omega_3+\Omega)\) {1`2{1{1```2``}2`}2}
\(\psi(\Omega_3+\psi_{\Omega_2}(\Omega_3))\) {1{1{1```2``}2`}3}
\(\psi(\Omega_3+\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_2}(\Omega_3)))\) {1{1{1```2``}2`}1{1{1```2``}2`}2}
\(\psi(\Omega_3+\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_2}(\Omega_3))))\) {1{1{1{1```2``}2`}2{1```2``}2`}2}
\(\psi(\Omega_3+\Omega_2)\) {1{1``2{1```2``}2`}2}
\(\psi(\Omega_3+\Omega_2^2)\) {1{1``1``2{1```2``}2`}2}
\(\psi(\Omega_3+\Omega_2^{\Omega_2})\) {1{1{1``2``}2{1```2``}2`}2}
\(\psi(\Omega_3+\Omega_2^{\Omega_2^{\Omega_2}})\) {1{1{1``1``2``}2{1```2``}2`}2}
\(\psi(\Omega_3+\Omega_2^{\Omega_2^{\Omega_2^{\Omega_2}}})\) {1{1{1{1``2``}2``}2{1```2``}2`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3))\) {1{1{1```2``}3`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3)+\Omega_2)\) {1{1``2{1```2``}3`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3)2)\) {1{1{1```2``}4`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3+1))\) {1{1{1```2``}1,2`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3+\Omega_2))\) {1{1{1```2``}1``2`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(\Omega_3)))\) {1{1{1```2``}1{1```2``}2`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(\Omega_3+1)))\) {1{1{2```2``}2`}2}
\(\psi(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(\Omega_3))))\) {1{1{1{1```2``}2```2``}2`}2}
\(\psi(\Omega_32)\) {1{1{1```3``}2`}2}
\(\psi(\Omega_33)\) {1{1{1```4``}2`}2}
\(\psi(\Omega_3\omega)\) {1{1{1```1,2``}2`}2}
\(\psi(\Omega_3\Omega)\) {1{1{1```1`2``}2`}2}
\(\psi(\Omega_3\Omega_2)\) {1{1{1```1``2``}2`}2}
\(\psi(\Omega_3^2)\) {1{1{1```1```2``}2`}2}
\(\psi(\Omega_3^3)\) {1{1{1```1```1```2``}2`}2}
\(\psi(\Omega_3^\omega)\) {1{1{1{2```}2``}2`}2}
\(\psi(\Omega_3^\Omega)\) {1{1{1{1`2```}2``}2`}2}
\(\psi(\Omega_3^{\Omega_2})\) {1{1{1{1``2```}2``}2`}2}
\(\psi(\Omega_3^{\Omega_3})\) {1{1{1{1```2```}2``}2`}2}
\(\psi(\Omega_3^{\Omega_3^{\Omega_3}})\) {1{1{1{1```1```2```}2``}2`}2}
\(\psi(\Omega_3^{\Omega_3^{\Omega_3^{\Omega_3}}})\) {1{1{1{1{1```2```}2```}2``}2`}2}
\(\psi(\varepsilon_{\Omega_3+1})=\psi(\Omega_4)\) {1{1{1{1````2```}2``}2`}2} or {1,,5}
\(\psi(\Omega_5)\) {1,,6}
\(\psi(\Omega_\omega)\) {1,,1,2}
\(\psi(\Omega_\omega)+\omega\) {1,2,,1,2}
\(\psi(\Omega_\omega)\) {1{1,,1,2}2}
\(\psi(\Omega_\omega+\Omega)\) {1`2{1,,1,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_2))\) {1{1``2`}2{1,,1,2}2} or {1``2{1,,1,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_3))\) {1```2{1,,1,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega))\) {1{1,,1,2}3}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)2)\) {1{1,,1,2}4}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+1))\) {1{1,,1,2}1,2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega))\) {1{1,,1,2}1`2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_2)))\) {1{1,,1,2}1``2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)))\) {1{1,,1,2}1{1,,1,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)2))\) {1{1,,1,2}1{1,,1,2}1{1,,1,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+1)))\) {1{2,,1,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega)))\) {1`2,,1,2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+1)))\) {1{2{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega))))\) {1{1{1{1,,1,2}2,,2}2{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\Omega_2)\) {1{1``2{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\Omega_2+\Omega)\) {1`2{1``2{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_\omega+\Omega_2))\) {1{1``2{1,,1,2}2,,2}3}
\(\psi(\Omega_\omega+\Omega_22)\) {1{1``3{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_3))\) {1{1```2{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega))\) {1{1{1,,1,2}3,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega)2)\) {1{1{1,,1,2}4,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+1))\) {1{1{1,,1,2}1,2,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega)))\) {1{1{1,,1,2}1{1,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+1)))\) {1{1{2,,1,2}2,,2}2}
\(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\Omega_2)))\) {1``2,,1,2}
\(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega+\Omega_3)))\) {1```2,,1,2}
\(\psi(\Omega_\omega+\psi_{\Omega_5}(\Omega_\omega+\psi_{\Omega_5}(\Omega_\omega+\Omega_4)))\) {1````2,,1,2}
\(\psi(\Omega_\omega2)\) {1{1,,1,2}2,,1,2}
\(\psi(\Omega_\omega3)\) {1{1,,1,2}3,,1,2}
\(\psi(\Omega_\omega\omega)\) {1{1,,1,2}1,2,,1,2}
\(\psi(\Omega_\omega\Omega)\) {1{1,,1,2}1`2,,1,2}
\(\psi(\Omega_\omega\psi_{\Omega_2}(\Omega_\omega))\) {1{1,,1,2}1{1{1,,1,2}2,,2}2,,1,2}
\(\psi(\Omega_\omega\psi_{\Omega_2}(\Omega_\omega2))\) {1{1,,1,2}1{1{1{1,,1,2}2,,1,2}2,,2}2,,1,2}
\(\psi(\Omega_\omega\psi_{\Omega_2}(\Omega_\omega\Omega))\) {1{1,,1,2}1{1{1{1,,1,2}1`2,,1,2}2,,2}2,,1,2}
\(\psi(\Omega_\omega\Omega_2)\) {1{1,,1,2}1``2,,1,2}
\(\psi(\Omega_\omega\Omega_3)\) {1{1,,1,2}1```2,,1,2}
\(\psi(\Omega_\omega^2)\) {1{1,,1,2}1{1,,1,2}2,,1,2}
\(\psi(\Omega_\omega^2+\Omega_\omega)\) {1{1,,1,2}2{1,,1,2}2,,1,2}
\(\psi(\Omega_\omega^22)\) {1{1,,1,2}1{1,,1,2}3,,1,2}
\(\psi(\Omega_\omega^3)\) {1{1,,1,2}1{1,,1,2}1{1,,1,2}2,,1,2}
\(\psi(\Omega_\omega^\omega)\) {1{2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^\omega+\Omega_\omega)\) {1{1,,1,2}2{2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^\omega2)\) {1{2,,1,2}3,,1,2}
\(\psi(\Omega_\omega^{\omega+1})\) {1{2,,1,2}1{1,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\omega2})\) {1{2,,1,2}1{2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\omega^2})\) {1{3,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\Omega})\) {1{1`2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\psi_{\Omega_2}(\Omega_\omega)})\) {1{1{1{1,,1,2}2,,2}2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\psi_{\Omega_2}(\Omega_\omega^\omega)})\) {1{1{1{1{2,,1,2}2,,1,2}2,,2}2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\Omega_2})\) {1{1{1,,3}2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\Omega_3})\) {1{1{1,,4}2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\Omega_\omega})\) {1{1{1,,1,2}2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\Omega_\omega^{\Omega_\omega}})\) {1{1{1,,1,2}1{1,,1,2}2,,1,2}2,,1,2}
\(\psi(\Omega_\omega^{\Omega_\omega^{\Omega_\omega^{\Omega_\omega}}})\) {1{1{1{1,,1,2}2,,1,2}2,,1,2}2,,1,2}
\(\psi(\varepsilon_{\Omega_\omega+1})=\psi(\Omega_{\omega+1})\) {1{1,,2,2}2,,1,2} or {1,,2,2}
\(\psi(\Omega_{\omega+1}+\Omega_\omega)\) {1{1,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\Omega_\omega2)\) {1{1,,1,2}3{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\Omega_\omega^2)\) {1{1,,1,2}1{1,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\Omega_\omega^{\Omega_\omega})\) {1{1{1,,1,2}2,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}))\) {1{1{1,,2,2}2,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1})2)\) {1{1{1,,2,2}2,,1,2}3{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}+1))\) {1{1{1,,2,2}2,,1,2}1,2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}+\Omega))\) {1{1{1,,2,2}2,,1,2}1`2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}+\Omega_\omega))\) {1{1{1,,2,2}2,,1,2}1{1,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1})))\) {1{1{1,,2,2}2,,1,2}1{1{1,,2,2}2,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}))))\) {1{1{1{1,,2,2}2,,1,2}2{1,,2,2}2,,1,2}2{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}2)\) {1{1,,2,2}3,,1,2}
\(\psi(\Omega_{\omega+1}\omega)\) {1{1,,2,2}1,2,,1,2}
\(\psi(\Omega_{\omega+1}\Omega)\) {1{1,,2,2}1`2,,1,2}
\(\psi(\Omega_{\omega+1}\Omega_\omega)\) {1{1,,2,2}1{1,,1,2}2,,1,2}
\(\psi(\Omega_{\omega+1}\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}))\) {1{1,,2,2}1{1{1,,2,2}2,,1,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^2)\) {1{1,,2,2}1{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^3)\) {1{1,,2,2}1{1,,2,2}1{1,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^\omega)\) {1{2,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^{\Omega_\omega})\) {1{1{1,,1,2}2,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^{\Omega_{\omega+1}})\) {1{1{1,,2,2}2,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^{\Omega_{\omega+1}^{\Omega_{\omega+1}}})\) {1{1{1,,2,2}1{1,,2,2}2,,2,2}2,,1,2}
\(\psi(\Omega_{\omega+1}^{\Omega_{\omega+1}^{\Omega_{\omega+1}^{\Omega_{\omega+1}}}})\) {1{1{1{1,,2,2}2,,2,2}2,,2,2}2,,1,2}
\(\psi(\varepsilon_{\Omega_{\omega+1}+1})=\psi(\Omega_{\omega+2})\) {1{1{1,,3,2}2,,2,2}2,,1,2} or {1,,3,2}
\(\psi(\Omega_{\omega+3})\) {1,,4,2}
\(\psi(\Omega_{\omega2})\) {1,,1,3}
\(\psi(\Omega_{\omega2+1})\) {1,,2,3}
\(\psi(\Omega_{\omega2+2})\) {1,,3,3}
\(\psi(\Omega_{\omega3})\) {1,,1,4}
\(\psi(\Omega_{\omega^2})\) {1,,1,1,2}
\(\psi(\Omega_{\omega^\omega})\) {1,,1{2}2}
\(\psi(\Omega_{\varepsilon_0})\) {1,,1{1`2}2}
\(\psi(\Omega_{\psi(\Omega_\omega)})\) {1,,1{1{1,,1,2}2}2}
\(\psi(\Omega_{\psi(\Omega_{\psi(\Omega_\omega)})})\) {1,,1{1{1,,1{1{1,,1,2}2}2}2}2}
\(\psi(\Omega_\Omega)\) {1,,1`2}
\(\psi(\Omega_\Omega+\Omega_\omega)\) {1{1,,1,2}2,,1`2}
\(\psi(\Omega_\Omega+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}))\) {1{1{1,,2,2}2,,1,2}2,,1`2}
\(\psi(\Omega_\Omega+\psi_{\Omega_{\omega+1}}(\Omega_\Omega))\) {1{1{1,,1`2}2,,1,2}2,,1`2}
\(\psi(\Omega_\Omega+\Omega_{\omega+1})\) {1{1,,2,2}2,,1`2}
\(\psi(\Omega_\Omega+\Omega_{\omega+2})\) {1{1,,3,2}2,,1`2}
\(\psi(\Omega_\Omega2)\) {1{1,,1`2}2,,1`2}
\(\psi(\Omega_\Omega^2)\) {1{1,,1`2}1{1,,1`2}2,,1`2}
\(\psi(\Omega_\Omega^{\Omega_\Omega})\) {1{1{1,,1`2}2,,1`2}2,,1`2}
\(\psi(\Omega_\Omega^{\Omega_\Omega^{\Omega_\Omega}})\) {1{1{1,,1`2}1{1,,1`2}2,,1`2}2,,1`2}
\(\psi(\Omega_{\Omega+1})\) {1,,2`2}
\(\psi(\Omega_{\Omega+\psi(\Omega_\omega)})\) {1,,1{1{1,,1,2}2}2`2}
\(\psi(\Omega_{\Omega+\psi(\Omega_\Omega)})\) {1,,1{1{1,,1`2}2}2`2}
\(\psi(\Omega_{\Omega+\psi(\Omega_{\Omega+1})})\) {1,,1{1{1,,2`2}2}2`2}
\(\psi(\Omega_{\Omega2})\) {1,,1`3}
\(\psi(\Omega_{\psi_{\Omega_2}(\Omega_2)})\) {1,,1{1``2,,2}2}
\(\psi(\Omega_{\psi_{\Omega_2}(\Omega_\Omega)})\) {1,,1{1{1,,1`2}2,,2}2}
\(\psi(\Omega_{\psi_{\Omega_2}(\Omega_{\psi_{\Omega_2}(\Omega_\Omega)})})\) {1,,1{1{1,,1{1{1,,1`2}2,,2}2}2,,2}2}
\(\psi(\Omega_{\Omega_2})\) {1,,1``2}
\(\psi(\Omega_{\Omega_3})\) {1,,1```2}
\(\psi(\Omega_{\Omega_\omega})\) {1,,1{1,,1,2}2}
\(\psi(\Omega_{\Omega_\Omega})\) {1,,1{1,,1`2}2}
\(\psi(\Omega_{\Omega_{\Omega_2}})\) {1,,1{1,,1``2}2}
\(\psi(\Omega_{\Omega_{\Omega_\omega}})\) {1,,1{1,,1{1,,1,2}2}2}
\(\psi(\Omega_{\Omega_{\Omega_{\Omega_\omega}}})\) {1,,1{1,,1{1,,1{1,,1,2}2}2}2}

As a result, \(\Omega_\alpha\) works as the 1-separator of \(\psi_{\Omega_\alpha}\) function.

Using a weakly inaccessible

Definition: let \(\Omega_0=0\), \(\Omega_\alpha\) is the \(\alpha\)-th uncountable cardinal, and \(I\) is the first weakly inaccessible cardinal (a "large" ordinal over cardinality). We use \(\pi\) to represent uncountable regular cardinals (at this point, uncountable regular cardinals can be written as \(\Omega_{\alpha+1}\) or \(I\)). Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0,I\} \\ 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_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\).

The only difference between this notation and the previous one is the \(I\) in \(C_0(\alpha,\beta)\).

Now look at \(\psi_\pi(0)\) again. \(C(0,0)\) contains not only 0, but also \(I\), \(I2\), \(I3\), \(\Omega_{I2}\), \(\Omega_{I2}+I\), \(\Omega_{I3}\), \(\Omega_{\Omega_{I2}}\), \(\Omega_{\Omega_{I2}+I}\), \(\Omega_{\Omega_{\Omega_{I2}}}\), etc. As a result, \(\psi_\pi(0)=1\) for \(\pi\le I\), but \(\psi_{\Omega_{I+1}}(0)=I\omega\), \(\psi_{\Omega_{I+2}}(0)=\Omega_{I+1}\omega\), and so on, analogous to the \(\psi_\pi(1)\)'s below \(I\).

Next, \(C(1,0)\) contains 0, 1, \(\Omega\), \(\Omega_\Omega\), \(\Omega_{\Omega_\Omega}\), etc. As a result, \(\psi_I(1)\) equals the omega-fixed-point. By the way, in Deedlit11's notation, \(\varphi(\gamma,\delta)\) appears in the construction of \(C(\alpha,\beta)\), so \(C(0,0)\) already contains 1, \(\Omega\), \(\Omega_\Omega\), \(\Omega_{\Omega_\Omega}\), etc. then \(\psi_I(0)\) equals the omega-fixed-point.

\(\psi_I\) enumerates fixed points of \(\alpha\mapsto\Omega_\alpha\), up to \(\psi_I(I)\). Then \(\psi_I(\psi_I(I)+1)=\psi_I(\psi_I(I))=\psi_I(I)\), and it gets unstuck at \(\psi_I(I+1)\) - \(I\) works as the 1-separator (or diagonalizer) of \(\psi_I\) function.

Here come comparisons between OCF and array notation.

Ordinal Array notation separator
\(\psi(\psi_I(1))\) {1,,1{1,,1,,2}2} or {1,,1,,2}
\(\psi(\psi_I(1)+\Omega_\omega)\) {1{1,,1,2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)+\Omega_\Omega)\) {1{1,,1`2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)+\Omega_{\Omega_\omega})\) {1{1,,1{1,,1,2}2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)+\Omega_{\Omega_{\Omega_\omega}})\) {1{1,,1{1,,1{1,,1,2}2}2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)2)\) {1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)\omega)\) {1{1,,1{1,,1,,2}2}1,2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)^2)\) {1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)^\omega)\) {1{2,,1{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\psi_I(1)^{\psi_I(1)})\) {1{1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\varepsilon_{\psi_I(1)+1})=\psi(\Omega_{\psi_I(1)+1})\) {1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2} or {1,,2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+1}+\Omega_\omega)\) {1{1,,1,2}2{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+1}+\psi_I(1))\) {1{1,,1{1,,1,,2}2}2{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+1}+\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\psi_I(1)+1}))\) {1{1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+1}2)\) {1{1,,2{1,,1,,2}2}3,,1{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+1}^2)\) {1{1,,2{1,,1,,2}2}1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+1}^{\Omega_{\psi_I(1)+1}})\) {1{1{1,,2{1,,1,,2}2}2,,2{1,,1,,2}2}2,,1{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+2})\) {1{1{1,,3{1,,1,,2}2}2,,2{1,,1,,2}2}2,,1{1,,1,,2}2} or {1,,3{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+\omega})\) {1,,1,2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+\Omega})\) {1,,1`2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)+\Omega_\omega})\) {1,,1{1,,1,2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)2})\) {1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)2+1})\) {1,,2{1,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)3})\) {1,,1{1,,1{1,,1,,2}2}3{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)\omega})\) {1,,1{1,,1{1,,1,,2}2}1,2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)^2})\) {1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_I(1)^{\psi_I(1)}})\) {1,,1{1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\psi_I(1)+1})})\) {1,,1{1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\psi_I(1)+2})})\) {1,,1{1{1,,3{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\psi_I(1)2})})\) {1,,1{1{1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+1}})\) {1,,1{1,,2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+1}+1})\) {1,,2{1,,2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+1}+\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\Omega_{\psi_I(1)+1}})})\) {1,,1{1,,1{1,,2{1,,1,,2}2}2{1,,1,,2}2}2{1,,2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+1}2})\) {1,,1{1,,2{1,,1,,2}2}3{1,,1,,2}2}
\(\psi(\Omega_{\psi_{\Omega_{\psi_I(1)+2}}(\Omega_{\psi_I(1)+2})})\) {1,,1{1{1,,3{1,,1,,2}2}2,,2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+2}})\) {1,,1{1,,3{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+\omega}})\) {1,,1{1,,1,2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)+\Omega_\omega}})\) {1,,1{1,,1{1,,1,2}2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\psi_I(1)2}})\) {1,,1{1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\Omega_{\Omega_{\Omega_{\psi_I(1)2}}})\) {1,,1{1,,1{1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2}2{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(\psi_I(2))\) {1,,1{1,,1,,2}3}
\(\psi(\Omega_{\psi_I(2)+1})\) {1,,2{1,,1,,2}3}
\(\psi(\Omega_{\psi_I(2)+\psi_I(1)})\) {1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}3}
\(\psi(\Omega_{\psi_I(2)2})\) {1,,1{1,,1{1,,1,,2}3}2{1,,1,,2}3}
\(\psi(\Omega_{\Omega_{\psi_I(2)2}})\) {1,,1{1,,1{1,,1{1,,1,,2}3}2{1,,1,,2}3}2{1,,1,,2}3}
\(\psi(\psi_I(3))\) {1,,1{1,,1,,2}4}
\(\psi(\psi_I(\omega))\) {1,,1{1,,1,,2}1,2}
\(\psi(\psi_I(\Omega))\) {1,,1{1,,1,,2}1`2}
\(\psi(\psi_I(\Omega_\omega))\) {1,,1{1,,1,,2}1{1,,1,2}2}
\(\psi(\psi_I(\psi_I(1)))\) {1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}
\(\psi(\Omega_{\psi_I(\psi_I(1))+1})\) {1,,2{1,,1,,2}1{1,,1{1,,1,,2}2}2}
\(\psi(\Omega_{\psi_I(\psi_I(1))2})\) {1,,1{1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}2{1,,1,,2}1{1,,1{1,,1,,2}2}2}
\(\psi(\psi_I(\psi_I(1)+1))\) {1,,1{1,,1,,2}2{1,,1{1,,1,,2}2}2}
\(\psi(\psi_I(\psi_I(1)2))\) {1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}3}
\(\psi(\psi_I(\Omega_{\psi_I(1)+1}))\) {1,,1{1,,1,,2}1{1,,2{1,,1,,2}2}2}
\(\psi(\psi_I(\psi_I(2)))\) {1,,1{1,,1,,2}1{1,,1{1,,1,,2}3}2}
\(\psi(\psi_I(\psi_I(\psi_I(1))))\) {1,,1{1,,1,,2}1{1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}2}
\(\psi(\psi_I(I))=\psi(I)\) {1,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\Omega_\omega)\) {1{1,,1,2}2,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(1))\) {1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(2))\) {1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(\psi_I(1)))\) {1{1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}2,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(I))\) {1{1,,1{1,,1,,2}1{1,,1,,2}2}2,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(I)2)\) {1{1,,1{1,,1,,2}1{1,,1,,2}2}3,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(I)\omega)=\psi(I+\psi_{\Omega_{\psi_I(I)+1}}(I+1))\) {1{1,,1{1,,1,,2}1{1,,1,,2}2}1,2,,1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\varepsilon_{\psi_I(I)+1})=\psi(I+\psi_{\Omega_{\psi_I(I)+1}}(I+\Omega_{\psi_I(I)+1}))\)

\(=\psi(I+\Omega_{\psi_I(I)+1})\)

{1,,2{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\Omega_{\psi_I(I)+2})\) {1,,3{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\Omega_{\psi_I(I)2})\) {1,,1{1,,1{1,,1,,2}1{1,,1,,2}2}2{1,,1,,2}1{1,,1,,2}2}
\(\psi(I+\psi_I(I+1))\) {1,,1{1,,1,,2}2{1,,1,,2}2}
\(\psi(I+\psi_I(I+\psi_I(I)))\) {1,,1{1,,1,,2}1{1,,1{1,,1,,2}1{1,,1,,2}2}2{1,,1,,2}2}
\(\psi(I2)\) {1,,1{1,,1,,2}1{1,,1,,2}3}
\(\psi(I\omega)\) {1,,1{1,,1,,2}1{1,,1,,2}1,2}
\(\psi(I\psi_I(I))\) {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1{1,,1,,2}1{1,,1,,2}2}2}
\(\psi(I^2)\) {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I^3)\) {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2}
\(\psi(I^\omega)\) {1,,1{2,,1,,2}2}
\(\psi(I^\Omega)\) {1,,1{1`2,,1,,2}2}
\(\psi(I^{\psi_I(1)})\) {1,,1{1{1,,1{1,,1,,2}2}2,,1,,2}2}
\(\psi(I^{\psi_I(I)})\) {1,,1{1{1,,1{1,,1,,2}1{1,,1,,2}2}2,,1,,2}2}
\(\psi(I^{\psi_I(I^\omega)})\) {1,,1{1{1,,1{2,,1,,2}2}2,,1,,2}2}
\(\psi(I^I)\) {1,,1{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+\psi_I(I^I))\) {1{1,,1{1{1,,1,,2}2,,1,,2}2}2,,1{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+\Omega_{\psi_I(I^I)+1})\) {1,,2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+\Omega_{\psi_I(I^I)2})\) {1,,1{1,,1{1{1,,1,,2}2,,1,,2}2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+\psi_I(I^I+1))\) {1,,1{1,,1,,2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+\psi_I(I^I+\psi_I(I^I)))\) {1,,1{1,,1,,2}1{1,,1{1{1,,1,,2}2,,1,,2}2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+I)\) {1,,1{1,,1,,2}1{1,,1,,2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+I2)\) {1,,1{1,,1,,2}1{1,,1,,2}3{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+I^2)\) {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+I^\omega)\) {1,,1{2,,1,,2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I+I^{\psi_I(I^I+I)})\) {1,,1{1 {1,,1{1,,1,,2}1{1,,1,,2}2{1{1,,1,,2}2,,1,,2}2} 2,,1,,2}2{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^I2)\) {1,,1{1{1,,1,,2}2,,1,,2}3}
\(\psi(I^{I+1})\) {1,,1{1{1,,1,,2}2,,1,,2}1{1,,1,,2}2}
\(\psi(I^{I2})\) {1,,1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}2}
\(\psi(I^{I\omega})\) {1,,1{2{1,,1,,2}2,,1,,2}2}
\(\psi(I^{I^2})\) {1,,1{1{1,,1,,2}3,,1,,2}2}
\(\psi(I^{I^\omega})\) {1,,1{1{1,,1,,2}1,2,,1,,2}2}
\(\psi(I^{I^I})\) {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}2}
\(\psi(I^{I^{I^I}})\) {1,,1{1{1{1,,1,,2}2,,1,,2}2,,1,,2}2}
\(\psi(\varepsilon_{I+1})=\psi(\Omega_{I+1})\) {1,,1{1{1,,2,,2}2,,1,,2}2} or {1,,2,,2}
\(\psi(\Omega_{I+1}+\psi_I(I))\) {1{1,,1{1,,1,,2}1{1,,1,,2}2}2,,1{1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}+\psi_I(\Omega_{I+1}))\) {1{1,,1{1{1,,2,,2}2,,1,,2}2}2,,1{1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}+\Omega_{\psi_I(\Omega_{I+1})+1})\) {1,,2{1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}+\psi_I(\Omega_{I+1}+1))\) {1,,1{1,,1,,2}2{1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}+I)\) {1,,1{1,,1,,2}1{1,,1,,2}2{1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1}))\) {1,,1{1{1,,2,,2}2,,1,,2}3}
\(\psi(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1}+1))\) {1,,1{1{1,,2,,2}2,,1,,2}1,2}
\(\psi(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1})))\) {1,,1{1{1,,2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1}+\psi_{\Omega_{I+1}}(\Omega_{I+1}))))\) {1,,1{1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}2)\) {1,,1{1{1,,2,,2}3,,1,,2}2}
\(\psi(\Omega_{I+1}\omega)\) {1,,1{1{1,,2,,2}1,2,,1,,2}2}
\(\psi(\Omega_{I+1}I)\) {1,,1{1{1,,2,,2}1{1,,1,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}\psi_{\Omega_{I+1}}(\Omega_{I+1}))\) {1,,1{1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}^2)\) {1,,1{1{1,,2,,2}1{1,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}^{\Omega_{I+1}})\) {1,,1{1{1{1,,2,,2}2,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+1}^{\Omega_{I+1}^{\Omega_{I+1}}})\) {1,,1{1{1{1,,2,,2}1{1,,2,,2}2,,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+2})\) {1,,1{1{1{1,,3,,2}2,,2,,2}2,,1,,2}2} or {1,,3,,2}
\(\psi(\Omega_{I+\omega})\) {1,,1,2,,2}
\(\psi(\Omega_{I+\omega}+\psi_I(\Omega_{I+\omega}))\) {1{1,,1{1,,1,2,,2}2}2,,1{1,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+\Omega_{\psi_I(\Omega_{I+\omega})+1})\) {1,,2{1,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+\psi_I(\Omega_{I+\omega}+1))\) {1,,1{1,,1,,2}2{1,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+I)\) {1,,1{1,,1,,2}1{1,,1,,2}2{1,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+1}))\) {1,,1{1,,2,,2}2{1,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}))\) {1,,1{1,,1,2,,2}3}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+1))\) {1,,1{1,,1,2,,2}1,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega})))\) {1,,1{1,,1,2,,2}1{1,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+1)))\) {1,,1{2,,1,2,,2}2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+I)))\) {1{1,,1,,2}2,,1,2,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}+1)))\) {1,,1{2{1,,1,2,,2}2,,1,,2}2}
\(\psi(\Omega_{I+\omega}+\Omega_{I+1})\) {1{1,,2,,2}2{1,,1,2,,2}2,,1,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+2}))\) {1{1,,3,,2}2{1,,1,2,,2}2,,1,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega}))\) {1{1,,1,2,,2}3,,1,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega})))\) {1{1,,1,2,,2}1{1,,1,2,,2}2,,1,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega}+1)))\) {1{2,,1,2,,2}2,,1,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega}+\psi_{\Omega_{I+2}}(\Omega_{I+\omega}+\Omega_{I+1})))\) {1{1,,2,,2}2,,1,2,,2}
\(\psi(\Omega_{I+\omega}+\psi_{\Omega_{I+3}}(\Omega_{I+\omega}+\psi_{\Omega_{I+3}}(\Omega_{I+\omega}+\Omega_{I+2})))\) {1{1,,3,,2}2,,1,2,,2}
\(\psi(\Omega_{I+\omega}2)\) {1{1,,1,2,,2}2,,1,2,,2}
\(\psi(\Omega_{I+\omega}^2)\) {1{1,,1,2,,2}1{1,,1,2,,2}2,,1,2,,2}
\(\psi(\Omega_{I+\omega}^{\Omega_{I+\omega}})\) {1{1{1,,1,2,,2}2,,1,2,,2}2,,1,2,,2}
\(\psi(\Omega_{I+\omega+1})\) {1,,2,2,,2}
\(\psi(\Omega_{I+\Omega})\) {1,,1`2,,2}
\(\psi(\Omega_{I+\psi_I(1)})\) {1,,1{1,,1{1,,1,,2}2}2,,2}
\(\psi(\Omega_{I+\psi_I(I)})\) {1,,1{1,,1{1,,1,,2}1{1,,1,,2}2}2,,2}
\(\psi(\Omega_{I+\psi_I(\Omega_{I+\omega})})\) {1,,1{1,,1{1,,1,2,,2}2}2,,2}
\(\psi(\Omega_{I2})\) {1,,1{1,,1,,2}2,,2}
\(\psi(\Omega_{I2+1})\) {1,,2{1,,1,,2}2,,2}
\(\psi(\Omega_{I3})\) {1,,1{1,,1,,2}3,,2}
\(\psi(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I+1})})\) {1,,1{1{1,,2,,2}2,,1,,2}2,,2}
\(\psi(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I2})})\) {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2}
\(\psi(\Omega_{\Omega_{I+1}})\) {1,,1{1,,2,,2}2,,2}
\(\psi(\Omega_{\Omega_{I2}})\) {1,,1{1,,1{1,,1,,2}2,,2}2,,2}
\(\psi(\Omega_{\Omega_{\Omega_{I2}}})\) {1,,1{1,,1{1,,1{1,,1,,2}2,,2}2,,2}2,,2}

Here're some approximations, to make the comparisons above more clear:

  • \(\Omega\) approximately corresponds to the grave accent (or {1,,2})
  • \(\Omega_2\) approximately corresponds to {1,,3}
  • \(\Omega_3\) approximately corresponds to {1,,4}
  • \(\Omega_\omega\) approximately corresponds to {1,,1,2}
  • \(\Omega_\Omega\) approximately corresponds to {1,,1`2}
  • \(\psi_I(1)\) approximately corresponds to {1,,1{1,,1,,2}2}
  • \(\Omega_{\psi_I(1)+1}\) approximately corresponds to {1,,2{1,,1,,2}2}
  • \(\Omega_{\psi_I(1)2}\) approximately corresponds to {1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2}
  • \(\Omega_{\Omega_{\psi_I(1)+1}}\) approximately corresponds to {1,,1{1,,2{1,,1,,2}2}2{1,,1,,2}2}
  • \(\Omega_{\Omega_{\psi_I(1)2}}\) approximately corresponds to {1,,1{1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2}2{1,,1,,2}2}
  • \(\psi_I(2)\) approximately corresponds to {1,,1{1,,1,,2}3}
  • \(\psi_I(\omega)\) approximately corresponds to {1,,1{1,,1,,2}1,2}
  • \(\psi_I(\psi_I(1))\) approximately corresponds to {1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}
  • \(\psi_I(I)\) approximately corresponds to {1,,1{1,,1,,2}1{1,,1,,2}2}
  • \(\Omega_{\psi_I(I)+1}\) approximately corresponds to {1,,2{1,,1,,2}1{1,,1,,2}2}
  • \(\Omega_{\psi_I(I)2}\) approximately corresponds to {1,,1{1,,1{1,,1,,2}1{1,,1,,2}2}2{1,,1,,2}1{1,,1,,2}2}
  • \(\psi_I(I+1)\) approximately corresponds to {1,,1{1,,1,,2}2{1,,1,,2}2}
  • \(\psi_I(I2)\) approximately corresponds to {1,,1{1,,1,,2}1{1,,1,,2}3}
  • \(\psi_I(I^2)\) approximately corresponds to {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2}
  • \(\psi_I(I^\omega)\) approximately corresponds to {1,,1{2,,1,,2}2}
  • \(\psi_I(I^I)\) approximately corresponds to {1,,1{1{1,,1,,2}2,,1,,2}2}
  • \(I\) approximately corresponds to {1,,1,,2}
  • \(\psi_{\Omega_{I+1}}(\Omega_{I+1})\) approximately corresponds to {1{1,,2,,2}2,,1,,2}
  • \(\Omega_{I+1}\) approximately corresponds to {1,,2,,2}
  • \(\Omega_{I+2}\) approximately corresponds to {1,,3,,2}
  • \(\Omega_{I+\omega}\) approximately corresponds to {1,,1,2,,2}
  • \(\psi_I(\Omega_{I+\omega})\) approximately corresponds to {1,,1{1,,1,2,,2}2}
  • \(\psi_{\Omega_{I+1}}(\Omega_{I+\omega})\) approximately corresponds to {1{1,,1,2,,2}2,,1,,2}
  • \(\Omega_{I2}\) approximately corresponds to {1,,1{1,,1,,2}2,,2}
  • \(\Omega_{\Omega_{I+1}}\) approximately corresponds to {1,,1{1,,2,,2}2,,2}
  • \(\Omega_{\Omega_{I2}}\) approximately corresponds to {1,,1{1,,1{1,,1,,2}2,,2}2,,2}

Using weakly inaccessibles

Definition: let \(\Omega_0=0\), \(\Omega_\alpha\) be the \(\alpha\)-th uncountable cardinal. \(I_0=0\), \(I_{\alpha+1}\) is the next weakly inaccessible cardinal after \(I_\alpha\), and \(I_\alpha=\sup\{I_\beta|\beta<\alpha\}\) for limit ordinal \(\alpha\). so the \(\omega\)-th weakly inaccessible cardinal is not \(I_\omega\) but \(I_{\omega+1}\), while \(I_\omega\) is not a regular cardinal.

We use \(\pi\) to represent uncountable regular cardinals (at this point, uncountable regular cardinals can be written as \(\Omega_{\alpha+1}\) or \(I_{\alpha+1}\)). 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& \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{I_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\), \(I\) is a shorthand for \(I_1\), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\).

Since \(I\notin C(0,0)\), we have \(\psi_\pi(0)=1\) again. Then \(\psi_I(1)\) equals the omega-fixed-point, \(\psi_{\Omega_{I+1}}(1)=I\omega\), \(\psi_{\Omega_{I+2}}(1)=\Omega_{I+1}\omega\), and \(\psi_{I_2}(1)\) equals the next omega fixed point after \(I\).

Here come comparisons between this OCF and pDAN.

Ordinal pDAN separator
\(\psi(\psi_{I_2}(1))\) {1,,1{1,,1,,3}2,,2} or {1,,1,,3}
\(\psi(\psi_{I_2}(1)+\psi_I(1))\) {1{1,,1{1,,1,,2}2}2,,1{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_I(\Omega_{I+1}))\) {1{1,,1{1,,2,,2}2}2,,1{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_I(\Omega_{I2}))\) {1{1,,1{1,,1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_I(\psi_{I_2}(1)))\) {1{1,,1{1,,1,,3}2}2,,1{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\Omega_{\psi_I(\psi_{I_2}(1))+1})\) {1,,2{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_I(\psi_{I_2}(1)+1))\) {1,,1{1,,1,,2}2{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+I)\) {1,,1{1,,1,,2}1{1,,1,,2}2{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\Omega_{I+1}))\) {1,,1{1,,2,,2}2{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\Omega_{I2}))\) {1,,1{1,,1{1,,1,,2}2,,2}2{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1)))\) {1,,1{1,,1,,3}3}
\(\psi(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1))))\) {1,,1{1,,1,,3}1{1,,1,,3}2}
\(\psi(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1)+1)))\) {1,,1{2,,1{1,,1,,3}2,,2}2}
\(\psi(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1)+\psi_{\Omega_{I+1}}(\psi_{I_2}(1)+I)))\) {1{1,,1,,2}2,,1{1,,1,,3}2,,2}
\(\psi(\psi_{I_2}(1)+\Omega_{I+\omega})\) {1{1,,1,2,,2}2,,1{1,,1,,3}2,,2}
\(\psi(\psi_{I_2}(1)+\Omega_{I2})\) {1{1,,1{1,,1,,2}2,,2}2,,1{1,,1,,3}2,,2}
\(\psi(\psi_{I_2}(1)2)\) {1{1,,1{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2}
\(\psi(\Omega_{\psi_{I_2}(1)+1})\) {1,,2{1,,1,,3}2,,2}
\(\psi(\Omega_{\psi_{I_2}(1)+I})\) {1,,1{1,,1,,2}2{1,,1,,3}2,,2}
\(\psi(\Omega_{\psi_{I_2}(1)+\Omega_{I+1}})\) {1,,1{1,,2,,2}2{1,,1,,3}2,,2}
\(\psi(\Omega_{\psi_{I_2}(1)2})\) {1,,1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}
\(\psi(\Omega_{\Omega_{\psi_{I_2}(1)2}})\) {1,,1{1,,1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2{1,,1,,3}2,,2}
\(\psi(\psi_{I_2}(2))\) {1,,1{1,,1,,3}3,,2}
\(\psi(\psi_{I_2}(I))\) {1,,1{1,,1,,3}1{1,,1,,2}2,,2}
\(\psi(\psi_{I_2}(\psi_{\Omega_{I+1}}(\psi_{I_2}(1))))\) {1,,1{1,,1,,3}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2}
\(\psi(\psi_{I_2}(\Omega_{I+1}))\) {1,,1{1,,1,,3}1{1,,2,,2}2,,2}
\(\psi(\psi_{I_2}(\Omega_{I2}))\) {1,,1{1,,1,,3}1{1,,1{1,,1,,2}2,,2}2,,2}
\(\psi(\psi_{I_2}(\psi_{I_2}(1)))\) {1,,1{1,,1,,3}1{1,,1{1,,1,,3}2,,2}2,,2}
\(\psi(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(1))))\) {1,,1{1,,1,,3}1{1,,1{1,,1,,3}1{1,,1{1,,1,,3}2,,2}2,,2}2,,2}
\(\psi(I_2)\) {1,,1{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\psi_I(I_2))\) {1{1,,1{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2} 2,,1{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2}
\(\psi(I_2+\Omega_{\psi_I(I_2)+1})\) {1,,2{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2}
\(\psi(I_2+\psi_I(I_2+1))\) {1,,1{1,,1,,2}2{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2}
\(\psi(I_2+\Omega_{I+1})\) {1,,1{1,,2,,2}2{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2}
\(\psi(I_2+\Omega_{I+\omega})\) {1{1,,1,2,,2}2,,1{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\psi_{I_2}(1))\) {1{1,,1{1,,1,,3}2,,2}2,,1{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\psi_{I_2}(\psi_{I_2}(1)))\) {1{1,,1{1,,1,,3}1{1,,1{1,,1,,3}2,,2}2,,2} 2,,1{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\psi_{I_2}(I_2))\) {1{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2,,1{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\varepsilon_{\psi_{I_2}(I_2)+1})=\psi(I_2+\Omega_{\psi_{I_2}(I_2)+1})\) {1,,2{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\Omega_{\psi_{I_2}(I_2)2})\) {1,,1{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2+\psi_{I_2}(I_2+1))\) {1,,1{1,,1,,3}2{1,,1,,3}2,,2}
\(\psi(I_2+\psi_{I_2}(I_2+\psi_{I_2}(I_2)))\) {1,,1{1,,1,,3}1{1,,1{1,,1,,3}1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}
\(\psi(I_22)\) {1,,1{1,,1,,3}1{1,,1,,3}3,,2}
\(\psi(I_2^2)\) {1,,1{1,,1,,3}1{1,,1,,3}1{1,,1,,3}2,,2}
\(\psi(I_2^{I_2})\) {1,,1{1{1,,1,,3}2,,1,,3}2,,2}
\(\psi(\Omega_{I_2+1})\) {1,,1{1{1,,2,,3}2,,1,,3}2,,2} or {1,,2,,3}
\(\psi(\Omega_{I_2+\omega})\) {1,,1,2,,3}
\(\psi(\Omega_{I_2+I})\) {1,,1{1,,1,,2}2,,3}
\(\psi(\Omega_{I_2+\psi_{I_2}(1)})\) {1,,1{1,,1{1,,1,,3}2,,2}2,,3}
\(\psi(\Omega_{I_2+\psi_{I_2}(\Omega_{I_2+I})})\) {1,,1{1,,1{1,,1{1,,1,,2}2,,3}2,,2}2,,3}
\(\psi(\Omega_{I_22})\) {1,,1{1,,1,,3}2,,3}
\(\psi(\Omega_{\Omega_{I_2+1}})\) {1,,1{1,,2,,3}2,,3}
\(\psi(\Omega_{\Omega_{I_22}})\) {1,,1{1,,1{1,,1,,3}2,,3}2,,3}
\(\psi(\psi_{I_3}(1))\) {1,,1{1,,1,,4}2,,3} or {1,,1,,4}
\(\psi(\Omega_{\psi_{I_3}(1)+1})\) {1,,2{1,,1,,4}2,,3}
\(\psi(\Omega_{\psi_{I_3}(1)2})\) {1,,1{1,,1{1,,1,,4}2,,3}2{1,,1,,4}2,,3}
\(\psi(\psi_{I_3}(2))\) {1,,1{1,,1,,4}3,,3}
\(\psi(\psi_{I_3}(\psi_{I_3}(1)))\) {1,,1{1,,1,,4}1{1,,1{1,,1,,4}2,,3}2,,3}
\(\psi(I_3)\) {1,,1{1,,1,,4}1{1,,1,,4}2,,3}
\(\psi(I_3+\psi_{I_3}(I_3))\) {1{1,,1{1,,1,,4}1{1,,1,,4}2,,3}2,,1{1,,1,,4}1{1,,1,,4}2,,3}
\(\psi(I_3+\psi_{I_3}(I_3+1))\) {1,,1{1,,1,,4}2{1,,1,,4}2,,3}
\(\psi(I_32)\) {1,,1{1,,1,,4}1{1,,1,,4}3,,3}
\(\psi(\Omega_{I_3+1})\) {1,,2,,4}
\(\psi(\Omega_{I_32})\) {1,,1{1,,1,,4}2,,4}
\(\psi(\psi_{I_4}(1))\) {1,,1{1,,1,,5}2,,4} or {1,,1,,5}
\(\psi(I_4)\) {1,,1{1,,1,,5}1{1,,1,,5}2,,4}
\(\psi(\psi_{I_5}(1))\) {1,,1,,6}
\(\psi(I_\omega)\) {1,,1,,1,2}
\(\psi(I_\omega+\psi_I(1))\) {1{1,,1{1,,1,,2}2}2,,1{1,,1,,1,2}2}
\(\psi(I_\omega+\psi_I(I_\omega))\) {1{1,,1{1,,1,,1,2}2}2,,1{1,,1,,1,2}2}
\(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+1})\) {1,,2{1,,1,,1,2}2}
\(\psi(I_\omega+\psi_I(I_\omega+1))\) {1,,1{1,,1,,2}2{1,,1,,1,2}2}
\(\psi(I_\omega+I)\) {1,,1{1,,1,,2}1{1,,1,,2}2{1,,1,,1,2}2}
\(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega))\) {1,,1{1,,1,,1,2}3}
\(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)))\) {1,,1{1,,1,,1,2}1{1,,1,,1,2}2}
\(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+1)))\) {1,,1{2,,1,,1,2}2}
\(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+I)))\) {1{1,,1,,2}2,,1,,1,2}
\(\psi(I_\omega+\Omega_{I+1})\) {1{1,,2,,2}1{1,,1{1,,1,,1,2}2,,2}2,,1,,2}
\(\psi(I_\omega+\Omega_{I+\omega})\) {1{1,,1,2,,2}2,,1{1,,1,,1,2}2,,2}
\(\psi(I_\omega+\psi_{I_2}(1))\) {1{1,,1{1,,1,,3}2,,2}2,,1{1,,1,,1,2}2,,2}
\(\psi(I_\omega+\psi_{I_2}(I_\omega))\) {1{1,,1{1,,1,,1,2}2,,2}2,,1{1,,1,,1,2}2,,2}
\(\psi(I_\omega+\Omega_{\psi_{I_2}(I_\omega)+1})\) {1,,2{1,,1,,1,2}2,,2}
\(\psi(I_\omega+I_2)\) {1,,1{1,,1,,3}1{1,,1,,3}2{1,,1,,1,2}2,,2}
\(\psi(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega))\) {1,,1{1,,1,,1,2}3,,2}
\(\psi(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega)))\) {1,,1{1,,1,,1,2}1{1,,1,,1,2}2,,2}
\(\psi(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega+1)))\) {1,,1{2,,1,,1,2}2,,2}
\(\psi(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega+\psi_{\Omega_{I_2+1}}(I_\omega+I_2)))\) {1{1,,1,,3}2,,1,,1,2}
\(\psi(I_\omega+\psi_{\Omega_{I_3+1}}(I_\omega+\psi_{\Omega_{I_3+1}}(I_\omega+I_3)))\) {1{1,,1,,4}2,,1,,1,2}
\(\psi(I_\omega2)\) {1{1,,1,,1,2}2,,1,,1,2}
\(\psi(\Omega_{I_\omega+1})\) {1,,2,,1,2}
\(\psi(\Omega_{I_\omega+I})\) {1,,1{1,,1,,2}2,,1,2}
\(\psi(\Omega_{I_\omega2})\) {1,,1{1,,1,,1,2}2,,1,2}
\(\psi(\psi_{I_{\omega+1}}(1))\) {1,,1,,2,2}
\(\psi(\psi_{I_{\omega+2}}(1))\) {1,,1,,3,2}
\(\psi(I_{\omega2})\) {1,,1,,1,3}
\(\psi(I_{\psi_I(1)})\) {1,,1,,1{1,,1{1,,1,,2}2}2}
\(\psi(I_{\psi_I(I_{\psi_I(1)})})\) {1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,2}2}2}2}2}
\(\psi(I_I)\) {1,,1,,1{1,,1,,2}2}
\(\psi(\Omega_{I_I+1})\) {1,,2,,1{1,,1,,2}2}
\(\psi(\psi_{I_{I+1}}(1))\) {1,,1,,2{1,,1,,2}2}
\(\psi(I_{I+\omega})\) {1,,1,,1,2{1,,1,,2}2}
\(\psi(I_{I2})\) {1,,1,,1{1,,1,,2}3}
\(\psi(I_{\Omega_{I+1}})\) {1,,1,,1{1,,2,,2}2}
\(\psi(I_{\psi_{I_2}(I_\omega)})\) {1,,1,,1{1,,1{1,,1,,1,2}2,,2}2}
\(\psi(I_{I_2})\) {1,,1,,1{1,,1,,3}2}
\(\psi(I_{I_3})\) {1,,1,,1{1,,1,,4}2}
\(\psi(I_{I_\omega})\) {1,,1,,1{1,,1,,1,2}2}
\(\psi(I_{I_{I_\omega}})\) {1,,1,,1{1,,1,,1{1,,1,,1,2}2}2}

Here're some approximations, to make the comparisons above more clear:

  • \(\psi_{I_2}(1)\) approximately corresponds to {1,,1{1,,1,,3}2,,2}
  • \(\Omega_{\psi_{I_2}(1)+1}\) approximately corresponds to {1,,2{1,,1,,3}2,,2}
  • \(\Omega_{\psi_{I_2}(1)2}\) approximately corresponds to {1,,1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}
  • \(\psi_{I_2}(2)\) approximately corresponds to {1,,1{1,,1,,3}3,,2}
  • \(\psi_{I_2}(I_2)\) approximately corresponds to {1,,1{1,,1,,3}1{1,,1,,3}2,,2}
  • \(I_2\) approximately corresponds to {1,,1,,3}
  • \(\Omega_{I_2+1}\) approximately corresponds to {1,,2,,3}
  • \(\Omega_{I_22}\) approximately corresponds to {1,,1{1,,1,,3}2,,3}
  • \(\psi_{I_3}(1)\) approximately corresponds to {1,,1{1,,1,,4}2,,3}
  • \(I_3\) approximately corresponds to {1,,1,,4}
  • \(I_\omega\) approximately corresponds to {1,,1,,1,2}
  • \(\psi_I(I_\omega)\) approximately corresponds to {1,,1{1,,1,,1,2}2}
  • \(\psi_{I_2}(I_\omega)\) approximately corresponds to {1,,1{1,,1,,1,2}2,,2}
  • \(\Omega_{I_\omega+1}\) approximately corresponds to {1,,2,,1,2}
  • \(I_{\omega+1}\) approximately corresponds to {1,,1,,2,2}
  • \(I_{\omega2}\) approximately corresponds to {1,,1,,1,3}
  • \(I_{\psi_I(1)}\) approximately corresponds to {1,,1,,1{1,,1{1,,1,,2}2}2}
  • \(I_I\) approximately corresponds to {1,,1,,1{1,,1,,2}2}
  • \(\Omega_{I_I+1}\) approximately corresponds to {1,,2,,1{1,,1,,2}2}
  • \(I_{I+1}\) approximately corresponds to {1,,1,,2{1,,1,,2}2}
  • \(I_{I2}\) approximately corresponds to {1,,1,,1{1,,1,,2}3}
  • \(I_{\Omega_{I+1}}\) approximately corresponds to {1,,1,,1{1,,2,,2}2}
  • \(I_{I_2}\) approximately corresponds to {1,,1,,1{1,,1,,3}2}
  • \(I_{I_\omega}\) approximately corresponds to {1,,1,,1{1,,1,,1,2}2}
  • \(I_{I_{I_\omega}}\) approximately corresponds to {1,,1,,1{1,,1,,1{1,,1,,1,2}2}2}

Collapsing higher inaccessibility

An ordinal is \(\alpha\)-weakly inaccessible if it's an uncountable regular cardinal and it's a limit of \(\gamma\)-weakly inaccessible cardinals for all \(\gamma<\alpha\). So 0-weakly inaccessible cardinals are just uncountable regular cardinals, and 1-weakly inaccessible cardinals are weakly inaccessible cardinals. But the inaccessibility can extend further: an ordinal \(\pi\) is \((\alpha,\beta)\)-weakly inaccessible if it's \((\gamma,\pi)\)-weakly inaccessible for all \(\gamma<\alpha\), and it's a limit of \((\alpha,\gamma)\)-weakly inaccessibles for all \(\gamma<\beta\) (combined with the previous, "\(\alpha\)-weakly inaccessible" is a shorthand of "\((0,\alpha)\)-weakly inaccessible"). And even more: an ordinal \(\pi\) is \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible if it's \((\alpha_1,\alpha_2,\cdots\alpha_i,\gamma,\pi,\underbrace{0,0,\cdots0}_{n-i-2})\)-weakly inaccessible for all \(\gamma<\alpha_{i+1}\) and \(0\le i\le n-2\), and it's a limit of \((\alpha_1,\alpha_2,\cdots\alpha_{n-1},\gamma)\)-weakly inaccessibles for all \(\gamma<\alpha_n\). ("\((0,\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible" is identical to "\((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible")

Let \(I(\alpha_1,\alpha_2\cdots,\alpha_n,0)\) be the first \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible cardinal, \(I(\alpha_1,\alpha_2\cdots,\alpha_n,\beta+1)\) be the next \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible cardinal after \(I(\alpha_1,\alpha_2\cdots,\alpha_n,\beta)\), and \(I(\alpha_1,\alpha_2\cdots,\alpha_n,\beta)=\sup\{I(\alpha_1,\alpha_2\cdots,\alpha_n,\gamma)|\gamma<\beta\}\) for limit ordinal \(\beta\). When \(\beta=0\) or \(\beta=\gamma+1\) the \(I(\alpha_1,\alpha_2\cdots,\alpha_n,\beta)\) is a \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible cardinal; when \(\beta\) is a limit ordinal (e.g. \(\beta=\omega\)), this doesn't hold. So \(I(0,\alpha)=\Omega_{1+\alpha}\) and \(I(1,\alpha)=I_{1+\alpha}\) and in general case \(I(\alpha,\beta)=\)the \((1+\beta)\)th ordinal in the set \(\{\gamma\in R|\forall\delta<\alpha(I(\delta,\gamma)=\gamma)\}\) where \(R\) denotes the set of all regular ordinals \(\xi>\omega\).

We use \(\pi\) to represent uncountable regular cardinals (at this point, uncountable regular cardinals can be written as \(I(\alpha_1,\alpha_2\cdots,\alpha_n,0)\) or \(I(\alpha_1,\alpha_2\cdots,\alpha_n,\beta+1)\)). 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& \{I(\gamma_1,\gamma_2\cdots,\gamma_k,\delta)|\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1=I(0,0)\), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\).

Now the "offset" of \(\psi_\pi(0)\) disappears. \(I(0,0)=\Omega\), \(I(0,I(0,0))=\Omega_\Omega\), and so on, so \(C(0,0)\) contains \(\Omega\), \(\Omega2\), \(\Omega_\Omega\), \(\Omega_{\Omega_\Omega}\), etc. Then \(\psi_{\Omega_2}(0)=\Omega\omega\), \(\psi_{\Omega_{\omega+1}}(0)=\Omega_\omega\omega\), \(\psi_I(0)=\Omega_{\Omega_{\Omega_\cdots}}\), \(\psi_{I(2,0)}(0)=I_{I_{I_\cdots}}\), etc.

Here come comparisons between this OCF and pDAN.

pDAN separator Ordinal
{1,,1,,1{1,,1,,1,,2}2} or {1,,1,,1,,2} \(\psi(\psi_{I(2,0)}(0))\)
{1{1,,1{1,,1,,1{1,,1,,1,,2}2}2} 2,,1{1,,1,,1{1,,1,,1,,2}2}2} \(\psi(\psi_{I(2,0)}(0)+\psi_I(\psi_{I(2,0)}(0)))\)
{1,,1{1,,1,,2}2{1,,1,,1{1,,1,,1,,2}2}2} \(\psi(\psi_{I(2,0)}(0)+\psi_I(\psi_{I(2,0)}(0)+1))\)
{1,,1{1,,1,,1{1,,1,,1,,2}2}3} \(\psi(\psi_{I(2,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(2,0)}(0)))\)
{1,,1{1,,1,,1{1,,1,,1,,2}2}1{1,,1,,1{1,,1,,1,,2}2}2} \(\psi(\psi_{I(2,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(2,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(2,0)}(0))))\)
{1{1,,1,,2}2,,1,,1{1,,1,,1,,2}2} \(\psi(\psi_{I(2,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(2,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(2,0)}(0)+I)))\)
{1{1,,1,,3}2,,1,,1{1,,1,,1,,2}2} \(\psi(\psi_{I(2,0)}(0)+\psi_{\Omega_{I_2+1}}(\psi_{I(2,0)}(0)+\psi_{\Omega_{I_2+1}}(\psi_{I(2,0)}(0)+I_2)))\)
{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}2} \(\psi(\psi_{I(2,0)}(0)+I_\omega)\)
{1{1,,1,,1{1,,1,,1,,2}2}2,,1,,1{1,,1,,1,,2}2} \(\psi(\psi_{I(2,0)}(0)2)\)
{1,,2,,1{1,,1,,1,,2}2} \(\psi(\Omega_{\psi_{I(2,0)}(0)+1})\)
{1,,1{1,,1,,1{1,,1,,1,,2}2}2,,1{1,,1,,1,,2}2} \(\psi(\Omega_{\psi_{I(2,0)}(0)2})\)
{1,,1{1,,1,,2{1,,1,,1,,2}2}2,,1{1,,1,,1,,2}2} or {1,,1,,2{1,,1,,1,,2}2} \(\psi(\psi_{I(1,\psi_{I(2,0)}(0)+1)}(1))\)
{1,,1{1,,1,,2{1,,1,,1,,2}2}3,,1{1,,1,,1,,2}2} \(\psi(\psi_{I(1,\psi_{I(2,0)}(0)+1)}(2))\)
{1,,1{1,,1,,2{1,,1,,1,,2}2}1 {1,,1,,2{1,,1,,1,,2}2}2 ,,1{1,,1,,1,,2}2} \(\psi(I(1,\psi_{I(2,0)}(0)+1))\)
{1,,2,,2{1,,1,,1,,2}2} \(\psi(\Omega_{I(1,\psi_{I(2,0)}(0)+1)+1})\)
{1,,1,,3{1,,1,,1,,2}2} \(\psi(\psi_{I(1,\psi_{I(2,0)}(0)+2)}(1))\)
{1,,1,,1,2{1,,1,,1,,2}2} \(\psi(I(1,\psi_{I(2,0)}(0)+\omega))\)
{1,,1,,1{1,,1,,2{1,,1,,1,,2}2}2{1,,1,,1,,2}2} \(\psi(I(1,\psi_{I(2,0)}(0)2))\)
{1,,1,,1 {1,,1,,1 {1,,1,,2{1,,1,,1,,2}2} 2{1,,1,,1,,2}2} 2{1,,1,,1,,2}2} \(\psi(I(1,I(1,\psi_{I(2,0)}(0)2)))\)
{1,,1,,1{1,,1,,1,,2}3} \(\psi(\psi_{I(2,0)}(1))\)
{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2} \(\psi(\psi_{I(2,0)}(I))\)
{1,,1,,1{1,,1,,1,,2}1{1,,1,,3}2} \(\psi(\psi_{I(2,0)}(I_2))\)
{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2} \(\psi(\psi_{I(2,0)}(\psi_{I(2,0)}(0)))\)
{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0))\)
{1{1,,1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2} 2,,1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2} \(\psi(I(2,0)+\psi_I(I(2,0)))\)
{1,,1{1,,1,,2}2{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2} \(\psi(I(2,0)+\psi_I(I(2,0)+1))\)
{1{1,,1,,2}2,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{\Omega_{I+1}}(I(2,0)+\psi_{\Omega_{I+1}}(I(2,0)+I)))\)
{1{1,,1,,3}2,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{\Omega_{I_2+1}}(I(2,0)+\psi_{\Omega_{I_2+1}}(I(2,0)+I_2)))\)
{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+I_\omega)\)
{1{1,,1,,1{1,,1,,1,,2}2} 2,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{I(2,0)}(0))\)
{1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} 2,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{I(2,0)}(I(2,0)))\)
{1,,2,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+\Omega_{\psi_{I(2,0)}(I(2,0))+1})\)
{1,,1,,2{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{I(1,\psi_{I(2,0)}(I(2,0))+1)}(I(2,0)+1))\)
{1,,1,,1 {1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} 2{1,,1,,1,,2}1{1,,1,,1,,2}2} \(\psi(I(2,0)+I(1,\psi_{I(2,0)}(I(2,0))2))\)
{1,,1,,1{1,,1,,1,,2}2{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{I(2,0)}(I(2,0)+1))\)
{1,,1,,1{1,,1,,1,,2}1 {1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} 2{1,,1,,1,,2}2} \(\psi(I(2,0)+\psi_{I(2,0)}(I(2,0)+\psi_{I(2,0)}(I(2,0))))\)
{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}3} \(\psi(I(2,0)2)\)
{1,,2,,1,,2} \(\psi(\Omega_{I(2,0)+1})\)
{1,,1{1,,1,,1{1,,1,,1,,2}2}2,,1,,2} \(\psi(\Omega_{I(2,0)+\psi_{I(2,0)}(0)})\)
{1,,1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2,,1,,2} \(\psi(\Omega_{I(2,0)+\psi_{I(2,0)}(I(2,0))})\)
{1,,1{1,,1,,1,,2}2,,1,,2} \(\psi(\Omega_{I(2,0)2})\)
{1,,1,,2,,2} \(\psi(\psi_{I(1,I(2,0)+1)}(0))\)
{1,,1,,1{1,,1,,1,,2}2,,2} \(\psi(I(1,I(2,0)2))\)
{1,,1,,1,,3} \(\psi(\psi_{I(2,1)}(0))\)
{1,,1,,1{1,,1,,1,,3}1{1,,1,,1,,3}2,,2} \(\psi(I(2,1))\)
{1,,1,,1,,4} \(\psi(\psi_{I(2,2)}(0))\)
{1,,1,,1,,1,2} \(\psi(I(2,\omega))\)
{1,,1,,1,,1{1,,1,,1,,2}2} \(\psi(I(2,I(2,0)))\)
{1,,1,,1,,1{1,,1,,1,,1,,2}2} or {1,,1,,1,,1,,2} \(\psi(\psi_{I(3,0)}(0))\)
{1,,2,,1,,1{1,,1,,1,,1,,2}2} \(\psi(\Omega_{\psi_{I(3,0)}(0)+1})\)
{1,,1,,2,,1{1,,1,,1,,1,,2}2} \(\psi(\psi_{I(1,\psi_{I(3,0)}(0)+1)}(1))\)
{1,,1,,1,,2{1,,1,,1,,1,,2}2} \(\psi(\psi_{I(2,\psi_{I(3,0)}(0)+1)}(1))\)
{1,,1,,1,,1 {1,,1,,1,,2{1,,1,,1,,1,,2}2} 2{1,,1,,1,,1,,2}2} \(\psi(I(2,\psi_{I(3,0)}(0)2))\)
{1,,1,,1,,1{1,,1,,1,,1,,2}3} \(\psi(\psi_{I(3,0)}(1))\)
{1,,1,,1,,1{1,,1,,1,,1,,2}1 {1,,1,,1,,1{1,,1,,1,,1,,2}2}2} \(\psi(\psi_{I(3,0)}(\psi_{I(3,0)}(0)))\)
{1,,1,,1,,1{1,,1,,1,,1,,2}1{1,,1,,1,,1,,2}2} \(\psi(I(3,0))\)
{1,,2,,1,,1,,2} \(\psi(\Omega_{I(3,0)+1})\)
{1,,1{1,,1,,1,,1,,2}2,,1,,1,,2} \(\psi(\Omega_{I(3,0)2})\)
{1,,1,,2,,1,,2} \(\psi(\psi_{I(1,I(3,0)+1)}(0))\)
{1,,1,,1{1,,1,,1,,1,,2}2,,1,,2} \(\psi(I(1,I(3,0)2))\)
{1,,1,,1,,2,,2} \(\psi(\psi_{I(2,I(3,0)+1)}(0))\)
{1,,1,,1,,1{1,,1,,1,,1,,2}2,,2} \(\psi(I(2,I(3,0)2))\)
{1,,1,,1,,1,,3} \(\psi(\psi_{I(3,1)}(0))\)
{1,,1,,1,,1,,1{1,,1,,1,,1,,2}2} \(\psi(I(3,I(3,0)))\)
{1,,1,,1,,1,,1{1,,1,,1,,1,,1,,2}2} or {1,,1,,1,,1,,1,,2} \(\psi(\psi_{I(4,0)}(0))\)
{1,,1,,1,,1,,1 {1,,1,,1,,1,,1,,2}1 {1,,1,,1,,1,,1,,2}2} \(\psi(I(4,0))\)
{1,,1,,1,,1,,1,,1{1,,1,,1,,1,,1,,1,,2}2} or {1,,1,,1,,1,,1,,1,,2} \(\psi(\psi_{I(5,0)}(0))\)
{1,,1,,1,,1,,1,,1 {1,,1,,1,,1,,1,,1,,2}1 {1,,1,,1,,1,,1,,1,,2}2} \(\psi(I(5,0))\)
{1{2,,}2} \(\psi(\psi_{I(\omega,0)}(0))\)
{1`2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\Omega)\)
{1{1,,3}2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\Omega_3))\)
{1{1,,1,,2}2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_I(0)))\)
{1{1,,1,,1,,2}2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(2,0)}(0)))\)
{1{1{2,,}2}3} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0)))\)
{1{1{2,,}2}1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0))))\)
{1{2{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0)+1)))\)
{1{1,,2}2{2,,}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_2}(\psi_{I(\omega,0)}(0)+\Omega)))\)
{1{1,,1{1,,1,,2}2}2,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_I(0))\)
{1{1,,1{1,,2,,2}2}2,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_I(\Omega_{I+1}))\)
{1{1,,1{1,,1,,3}2}2,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_I(\psi_{I_2}(0)))\)
{1{1,,1{1,,1,,1,,2}2}2,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_I(\psi_{I(2,0)}(0)))\)
{1{1,,1{1{2,,}2}2}2,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_I(\psi_{I(\omega,0)}(0)))\)
{1,,2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\Omega_{\psi_I(\psi_{I(\omega,0)}(0))+1})\)
{1,,1{1,,1,,2}2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_I(\psi_{I(\omega,0)}(0)+1))\)
{1,,1{1,,1,,2}1{1,,1,,2}2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+I)\)
{1,,1{1,,2,,2}2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I+1}}(\Omega_{I+1}))\)
{1,,1{1{2,,}2}3} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(\omega,0)}(0)))\)
{1,,1{1{2,,}2}1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(\omega,0)}(0))))\)
{1{1,,1,,2}2{2,,}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I+1}}(\psi_{I(\omega,0)}(0)+I)))\)
{1{1,,1,,1,2}2,,1,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+I_\omega)\)
{1{1,,1,,1{1{2,,}2}2}2,,1,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{I(2,0)}(\psi_{I(\omega,0)}(0)))\)
{1,,2,,1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\Omega_{\psi_{I(2,0)}(\psi_{I(\omega,0)}(0))+1})\)
{1,,1,,2{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{I(1,\psi_{I(2,0)}(\psi_{I(\omega,0)}(0))+1)}(\psi_{I(\omega,0)}(0)+1))\)
{1,,1,,1{1,,1,,1,,2}2{1{2,,}3}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{I(2,0)}(\psi_{I(\omega,0)}(0)+1))\)
{1,,1,,1{1{2,,}2}3} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I(2,0)+1}}(\psi_{I(\omega,0)}(0)))\)
{1{1,,1,,1,,2}2{2,,}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I(2,0)+1}}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I(2,0)+1}}(\psi_{I(\omega,0)}(0)+I(2,0))))\)
{1{1,,1,,1,,1,,2}2{2,,}2} \(\psi(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I(3,0)+1}}(\psi_{I(\omega,0)}(0)+\psi_{\Omega_{I(3,0)+1}}(\psi_{I(\omega,0)}(0)+I(3,0))))\)
{1{1{2,,}2}2{2,,}2} \(\psi(\psi_{I(\omega,0)}(0)2)\)
{1,,2{2,,}2} \(\psi(\Omega_{\psi_{I(\omega,0)}(0)+1})\)
{1,,1{1{2,,}2}2{2,,}2} \(\psi(\Omega_{\psi_{I(\omega,0)}(0)2})\)
{1,,1,,2{2,,}2} \(\psi(\psi_{I(1,\psi_{I(\omega,0)}(0)+1)}(1))\)
{1,,1,,1{1{2,,}2}2{2,,}2} \(\psi(I(1,\psi_{I(\omega,0)}(0)2))\)
{1,,1,,1,,2{2,,}2} \(\psi(\psi_{I(2,\psi_{I(\omega,0)}(0)+1)}(1))\)
{1,,1,,1,,1{1{2,,}2}2{2,,}2} \(\psi(I(2,\psi_{I(\omega,0)}(0)2))\)
{1{2,,}3} \(\psi(\psi_{I(\omega,0)}(1))\)
{1{2,,}4} \(\psi(\psi_{I(\omega,0)}(2))\)
{1{2,,}1{1{2,,}2}2} \(\psi(\psi_{I(\omega,0)}(\psi_{I(\omega,0)}(0)))\)
{1{2,,}1{1{2,,}1,,2}2} or {1{2,,}1,,2} \(\psi(I(\omega,0))\)
{1{1{2,,}2}2{2,,}1{1{2,,}1,,2}2} \(\psi(I(\omega,0)+\psi_{I(\omega,0)}(0))\)
{1{1{2,,}1{1{2,,}1,,2}2}2{2,,}1{1{2,,}1,,2}2} \(\psi(I(\omega,0)+\psi_{I(\omega,0)}(I(\omega,0)))\)
{1,,2{2,,}1{1{2,,}1,,2}2} \(\psi(I(\omega,0)+\Omega_{\psi_{I(\omega,0)}(I(\omega,0))+1})\)
{1,,1,,2{2,,}1{1{2,,}1,,2}2} \(\psi(I(\omega,0)+\psi_{I(1,\psi_{I(\omega,0)}(I(\omega,0))+1)}(I(\omega,0)+1))\)
{1,,1,,1,,2{2,,}1{1{2,,}1,,2}2} \(\psi(I(\omega,0)+\psi_{I(2,\psi_{I(\omega,0)}(I(\omega,0))+1)}(I(\omega,0)+1))\)
{1{2,,}2{1{2,,}1,,2}2} \(\psi(I(\omega,0)+\psi_{I(\omega,0)}(I(\omega,0)+1))\)
{1{2,,}1{1{2,,}1,,2}3} \(\psi(I(\omega,0)2)\)
{1,,2{2,,}1,,2} \(\psi(\Omega_{I(\omega,0)+1})\)
{1,,1,,2{2,,}1,,2} \(\psi(\psi_{I(1,I(\omega,0)+1)}(0))\)
{1,,1,,1,,2{2,,}1,,2} \(\psi(\psi_{I(2,I(\omega,0)+1)}(0))\)
{1{2,,}2,,2} \(\psi(\psi_{I(\omega,1)}(0))\)
{1{2,,}1,,3} \(\psi(I(\omega,1))\)
{1{2,,}1,,4} \(\psi(I(\omega,2))\)
{1{2,,}1,,1{1{2,,}2}2} \(\psi(I(\omega,\psi_{I(\omega,0)}(0)))\)
{1{2,,}1,,1{1{2,,}1{1{2,,}1,,2}2}2} \(\psi(I(\omega,\psi_{I(\omega,0)}(I(\omega,0))))\)
{1{2,,}1,,1 {1{2,,}1{1{2,,}1,,1 {1{2,,}2} 2}2} 2} \(\psi(I(\omega,\psi_{I(\omega,0)}(I(\omega,\psi_{I(\omega,0)}(0)))))\)
{1{2,,}1,,1{1{2,,}1,,2}2} \(\psi(I(\omega,I(\omega,0)))\)
{1{2,,}1,,1 {1{2,,}1,,1 {1{2,,}1,,2}2}2} \(\psi(I(\omega,I(\omega,I(\omega,0))))\)
{1{2,,}1,,1,,2} \(\psi(\psi_{I(\omega+1,0)}(0))\)
{1{2,,}1,,1,,1,,2} \(\psi(\psi_{I(\omega+2,0)}(0))\)
{1{2,,}1{2,,}2} \(\psi(\psi_{I(\omega2,0)}(0))\)
{1{3,,}2} \(\psi(\psi_{I(\omega^2,0)}(0))\)
{1{1{1`2}2,,}2} \(\psi(\psi_{I(\varepsilon_0,0)}(0))\)
{1{1`2,,}2} \(\psi(\psi_{I(\Omega,0)}(0))\)
{1{1{1,,3}2,,}2} \(\psi(\psi_{I(\Omega_2,0)}(0))\)
{1{1{1,,1,,2}2,,}2} \(\psi(\psi_{I(I,0)}(0))\)
{1{1{1,,1,,1,,2}2,,}2} \(\psi(\psi_{I(I(2,0),0)}(0))\)
{1{1{1{2,,}2}2,,}2} \(\psi(\psi_{I(\psi_{I(\omega,0)}(0),0)}(0))\)
{1{1{1{2,,}1,,2}2,,}1,,2} \(\psi(I(I(\omega,0),0))\)
{1{1{1{1{1{2,,}2}2,,}2}2,,}2} \(\psi(\psi_{I(\psi_{I(\psi_{I(\omega,0)}(0),0)}(0),0)}(0))\)
{1{1{1{1{1{2,,}1,,2}2,,}1,,2}2,,}1,,2} \(\psi(I(I(I(\omega,0),0),0))\)
{1{1{1{1,,2,,}2}2,,}2} or {1{1,,2,,}2} \(\psi(\psi_{I(1,0,0)}(0))\)

Let separator ♦ = {1{1,,2,,}2} in the table shown below. The ♦ is the 1-separator in {1{1 ____ 2,,}2}.

Ordinal pDAN separator
\(\psi(\psi_{I(1,0,0)}(0))\) {1{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(0)+\psi_{I(\omega,0)}(0))\) {1{1{2,,}2}2{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(0)+\psi_{I(\omega,0)}(\psi_{I(1,0,0)}(0)))\) {1{1{2,,}1{1{1♦2,,}2}2}2{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(0)+I(\omega,0))\) {1{1{2,,}1,,2}2{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(0)+\psi_{I(\psi_{I(\omega,0)}(0),0)}(0))\) {1{1{1{1{2,,}2}2,,}2}2{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(0)+I(I(\omega,0),0))\) {1{1{1{1{2,,}1,,2}2,,}1,,2}2{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(0)2)\) {1{1{1♦2,,}2}2{1♦2,,}2}
\(\psi(\Omega_{\psi_{I(1,0,0)}(0)+1})\) {1,,2{1♦2,,}2}
\(\psi(\psi_{I(1,\psi_{I(1,0,0)}(0)+1)}(1))\) {1,,1,,2{1♦2,,}2}
\(\psi(\psi_{I(\omega,\psi_{I(1,0,0)}(0)+1)}(1))\) {1{2,,}2{1♦2,,}2}
\(\psi(I(\omega,\psi_{I(1,0,0)}(0)+1))\) {1{2,,}1,,2{1♦2,,}2}
\(\psi(I(\psi_{I(\omega,0)}(0),\psi_{I(1,0,0)}(0)+1))\) {1{1{1{2,,}2}2,,}1,,2{1♦2,,}2}
\(\psi(I(I(\omega,0),\psi_{I(1,0,0)}(0)+1))\) {1{1{1{2,,}1,,2}2,,}1,,2{1♦2,,}2}
\(\psi(I(\psi_{I(\psi_{I(\omega,0)}(0),0)}(0),\psi_{I(1,0,0)}(0)+1))\) {1{1{1{1{1{2,,}2}2,,}2}2,,}1,,2{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0),1)}(1))\) {1{1{1{1♦2,,}2}2,,}2{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0),1)}(2))\) {1{1{1{1♦2,,}2}2,,}3{1♦2,,}2}
\(\psi(I(\psi_{I(1,0,0)}(0),1))\) {1{1{1{1♦2,,}2}2,,}1,,2{1♦2,,}2}
\(\psi(I(\omega,I(\psi_{I(1,0,0)}(0),1)+1))\) {1{2,,}1,,2{1{1{1♦2,,}2}2,,}1,,2{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0),2)}(1))\) {1{1{1{1♦2,,}2}2,,}2,,2{1♦2,,}2}
\(\psi(I(\psi_{I(1,0,0)}(0),2))\) {1{1{1{1♦2,,}2}2,,}1,,3{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0)+1,0)}(1))\) {1{1{1{1♦2,,}2}2,,}1,,1,,2{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0)\omega,0)}(1))\) {1{2{1{1♦2,,}2}2,,}2{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0)^2,0)}(1))\) {1{1{1{1♦2,,}2}3,,}2{1♦2,,}2}
\(\psi(\psi_{I(\Omega_{\psi_{I(1,0,0)}(0)+1},0)}(1))\) {1{1{1,,2{1♦2,,}2}2,,}2{1♦2,,}2}
\(\psi(\psi_{I(I(\omega,\psi_{I(1,0,0)}(0)+1),0)}(1))\) {1{1{1{2,,}1,,2{1♦2,,}2}2,,}2{1♦2,,}2}
\(\psi(\psi_{I(\psi_{I(\psi_{I(1,0,0)}(0),1)}(0),0)}(1))\) {1{1 {1{1 {1{1♦2,,}2} 2,,}2{1♦2,,}2} 2,,}2{1♦2,,}2}
\(\psi(\psi_{I(1,0,0)}(1))\) {1{1♦2,,}3}
\(\psi(\psi_{I(1,0,0)}(\psi_{I(1,0,0)}(0)))\) {1{1♦2,,}1{1{1♦2,,}2}2}
\(\psi(I(1,0,0))\) {1{1♦2,,}1,,2}
\(\psi(\Omega_{I(1,0,0)+1})\) {1,,2{1♦2,,}1,,2}
\(\psi(\psi_{I(\omega,I(1,0,0)+1)}(0))\) {1{2,,}2{1♦2,,}1,,2}
\(\psi(\psi_{I(I(1,0,0),1)}(0))\) {1{1{1{1♦2,,}1,,2}2,,}2{1♦2,,}1,,2}
\(\psi(\psi_{I(1,0,1)}(0))\) {1{1♦2,,}2,,2}
\(\psi(I(1,0,1))\) {1{1♦2,,}1,,3}
\(\psi(\psi_{I(1,1,0)}(0))\) {1{1♦2,,}1,,1,,2}
\(\psi(\psi_{I(1,\omega,0)}(0))\) {1{1♦2,,}1{2,,}2}
\(\psi(\psi_{I(1,\psi_{I(1,0,0)}(0),0)}(0))\) {1{1♦2,,}1{1{1{1♦2,,}2}2,,}2}
\(\psi(I(1,I(1,0,0),0))\) {1{1♦2,,}1{1{1{1♦2,,}1,,2}2,,}1,,2}
\(\psi(\psi_{I(2,0,0)}(0))\) {1{1♦2,,}1{1♦2,,}2}
\(\psi(I(2,0,0))\) {1{1♦2,,}1{1♦2,,}1,,2}
\(\psi(\psi_{I(\omega,0,0)}(0))\) {1{2♦2,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0)}(0),0,0)}(0))\) {1{1{1{1♦2,,}2}2♦2,,}2}
\(\psi(I(I(1,0,0),0,0))\) {1{1{1{1♦2,,}1,,2}2♦2,,}1,,2}
\(\psi(\psi_{I(1,0,0,0)}(0))\) {1{1♦3,,}2}
\(\psi(I(1,0,0,0))\) {1{1♦3,,}1,,2}
\(\psi(\psi_{I(1,1,0,0)}(0))\) {1{1♦3,,}1{1♦2,,}2}
\(\psi(\psi_{I(2,0,0,0)}(0))\) {1{1♦3,,}1{1♦3,,}2}
\(\psi(\psi_{I(\omega,0,0,0)}(0))\) {1{2♦3,,}2}
\(\psi(\psi_{I(\psi_{I(1,0,0,0)}(0),0,0,0)}(0))\) {1{1{1{1♦3,,}2}2♦3,,}2}
\(\psi(I(I(1,0,0,0),0,0,0))\) {1{1{1{1♦3,,}1,,2}2♦3,,}1,,2}
\(\psi(\psi_{I(1,0,0,0,0)}(0))\) {1{1♦4,,}2}
\(\psi(I(1,0,0,0,0))\) {1{1♦4,,}1,,2}
\(\psi(\psi_{I(1,0,0,0,0,0)}(0))\) {1{1♦5,,}2}
\(\psi(I(1,0,0,0,0,0))\) {1{1♦5,,}1,,2}

Here're some approximations, to make the comparisons above more clear:

  • \(\psi_{I(2,0)}(0)\) approximately corresponds to {1,,1,,1{1,,1,,1,,2}2}
  • \(\Omega_{\psi_{I(2,0)}(0)+1}\) approximately corresponds to {1,,2,,1{1,,1,,1,,2}2}
  • \(I(1,\psi_{I(2,0)}(0)+1)\) approximately corresponds to {1,,1,,2{1,,1,,1,,2}2}
  • \(\psi_{I(2,0)}(1)\) approximately corresponds to {1,,1,,1{1,,1,,1,,2}3}
  • \(\psi_{I(2,0)}(I(2,0))\) approximately corresponds to {1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}
  • \(I(2,0)\) approximately corresponds to {1,,1,,1,,2}
  • \(\Omega_{I(2,0)+1}\) approximately corresponds to {1,,2,,1,,2}
  • \(I(1,I(2,0)+1)\) approximately corresponds to {1,,1,,2,,2}
  • \(\psi_{I(2,1)}(0)\) approximately corresponds to {1,,1,,1{1,,1,,1,,3}2,,2}
  • \(I(2,1)\) approximately corresponds to {1,,1,,1,,3}
  • \(\psi_{I(3,0)}(0)\) approximately corresponds to {1,,1,,1,,1{1,,1,,1,,1,,2}2}
  • \(\Omega_{\psi_{I(3,0)}(0)+1}\) approximately corresponds to {1,,2,,1,,1{1,,1,,1,,1,,2}2}
  • \(I(1,\psi_{I(3,0)}(0)+1)\) approximately corresponds to {1,,1,,2,,1{1,,1,,1,,1,,2}2}
  • \(I(2,\psi_{I(3,0)}(0)+1)\) approximately corresponds to {1,,1,,1,,2{1,,1,,1,,1,,2}2}
  • \(\psi_{I(3,0)}(1)\) approximately corresponds to {1,,1,,1,,1{1,,1,,1,,1,,2}3}
  • \(I(3,0)\) approximately corresponds to {1,,1,,1,,1,,2}
  • \(\Omega_{I(3,0)+1}\) approximately corresponds to {1,,2,,1,,1,,2}
  • \(I(1,I(3,0)+1)\) approximately corresponds to {1,,1,,2,,1,,2}
  • \(I(2,I(3,0)+1)\) approximately corresponds to {1,,1,,1,,2,,2}
  • \(I(3,1)\) approximately corresponds to {1,,1,,1,,1,,3}
  • \(I(4,0)\) approximately corresponds to {1,,1,,1,,1,,1,,2}
  • \(I(5,0)\) approximately corresponds to {1,,1,,1,,1,,1,,1,,2}
  • \(\psi_{I(\omega,0)}(0)\) approximately corresponds to {1{2,,}2}
  • \(\Omega_{\psi_{I(\omega,0)}(0)+1}\) approximately corresponds to {1,,2{2,,}2}
  • \(I(1,\psi_{I(\omega,0)}(0)+1)\) approximately corresponds to {1,,1,,2{2,,}2}
  • \(I(2,\psi_{I(\omega,0)}(0)+1)\) approximately corresponds to {1,,1,,1,,2{2,,}2}
  • \(\psi_{I(\omega,0)}(1)\) approximately corresponds to {1{2,,}3}
  • \(I(\omega,0)\) approximately corresponds to {1{2,,}1,,2}
  • \(\Omega_{I(\omega,0)+1}\) approximately corresponds to {1,,2{2,,}1,,2}
  • \(I(1,I(\omega,0)+1)\) approximately corresponds to {1,,1,,2{2,,}1,,2}
  • \(\psi_{I(\omega,1)}(0)\) approximately corresponds to {1{2,,}2,,2}
  • \(I(\omega,1)\) approximately corresponds to {1{2,,}1,,3}
  • \(I(\omega+1,0)\) approximately corresponds to {1{2,,}1,,1,,2}
  • \(\psi_{I(\omega2,0)}(0)\) approximately corresponds to {1{2,,}1{2,,}2}
  • \(\psi_{I(\omega^2,0)}(0)\) approximately corresponds to {1{3,,}2}
  • \(\psi_{I(\psi_{I(\omega,0)}(0),0)}(0)\) approximately corresponds to {1{1{1{2,,}2}2,,}2}
  • \(I(I(\omega,0),0)\) approximately corresponds to {1{1{1{2,,}1,,2}2,,}1,,2}
  • \(\psi_{I(1,0,0)}(0)\) approximately corresponds to {1{1♦2,,}2} (where ♦ = {1{1,,2,,}2})
  • \(\Omega_{\psi_{I(1,0,0)}(0)+1}\) approximately corresponds to {1,,2{1♦2,,}2}
  • \(\psi_{I(\omega,\psi_{I(1,0,0)}(0)+1)}(1)\) approximately corresponds to {1{2,,}2{1♦2,,}2}
  • \(I(\omega,\psi_{I(1,0,0)}(0)+1)\) approximately corresponds to {1{2,,}1,,2{1♦2,,}2}
  • \(\psi_{I(\psi_{I(1,0,0)}(0),1)}(1)\) approximately corresponds to {1{1{1{1♦2,,}2}2,,}2{1♦2,,}2}
  • \(I(\psi_{I(1,0,0)}(0),1)\) approximately corresponds to {1{1{1{1♦2,,}2}2,,}1,,2{1♦2,,}2}
  • \(\psi_{I(1,0,0)}(1)\) approximately corresponds to {1{1♦2,,}3}
  • \(I(1,0,0)\) approximately corresponds to {1{1♦2,,}1,,2}
  • \(I(\omega,I(1,0,0)+1)\) approximately corresponds to {1{2,,}1,,2{1♦2,,}1,,2}
  • \(I(I(1,0,0),1)\) approximately corresponds to {1{1{1{1♦2,,}1,,2}2,,}1,,2{1♦2,,}1,,2}
  • \(\psi_{I(1,0,1)}(0)\) approximately corresponds to {1{1♦2,,}2,,2}
  • \(I(1,0,1)\) approximately corresponds to {1{1♦2,,}1,,3}
  • \(I(1,1,0)\) approximately corresponds to {1{1♦2,,}1,,1,,2}
  • \(\psi_{I(2,0,0)}(0)\) approximately corresponds to {1{1♦2,,}1{1♦2,,}2}
  • \(I(2,0,0)\) approximately corresponds to {1{1♦2,,}1{1♦2,,}1,,2}
  • \(\psi_{I(\omega,0,0)}(0)\) approximately corresponds to {1{2♦2,,}2}
  • \(\psi_{I(\psi_{I(1,0,0)}(0),0,0)}(0)\) approximately corresponds to {1{1{1{1♦2,,}2}2♦2,,}2}
  • \(I(I(1,0,0),0,0)\) approximately corresponds to {1{1{1{1♦2,,}1,,2}2♦2,,}1,,2}
  • \(I(1,0,0,0)\) approximately corresponds to {1{1♦3,,}1,,2}
  • \(I(1,1,0,0)\) approximately corresponds to {1{1♦3,,}1{1♦2,,}1,,2}
  • \(I(2,0,0,0)\) approximately corresponds to {1{1♦3,,}1{1♦3,,}1,,2}
  • \(I(1,0,0,0,0)\) approximately corresponds to {1{1♦4,,}1,,2}
  • \(I(1,0,0,0,0,0)\) approximately corresponds to {1{1♦5,,}1,,2}

Using weakly Mahlos

Now we need another ordinal collapsing function to generate regular cardinals (or weakly inaccessible cardinals). In that function, we need a "large" ordinal just like the \(\Omega\) in \(\psi()\) function. An ordinal is weakly Mahlo if it's an uncountable regular cardinal, and regular cardinals in it (in another word, less than it) are stationary. Weakly Mahlo cardinals are weakly inaccessible, \(\alpha\)-weakly inaccessible, \((\alpha,\beta)\)-weakly inaccessible, \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly inaccessible, and so on. They're large enough for the new type of collapsing function.

Let \(M_0=0\), \(M_{\alpha+1}\) be the next weakly Mahlo cardinal after \(M_\alpha\), and \(M_\alpha=\sup\{M_\beta|\beta<\alpha\}\) 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& \{M_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is weakly Mahlo}\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is uncountable regular}\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \text{for weakly Mahlo }\pi,\ \chi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular}\} \\ \text{for uncountable regular }\pi,\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\) (first uncountable cardinal), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(M\) is a shorthand for \(M_1\), \(\chi(\alpha)\) is a shorthand for \(\chi_M(\alpha)\). At this point, weakly Mahlo cardinals can be written as \(M_{\alpha+1}\), uncountable regular cardinals can be written as \(\chi_\pi(\alpha)\) or \(M_{\alpha+1}\).

This time, again \(\psi_\pi(0)=1\). But \(\chi_\pi(0)\) is restricted to uncountable regular cardinals, so \(\chi_\pi(0)\ge\Omega\), then \(C(0,\beta)\) can contain larger ordinals. \(\chi(0)=\Omega\), \(\chi_{M_2}(0)=\Omega_{M+1}\), \(\chi_{M_{\omega+1}}(0)=\Omega_{M_\omega+1}\), etc.

Next, \(C(1,0)\) contains all natural numbers, \(M\), \(M_2\), \(M_M\), \(M_{M_M}\), etc. and \(\Omega\), \(\Omega2\), \(\Omega3\), \(M_\Omega\), etc. Then \(\psi(1)=\omega\), \(\psi_{\Omega_2}(1)=\Omega\omega\), but \(\psi_\pi(1)\) for \(\Omega_2\le\pi\le M\) is still \(\Omega\omega\). \(\psi_\pi(1)=M\omega\) for \(\Omega_{M+1}\le\pi\le M_2\), \(\psi_\pi(1)=M_2\omega\) for \(\Omega_{M_2+1}\le\pi\le M_3\), \(\psi_\pi(1)=M_\omega\) for \(\Omega_{M_\omega+1}\le\pi\le M_{\omega+1}\), \(\psi_\pi(1)=M_\Omega\omega\) for \(\Omega_{M_\Omega+1}\le\pi\le M_{\Omega+1}\), etc. \(\chi_\pi(1)\) is restricted to uncountable regular cardinals, so \(\chi(1)=\Omega_2\), \(\chi_{M_2}(1)=\Omega_{M+2}\), \(\chi_{M_{\omega+1}}(1)=\Omega_{M_\omega+2}\), etc.

It follows that \(\chi(2)=\Omega_3\), \(\chi(3)=\Omega_4\), etc. and \(\chi(\omega)=\Omega_{\omega+1}\). \(\psi_{\Omega_3}(1)=\Omega\omega\), \(\psi_{\Omega_3}(2)=\Omega_2\omega\), \(\psi_{\Omega_3}(3)=\Omega_2\omega^2\); \(\psi_{\Omega_4}(2)=\Omega_2\omega\), \(\psi_{\Omega_4}(3)=\Omega_3\omega\), \(\psi_{\Omega_4}(4)=\Omega_3\omega^2\). For \(\pi<M\), \(\psi_\pi(\alpha)\) approximately follows \(\chi(\beta)\) (times \(\omega\), or get a limit such as \(\psi_{\Omega_{\omega+1}}(\omega)=\Omega_\omega\)) until \(\pi\) is generated (this point for \(\chi(\alpha)\) is \(\psi_{\chi(\alpha)}(\alpha)\)). After that, it works in previous ways. But \(\psi_M(\alpha)\) approximately follows \(\chi(\beta)\) all the time. So for example \(\psi_M(1)=\Omega\omega\), \(\psi_M(2)=\Omega_2\omega\), \(\psi_M(3)=\Omega_3\omega\), \(\psi_M(\omega)=\Omega_\omega\), \(\psi_M(\omega+1)=\Omega_{\omega+1}\omega\), etc. Similar things happen between any \(M_\alpha\) and \(M_{\alpha+1}\).

Next important points are the omega-fixed-point and the \(I\). Let \(\omega_*\) denote the omega-fixed-point. \(\psi_{\Omega_{\omega_*+1}}(\omega_*)=\psi_M(\omega_*)=\omega_*\), and \(\chi(\omega_*)=\Omega_{\omega_*+1}\). Now think about \(\psi_\pi(\omega_*+1)\). In \(C(\omega_*+1,\omega_*)\), from ordinals less than \(\omega_*\) we cannot get \(\omega_*\) by addition, \(\psi_M\) and \(\chi_M\). So \(\psi_\pi(\omega_*+1)=\omega_*\) for \(\Omega_{\omega_*+1}\le\pi\le M\). But \(\chi(\omega_*+1)\) needs to be uncountable regular, so it's at least \(\Omega_{\omega_*+1}\); in \(C(\omega_*+1,\Omega_{\omega_*+1})\), we have \(\omega_*\), and we can apply \(\chi(\omega_*)=\Omega_{\omega_*+1}\), so \(\chi(\omega_*+1)=\Omega_{\omega_*+2}\) - \(\psi_\pi\) get stuck but \(\chi_M\) doesn't. To see the point where \(\chi_M\) get stuck we need a fixed point of it - \(\chi(I)=I\). Then \(\chi(I+1)=I\) - now \(\chi_M\) also get stuck.

They get unstuck beyond \(M\). \(\psi_\pi(M)=\omega_*\) for \(\Omega_{\omega_*+1}\le\pi\le M\), and \(\chi(M)=I\). In \(C(M+1,0)\) we can apply \(\psi_\pi(M)=\omega_*\) and \(\chi(M)=I\), then \(\psi_{\Omega_{\omega_*+1}}(M+1)=\omega_*\omega\), and \(\chi(M+1)=\Omega_{I+1}\). We can also apply \(\chi(\omega_*)=\Omega_{\omega_*+1}\), \(\chi(\Omega_{\omega_*+1})=\Omega_{\Omega_{\omega_*+1}+1}\), until the next omega-fixed-point, which equals \(\psi_I(M+1)\), and \(\psi_M(M+1)=I\omega\).

Continue upward, \(\chi(M+2)=\Omega_{I+2}\), \(\chi(M+\omega)=\Omega_{I+\omega+1}\), \(\chi(M+I)=\Omega_{I2+1}\), \(\chi(M2)=I_2\), \(\chi(M2+1)=\Omega_{I_2+1}\), \(\chi(M\omega)=\Omega_{I_\omega+1}\), \(\chi(M\omega+M)=I_{\omega+1}\), \(\chi(MI)=\Omega_{I_I+1}\), \(\chi(M^2)=I(2,0)\), \(\chi(M^3)=I(3,0)\), \(\chi(M^\omega)=\Omega_{\sup\{I(n,0)|n<\omega\}+1}\), \(\chi(M^{\omega+1})=I(\omega,0)\), \(\chi(M^M)=I(1,0,0)\), \(\chi(M^M2)=I(1,0,1)\), \(\chi(M^{M+1})=I(1,1,0)\), \(\chi(M^{M2})=I(2,0,0)\), \(\chi(M^{M^2})=I(1,0,0,0)\), \(\chi(M^{M^3})=I(1,0,0,0,0)\), etc.

Here come comparisons between this OCF and pDAN.

pDAN separator Ordinal
{1{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega})\)
{1{1,,1,,1,2}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\psi_{\chi(M\omega)}(M\omega))\)
{1{1{2,,}2}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\psi_{\chi(M^\omega)}(M^\omega))\)
{1 {1{1{1{1,,2,,}2}2,,}2} 2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\psi_{\chi(M^M)}(M^M))\)
{1 {1{1{1{1,,2,,}2}2,,}1,,2} 2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\chi(M^M))\)
{1 {1{1{1{1,,2,,}2}3,,}1,,2} 2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\chi(M^{M^2}))\)
{1 {1{1{1{1,,2,,}2}1,2,,}2} 2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\psi_{\chi(M^{M^\omega})}(M^{M^\omega}))\)
{1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\chi(M^{M^\omega}))\)
{1,,3{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\chi(M^{M^\omega}+1))\)
{1,,1 {1,,2{1{1{1,,2,,}2}1,2,,}2} 2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+\chi(M^{M^\omega}+\chi(M^{M^\omega})))\)
{1,,1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M)\)
{1,,1 {1,,1,,2{1{1{1,,2,,}2}1,2,,}2} 3{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M+\psi_{\chi(M^{M^\omega}+M)}(M^{M^\omega}+M+1))\)
{1,,1 {1,,1,,2{1{1{1,,2,,}2}1,2,,}2}1 {1,,1,,2{1{1{1,,2,,}2}1,2,,}2} 2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M+\chi(M^{M^\omega}+M))\)
{1,,2,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M+\chi(M^{M^\omega}+M+1))\)
{1,,1,,3{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M2)\)
{1,,1,,1,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M\omega)\)
{1,,1,,1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^2)\)
{1{2,,}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^\omega)\)
{1,,2{2,,}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^\omega+\chi(M^{M^\omega}+M^\omega))\)
{1,,1,,2{2,,}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^\omega+M)\)
{1{2,,}3{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^\omega2)\)
{1{2,,}1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^{\omega+1})\)
{1{1{1{1,,2,,}2}2,,}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^M)\)
{1{1{1{1,,2,,}2}2,,}3{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^M+\psi_{\chi(M^{M^\omega}+M^M)}(M^{M^\omega}+M^M+1))\)
{1{1{1{1,,2,,}2}2,,}1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^M+\chi(M^{M^\omega}+M^M))\)
{1,,2{1{1{1,,2,,}2}2,,}1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^M+\chi(M^{M^\omega}+M^M+1))\)
{1,,1,,2{1{1{1,,2,,}2}2,,}1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^M+M)\)
{1{1{1{1,,2,,}2}2,,}2,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^M2)\)
{1{1{1{1,,2,,}2}2,,}1,,1,,2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^{M+1})\)
{1{1{1{1,,2,,}2}3,,}2{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega}+M^{M^2})\)
{1{1{1{1,,2,,}2}1,2,,}3} \(\psi(M^{M^\omega}2)\)
{1{1{1{1,,2,,}2}1,2,,}1,,2} \(\psi(M^{M^\omega+1})\)
{1{1{1{1,,2,,}2}1,2,,}1{1{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega2})\)
{1{2{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega\omega})\)
{1{1{1{1{1{1,,2,,}2}1,2,,}2}2{1{1,,2,,}2}1,2,,}2} \(\psi(M^{M^\omega\chi(M^{M^\omega})})\)
{1{1{1{1,,2,,}2}2,2,,}2} \(\psi(M^{M^{\omega+1}})\)
{1{1{1{1,,2,,}2}1,1,2,,}2} \(\psi(M^{M^{\omega^2}})\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M})\)
{1 {1{1{1{1,,2,,}2}1,2,,}2} 2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\psi_{\chi(M^{M^\omega})}(M^{M^\omega}))\)
{1 {1,,2{1{1{1,,2,,}2}1,2,,}2} 2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\chi(M^{M^\omega}))\)
{1 {1,,2{1{1{1,,2,,}2}1 {1,,2{1{1{1,,2,,}2}1,2,,}2} 2,,}2} 2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\chi(M^{M^{\chi(M^{M^\omega})}}))\)
{1 {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} 2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\psi_{\chi(M^{M^M})}(M^{M^M}))\)
{1 {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} 3{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\psi_{\chi(M^{M^M})}(M^{M^M})2)\)
{1,,2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\chi(M^{M^{\psi_{\chi(M^{M^M})}(M^{M^M})}}))\)
{1,,2 {1{1{1,,2,,}2}1 {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} 2,,}2 {1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M}+\chi(M^{M^{\psi_{\chi(M^{M^M})}(M^{M^M})}}2))\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}3} \(\psi(M^{M^M}+\psi_{\chi(M^{M^M})}(M^{M^M}+1))\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2} \(\psi(M^{M^M}+\chi(M^{M^M}))\)
{1,,2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2} \(\psi(M^{M^M}+\chi(M^{M^M}+1))\)
{1,,1,,2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2} \(\psi(M^{M^M}+M)\)
{1{1{1{1,,2,,}2}2,,}2 {1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2} \(\psi(M^{M^M}+M^M)\)
{1{1{1{1,,2,,}2}1,2,,}2 {1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2} \(\psi(M^{M^M}+M^{M^\omega})\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2,,2} \(\psi(M^{M^M}2)\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,1,,2} \(\psi(M^{M^M+1})\)
{1{2{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^M\omega})\)
{1{1{1{1,,2,,}2}2{1{1,,2,,}2}2,,}2} \(\psi(M^{M^{M+1}})\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}3,,}2} \(\psi(M^{M^{M2}})\)
{1{1{1{1,,2,,}2}1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} \(\psi(M^{M^{M^2}})\)
{1{1{2{1,,2,,}2}2,,}2} \(\psi(M^{M^{M^\omega}})\)
{1{1{1,,2,,}2}2{1,,2,,}2} \(\psi(M^{M^{M^M}})\)
{1{1{1,,2,,}2}1{1{1,,2,,}2}2{1,,2,,}2} \(\psi(M^{M^{M^{M^M}}})\)
{1{1{1{1,,2,,}2}2{1,,2,,}2}2{1,,2,,}2} \(\psi(M^{M^{M^{M^{M^M}}}})\)
{1,,2{1,,2,,}2} \(\psi(\varepsilon_{M+1})=\psi(\Omega_{M+1})=\psi(\chi_{M_2}(0))\)
{1 {1{1{1{1,,2,,}2}2,,}1,,2} 2{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\chi(M^M))\)
{1 {1{1{1{1{1,,2,,}2}2{1,,2,,}2}2,,}1,,2} 2{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\chi(M^{M^{M^M}}))\)
{1 {1{1{1,,2{1,,2,,}2}2,,}2} 2{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\psi_{\chi(\Omega_{M+1})}(\Omega_{M+1}))\)
{1 {1{1{1,,2{1,,2,,}2}2,,}2} 1,2{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\psi_{\chi(\Omega_{M+1})}(\Omega_{M+1}+1))\)
{1,,2{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\chi(\Omega_{M+1}))\)
{1,,3{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\chi(\Omega_{M+1}+1))\)
{1,,1,,2{1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+M)\)
{1{1{1{1,,2,,}2}2,,}2 {1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+M^M)\)
{1{1{1{1{1,,2,,}2}2{1,,2,,}2}2,,}2 {1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+M^{M^{M^M}})\)
{1{1{1,,2{1,,2,,}2}2,,}3} \(\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}))\)
{1{1{1,,2{1,,2,,}2}2,,}1,2} \(\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}+1))\)

\(=\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1})\omega)\)

{1{1{1,,2{1,,2,,}2}2,,}1,,2} \(\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}+M))\)

\(=\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1})M)\)

{1{1{1,,2{1,,2,,}2}2,,}1 {1{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1})))\)

\(=\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1})^2)\)

{1{2{1,,2{1,,2,,}2}2,,}2} \(\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}+1)))\)

\(=\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1})^\omega)\)

{1{1{1,,2{1,,2,,}2}1,2,,}2} \(\psi(\Omega_{M+1}+\omega^{\omega^{\omega^{\psi_{\Omega_{M+1}}(\Omega_{M+1})+1}}})\)
{1{1 {2{1,,2{1,,2,,}2}2{1,,2,,}2} 2,,}2} \(\psi(\Omega_{M+1}+\omega^{\omega^{\omega^{\omega^{\psi_{\Omega_{M+1}}(\Omega_{M+1})+1}}}})\)
{1{1{1,,2{1,,2,,}2}2{1,,2,,}2}1,2{1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(\Omega_{M+1}+\omega^{\omega^{\omega^{\omega^{\omega^{\psi_{\Omega_{M+1}}(\Omega_{M+1})+1}}}}})\)
{1{1,,2{1,,2,,}2}3{1,,2,,}2} \(\psi(\Omega_{M+1}2)\)
{1{1,,2{1,,2,,}2}1{1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(\Omega_{M+1}^2)\)
{1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(\Omega_{M+1}^{\Omega_{M+1}})\)
{1,,3{1,,2,,}2} \(\psi(\Omega_{M+2})\)
{1,,1,2{1,,2,,}2} \(\psi(\Omega_{M+\omega})\)
{1,,1`2{1,,2,,}2} \(\psi(\Omega_{M+\Omega})\)
{1,,1{1,,1{1,,1,,2}2}2{1,,2,,}2} \(\psi(\Omega_{M+\psi_{\chi(M)}(M)})\)
{1,,1{1,,1{1,,2{1,,2,,}2}2}2{1,,2,,}2} \(\psi(\Omega_{M+\psi_{\chi(M)}(\Omega_{M+1})})\)
{1,,1{1,,1,,2}2{1,,2,,}2} \(\psi(\Omega_{M+\chi(M)})\)
{1,,1{1{1{1{1,,2,,}2}2,,}2}2{1,,2,,}2} \(\psi(\Omega_{M+\chi(M^M)})\)
{1,,1{1{1{1,,2{1,,2,,}2}2,,}2}2{1,,2,,}2} \(\psi(\Omega_{M+\chi(\Omega_{M+1})})\)
{1,,1{1{1,,2,,}2}2{1,,2,,}2} \(\psi(\Omega_{M2})=\psi(\chi_{M_2}(M))\)
{1,,1{1,,1{1{1,,2,,}2}2{1,,2,,}2}2{1,,2,,}2} \(\psi(\Omega_{\Omega_{M2}})=\psi(\chi_{M_2}(\chi_{M_2}(M)))\)
{1,,1,,2{1,,2,,}2} \(\psi(M_2)\)
{1{1{1,,1,,2{1,,2,,}2}2,,}3} \(\psi(M_2+\psi_{\Omega_{M+1}}(M_2))\)
{1{1,,2{1,,2,,}2}2 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 2{1,,2,,}2} \(\psi(M_2+\Omega_{M+1})=\psi(M_2+\chi_{M_2}(0))\)
{1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 3{1,,2,,}2} \(\psi(M_2+\psi_{\Omega_{M+2}}(M_2))\)
{1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2}1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 2{1,,2,,}2} \(\psi(M_2+\psi_{\Omega_{M+2}}(M_2+\psi_{\Omega_{M+2}}(M_2)))\)
{1 {1,,2{1,,2,,}2} 2,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\psi_{\Omega_{M+2}}(M_2+\psi_{\Omega_{M+2}}(M_2+\Omega_{M+1})))\)
{1 {1,,1,2{1,,2,,}2} 2,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\Omega_{M+\omega})\)
{1 {1,,1{1{1,,2,,}2}2{1,,2,,}2} 2,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(M))\)
{1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 2,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\psi_{\chi_{M_2}(M_2)}(M_2))\)
{1,,2{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(\psi_{\chi_{M_2}(M_2)}(M_2)))\)
{1,,3{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(\psi_{\chi_{M_2}(M_2)}(M_2)+1))\)
{1,,1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 2{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(\psi_{\chi_{M_2}(M_2)}(M_2)2))\)
{1,,1{1,,1,,2{1,,2,,}2}3{1,,2,,}2} \(\psi(M_2+\psi_{\chi_{M_2}(M_2)}(M_2+1))\)
{1,,1{1,,1,,2{1,,2,,}2}1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 2{1,,2,,}2} \(\psi(M_2+\psi_{\chi_{M_2}(M_2)}(M_2+\psi_{\chi_{M_2}(M_2)}(M_2)))\)
{1,,1 {1,,1,,2{1,,2,,}2}1 {1,,1,,2{1,,2,,}2} 2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(M_2))\)
{1,,2,,2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(M_2+1))\)
{1,,1{1,,1,,2{1,,2,,}2}2,,2{1,,2,,}2} \(\psi(M_2+\chi_{M_2}(M_2+\chi_{M_2}(M_2)))\)
{1,,1,,3{1,,2,,}2} \(\psi(M_22)\)
{1,,1,,1{1{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2M)\)
{1,,1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} \(\psi(M_2\chi_{M_2}(M_2))\)
{1,,1,,1,,2{1,,2,,}2} \(\psi(M_2^2)\)
{1,,1,,2,,2{1,,2,,}2} \(\psi(M_2^2+M_2)\)
{1,,1,,1,,3{1,,2,,}2} \(\psi(M_2^22)\)
{1,,1,,1,,1,,2{1,,2,,}2} \(\psi(M_2^3)\)
{1{2,,}2{1,,2,,}2} \(\psi(M_2^\omega)\)
{1{1 {1{1,,2,,}2} 2,,}2{1,,2,,}2} \(\psi(M_2^M)\)
{1{1 {1,,1,,2{1,,2,,}2} 2,,}2{1,,2,,}2} \(\psi(M_2^{\chi_{M_2}(M_2)})\)
{1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} or {1{1,,2,,}3} \(\psi(M_2^{M_2})\)
{1{1 {1{1,,2,,}3} 2,,}3} \(\psi(M_2^{M_2}+\psi_{\Omega_{M+1}}(M_2^{M_2}))\)
{1{1,,2{1,,2,,}2}2 {1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} 2{1,,2,,}2} \(\psi(M_2^{M_2}+\Omega_{M+1})\)
{1 {1,,1,2{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\Omega_{M+\omega})\)
{1 {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\psi_{\chi_{M_2}(M_2)}(M_2))\)
{1 {1,,1 {1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} 2{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\psi_{\chi_{M_2}(M_2)}(M_2^{M_2}))\)
{1 {1,,1,,2{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2))\)
{1 {1,,1,,3{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_22))\)
{1 {1,,1,,1,,2{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^2))\)
{1 {1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} 2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2}))\)
{1,,2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})}))\)
{1,,3{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})}+1))\)
{1,,1,,2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})}+M_2))\)
{1{1 {1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} 2,,}2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})}2))\)
{1{1 {1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} 2,,}1,,2{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})+1}))\)
{1{1 {1{1,,2,,}3} 2,,}3{1,,2,,}2} \(\psi(M_2^{M_2}+\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2}+1))\)
{1{1 {1{1,,2,,}3} 2,,}1,,2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{M_2}))\)
{1,,2{1 {1{1,,2,,}3} 2,,}1,,2{1,,2,,}2} \(\psi(M_2^{M_2}+\chi_{M_2}(M_2^{M_2}+1))\)
{1,,1,,2{1 {1{1,,2,,}3} 2,,}1,,2{1,,2,,}2} \(\psi(M_2^{M_2}+M_2)\)
{1{1 {1{1,,2,,}3} 2,,}2,,2{1,,2,,}2} \(\psi(M_2^{M_2}2)\)
{1{1 {1{1,,2,,}3} 2,,}1,,3{1,,2,,}2} \(\psi(M_2^{M_2}2+\chi_{M_2}(M_2^{M_2}2))\)
{1{1 {1{1,,2,,}3} 2,,}1,,1,2{1,,2,,}2} \(\psi(M_2^{M_2}\omega)\)
{1{1 {1{1,,2,,}3} 2,,}1,,1,,2{1,,2,,}2} \(\psi(M_2^{M_2+1})\)
{1{1 {1{1,,2,,}3} 2,,}1{1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_22})\)
{1{2 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2\omega})\)
{1{1 {1{1,,2,,}3} 3,,}2{1,,2,,}2} \(\psi(M_2^{M_2^2})\)
{1{1 {1{1,,2,,}3} 1,2,,}2{1,,2,,}2} \(\psi(M_2^{M_2^\omega})\)
{1{1 {1{1,,2,,}3}1 {1{1,,2,,}3} 2,,}2{1,,2,,}2} \(\psi(M_2^{M_2^{M_2}})\)
{1{1{1,,2,,}3}2{1,,2,,}3} \(\psi(M_2^{M_2^{M_2^{M_2}}})\)
{1,,2{1,,2,,}3} \(\psi(\Omega_{M_2+1})\)
{1,,3{1,,2,,}3} \(\psi(\Omega_{M_2+2})\)
{1,,1{1{1,,2,,}3}2{1,,2,,}3} \(\psi(\Omega_{M_22})=\psi(\chi_{M_3}(M_2))\)
{1,,1,,2{1,,2,,}3} \(\psi(M_3)\)
{1,,1{1,,1,,2{1,,2,,}3}3{1,,2,,}3} \(\psi(M_3+\psi_{\chi_{M_3}(M_3)}(M_3+1))\)
{1,,1 {1,,1,,2{1,,2,,}3}1 {1,,1,,2{1,,2,,}3} 2{1,,2,,}3} \(\psi(M_3+\chi_{M_3}(M_3))\)
{1,,2,,2{1,,2,,}3} \(\psi(M_3+\chi_{M_3}(M_3+1))\)
{1,,1,,3{1,,2,,}3} \(\psi(M_32)\)
{1,,1,,1,,2{1,,2,,}3} \(\psi(M_3^2)\)
{1{2,,}2{1,,2,,}3} \(\psi(M_3^\omega)\)
{1{1 {1{1,,2,,}4} 2,,}2{1,,2,,}3} or {1{1,,2,,}4} \(\psi(M_3^{M_3})\)
{1{1 {1{1,,2,,}4} 2,,}3{1,,2,,}3} \(\psi(M_3^{M_3}+\psi_{\chi_{M_3}(M_3^{M_3})}(M_3^{M_3}+1))\)
{1{1 {1{1,,2,,}4} 2,,}1,,2{1,,2,,}3} \(\psi(M_3^{M_3}+\chi_{M_3}(M_3^{M_3}))\)
{1,,2{1 {1{1,,2,,}4} 2,,}1,,2{1,,2,,}3} \(\psi(M_3^{M_3}+\chi_{M_3}(M_3^{M_3}+1))\)
{1,,1,,2{1 {1{1,,2,,}4} 2,,}1,,2{1,,2,,}3} \(\psi(M_3^{M_3}+M_3)\)
{1{1 {1{1,,2,,}4} 2,,}2,,2{1,,2,,}3} \(\psi(M_3^{M_3}2)\)
{1{1 {1{1,,2,,}4} 2,,}1,,1,,2{1,,2,,}3} \(\psi(M_3^{M_3+1})\)
{1{2 {1{1,,2,,}4} 2,,}2{1,,2,,}3} \(\psi(M_3^{M_3\omega})\)
{1{1 {1{1,,2,,}4} 3,,}2{1,,2,,}3} \(\psi(M_3^{M_3^2})\)
{1{1 {1{1,,2,,}4}1 {1{1,,2,,}4} 2,,}2{1,,2,,}3} \(\psi(M_3^{M_3^{M_3}})\)
{1{1{1,,2,,}4}2{1,,2,,}4} \(\psi(M_3^{M_3^{M_3^{M_3}}})\)
{1,,2{1,,2,,}4} \(\psi(\Omega_{M_3+1})\)
{1,,1,,2{1,,2,,}4} \(\psi(M_4)\)
{1{1,,2,,}5} \(\psi(M_4^{M_4})\)
{1,,1,,2{1,,2,,}5} \(\psi(M_5)\)
{1{1,,2,,}6} \(\psi(M_5^{M_5})\)
{1{1,,2,,}1,2} \(\psi(M_\omega)\)
{1{1 {1{1,,2,,}1,2} 2,,}3} \(\psi(M_\omega+\psi_{\Omega_{M+1}}(M_\omega))\)
{1{1 {1{1,,2,,}1,2} 2,,}1,2} \(\psi(M_\omega+\omega^{\psi_{\Omega_{M+1}}(M_\omega)+1})\)
{1{2 {1{1,,2,,}1,2} 2,,}2} \(\psi(M_\omega+\omega^{\omega^{\psi_{\Omega_{M+1}}(M_\omega)+1}})\)
{1{1 {1{1,,2,,}1,2} 1,2,,}2} \(\psi(M_\omega+\omega^{\omega^{\omega^{\psi_{\Omega_{M+1}}(M_\omega)+1}}})\)
{1{1 {2{1,,2,,}1,2} 2,,}2} \(\psi(M_\omega+\omega^{\omega^{\omega^{\omega^{\psi_{\Omega_{M+1}}(M_\omega)+1}}}})\)
{1{1{1,,2,,}2}2{1,,2,,}1,2} \(\psi(M_\omega+\omega^{\omega^{\omega^{\omega^{\psi_{\Omega_{M+1}}(M_\omega)+M}}}})\)
{1{1 {1{1,,2,,}1,2} 2,,}3{1,,2,,}2} \(\psi(M_\omega+\psi_{\Omega_{M_2+1}}(M_\omega))\)
{1{1{1,,2,,}3}2{1,,2,,}1,2} \(\psi(M_\omega+\omega^{\omega^{\omega^{\omega^{\psi_{\Omega_{M_2+1}}(M_\omega)+M_2}}}})\)
{1{1{1,,2,,}4}2{1,,2,,}1,2} \(\psi(M_\omega+\omega^{\omega^{\omega^{\omega^{\psi_{\Omega_{M_3+1}}(M_\omega)+M_3}}}})\)
{1{1{1,,2,,}1,2}2{1,,2,,}1,2} \(\psi(M_\omega2)\)
{1,,2{1,,2,,}1,2} \(\psi(\Omega_{M_\omega+1})=\psi(\chi_{M_{\omega+1}}(0))\)
{1,,1,,2{1,,2,,}1,2} \(\psi(M_{\omega+1}))\)
{1{1,,2,,}2,2} \(\psi(M_{\omega+1}^{M_{\omega+1}}))\)
{1,,1,,2{1,,2,,}2,2} \(\psi(M_{\omega+2}))\)
{1{1,,2,,}1,3} \(\psi(M_{\omega2}))\)
{1{1,,2,,}1{1{1,,2,,}2}2} \(\psi(M_M))\)
{1{1,,2,,}1{1{1,,2,,}1{1{1,,2,,}2}2}2} \(\psi(M_{M_M}))\)

Here're some approximations, to make the comparisons above more clear:

  • \(\psi_{\chi(M^{M^\omega})}(M^{M^\omega})\) approximately corresponds to {1{1{1{1,,2,,}2}1,2,,}2}
  • \(\chi(M^{M^\omega})\) approximately corresponds to {1,,2{1{1{1,,2,,}2}1,2,,}2}
  • \(\chi(M^{M^\omega}+M)\) approximately corresponds to {1,,1,,2{1{1{1,,2,,}2}1,2,,}2}
  • \(\chi(M^{M^\omega}+M^M)\) approximately corresponds to {1{1{1{1,,2,,}2}2,,}1,,2{1{1{1,,2,,}2}1,2,,}2}
  • \(\chi(M^{M^\omega}2)\) approximately corresponds to {1,,2{1{1{1,,2,,}2}1,2,,}3}
  • \(\chi(M^{M^\omega+1})\) approximately corresponds to {1{1{1{1,,2,,}2}1,2,,}1,,2}
  • \(\chi(M^{M^{\omega+1}})\) approximately corresponds to {1{1{1{1,,2,,}2}2,2,,}1,,2}
  • \(\psi_{\chi(M^{M^M})}(M^{M^M})\) approximately corresponds to {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}
  • \(\chi(M^{M^{\psi_{\chi(M^{M^M})}(M^{M^M})}})\) approximately corresponds to {1,,2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}
  • \(\chi(M^{M^{\psi_{\chi(M^{M^M})}(M^{M^M})}}2)\) approximately corresponds to {1,,2{1{1{1,,2,,}2}1 {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2} 2,,}2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}
  • \(\psi_{\chi(M^{M^M})}(M^{M^M}+1)\) approximately corresponds to {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}3}
  • \(\chi(M^{M^M})\) approximately corresponds to {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2}
  • \(\chi(M^{M^M}+1)\) approximately corresponds to {1,,2{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2}
  • \(\chi(M^{M^M}2)\) approximately corresponds to {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,3}
  • \(\chi(M^{M^M+1})\) approximately corresponds to {1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,1,,2}
  • \(\chi(M^{M^M\omega})\) approximately corresponds to {1,,2{2{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}
  • \(\chi(M^{M^{M+1}})\) approximately corresponds to {1{1{1{1,,2,,}2}2{1{1,,2,,}2}2,,}1,,2}
  • \(\chi(M^{M^{M2}})\) approximately corresponds to {1{1{1{1,,2,,}2}1{1{1,,2,,}2}3,,}1,,2}
  • \(\chi(M^{M^{M^M}})\) approximately corresponds to {1{1{1{1{1,,2,,}2}2{1,,2,,}2}2,,}1,,2}
  • \(\chi(M^{M^{M^{M^M}}})\) approximately corresponds to {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2{1,,2,,}2}2,,}1,,2}
  • \(\chi(M^{M^{M^{M^{M^M}}}})\) approximately corresponds to {1{1{1{1{1{1,,2,,}2}2{1,,2,,}2}2{1,,2,,}2}2,,}1,,2}
  • \(M\) approximately corresponds to {1{1,,2,,}2}
  • \(\chi(\Omega_{M+1})\) approximately corresponds to {1,,2{1{1,,2{1,,2,,}2}2,,}2}
  • \(\psi_{\Omega_{M+1}}(\Omega_{M+1})\) approximately corresponds to {1{1,,2{1,,2,,}2}2{1,,2,,}2}
  • \(\chi_{M_2}(0)=\Omega_{M+1}\) (next cardinal after \(M\)) approximately corresponds to {1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M)=\Omega_{M2}\) approximately corresponds to {1,,1{1{1,,2,,}2}2{1,,2,,}2}
  • \(\psi_{\chi_{M_2}(M_2)}(M_2)\) (next omega-fixed-point after \(M\)) approximately corresponds to {1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2}
  • \(\chi_{M_2}(\psi_{\chi_{M_2}(M_2)}(M_2))\) (next cardinal after \(\psi_{\chi_{M_2}(M_2)}(M_2)\)) approximately corresponds to {1,,2{1,,1,,2{1,,2,,}2}2{1,,2,,}2}
  • \(\chi_{M_2}(\psi_{\chi_{M_2}(M_2)}(M_2)+1)\) approximately corresponds to {1,,3{1,,1,,2{1,,2,,}2}2{1,,2,,}2}
  • \(\chi_{M_2}(\psi_{\chi_{M_2}(M_2)}(M_2)2)\) approximately corresponds to {1,,1{1,,1{1,,1,,2{1,,2,,}2}2{1,,2,,}2}2{1,,1,,2{1,,2,,}2}2{1,,2,,}2}
  • \(\psi_{\chi_{M_2}(M_2)}(M_2+1)\) (next omega-fixed-point after \(\psi_{\chi_{M_2}(M_2)}(M_2)\)) approximately corresponds to {1,,1{1,,1,,2{1,,2,,}2}3{1,,2,,}2}
  • \(\psi_{\chi_{M_2}(M_2)}(M_2+\chi_{M_2}(M_2))\) approximately corresponds to {1,,1{1,,1,,2{1,,2,,}2}1{1,,1,,2{1,,2,,}2}2{1,,2,,}2}
  • \(\chi_{M_2}(M_2)\) (next weakly inaccessible after \(M\)) approximately corresponds to {1,,1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_2+1)\) approximately corresponds to {1,,2,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_22)\) approximately corresponds to {1,,1,,3{1,,2,,}2}
  • \(\chi_{M_2}(M_2^2)\) approximately corresponds to {1,,1,,1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_2^\omega)\) approximately corresponds to {1,,2{2,,}2{1,,2,,}2}
  • \(\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})\) approximately corresponds to {1{1{1{1,,2,,}3}2,,}2{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2})})\) approximately corresponds to {1,,2{1{1{1,,2,,}3}2,,}2{1,,2,,}2}
  • \(\psi_{\chi_{M_2}(M_2^{M_2})}(M_2^{M_2}+1)\) approximately corresponds to {1{1{1{1,,2,,}3}2,,}3{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{M_2})\) approximately corresponds to {1{1{1{1,,2,,}3}2,,}1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{M_2}+1)\) approximately corresponds to {1,,2{1{1{1,,2,,}3}2,,}1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{M_2}2)\) approximately corresponds to {1{1{1{1,,2,,}3}2,,}1,,3{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{M_2^2})\) approximately corresponds to {1{1{1{1,,2,,}3}3,,}1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{M_2^{M_2}})\) approximately corresponds to {1{1{1{1,,2,,}3}1{1{1,,2,,}3}2,,}1,,2{1,,2,,}2}
  • \(\chi_{M_2}(M_2^{M_2^{M_2^{M_2}}})\) approximately corresponds to {1{1{1{1{1,,2,,}3}2{1,,2,,}3}2,,}1,,2{1,,2,,}2}
  • \(M_2\) approximately corresponds to {1{1,,2,,}3}
  • \(\chi_{M_2}(\Omega_{M_2+1})\) approximately corresponds to {1{1{1,,2{1,,2,,}3}2,,}1,,2{1,,2,,}2}
  • \(\psi_{\Omega_{M_2+1}}(\Omega_{M_2+1})\) approximately corresponds to {1{1,,2{1,,2,,}3}2{1,,2,,}3}
  • \(\chi_{M_3}(1)=\Omega_{M_2+1}\) approximately corresponds to {1,,2{1,,2,,}3}
  • \(\chi_{M_3}(M_3)\) approximately corresponds to {1,,1,,2{1,,2,,}3}
  • \(\psi_{\chi_{M_3}(M_3^{M_3})}(M_3^{M_3})\) approximately corresponds to {1{1{1{1,,2,,}4}2,,}2{1,,2,,}3}
  • \(\psi_{\chi_{M_3}(M_3^{M_3})}(M_3^{M_3}+1)\) approximately corresponds to {1{1{1{1,,2,,}4}2,,}3{1,,2,,}3}
  • \(\chi_{M_3}(M_3^{M_3})\) approximately corresponds to {1{1{1{1,,2,,}4}2,,}1,,2{1,,2,,}3}
  • \(\chi_{M_3}(M_3^{M_3^{M_3}})\) approximately corresponds to {1{1{1{1,,2,,}4}1{1{1,,2,,}4}2,,}1,,2{1,,2,,}3}
  • \(\chi_{M_3}(M_3^{M_3^{M_3^{M_3}}})\) approximately corresponds to {1{1{1{1{1,,2,,}4}2{1,,2,,}4}2,,}1,,2{1,,2,,}3}
  • \(M_3\) approximately corresponds to {1{1,,2,,}4}
  • \(\chi_{M_4}(M_4)\) approximately corresponds to {1,,1,,2{1,,2,,}4}
  • \(M_4\) approximately corresponds to {1{1,,2,,}5}
  • \(M_5\) approximately corresponds to {1{1,,2,,}6}
  • \(M_\omega\) approximately corresponds to {1{1,,2,,}1,2}
  • \(\chi_{M_{\omega+1}}(1)=\Omega_{M_\omega+1}\) approximately corresponds to {1,,2{1,,2,,}1,2}
  • \(M_{\omega+1}\) approximately corresponds to {1{1,,2,,}2,2}
  • \(M_{\omega2}\) approximately corresponds to {1{1,,2,,}1,3}
  • \(M_M\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}2}2}
  • \(M_{M_M}\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1{1{1,,2,,}2}2}2}

Inaccessibility over weakly Mahlos

The previous section we reach a "weakly Mahlo-fixed-point" - denoted by \(M_*\) here. Similar to the "Using a weakly inaccessible" section, the next step is adding a weakly inaccessible cardinal which is a limit of weakly Mahlo cardinals - and it'll approximately correspond to {1{1,,2,,}1,,2} in pDAN. We can also add more weakly inaccessible cardinals which are limits of weakly Mahlo cardinals, then weakly inaccessible cardinals which are limits of that kind of inaccessible cardinals... And we can go further like previous \(I(\alpha_1,\alpha_2,\cdots\alpha_n,\beta)\).

But there's a stronger version of this. Now think of a weakly Mahlo cardinal which is a limit of weakly Mahlo cardinals, denoted by \(M(1,0)\) - the \(\chi_{M(1,0)}\) will diagonalize over all the weakly inaccessibility whose ordinals are limits of weakly Mahlo cardinals; and \(M(1,0)\) will approximately correspond to {1{1,,2,,}1{1,,2,,}2} in pDAN. We can go further: let \(M(\alpha_1,\alpha_2,\cdots\alpha_n,\beta,0,0,\cdots0,0)\) be the first weakly Mahlo cardinal \(\pi\) such that \(\pi=M(\alpha_1,\alpha_2,\cdots\alpha_n,\delta,\pi,0,\cdots0,0)\) for all \(\delta<\beta\), \(M(\alpha_1,\alpha_2,\cdots\alpha_n,\beta,0,0,\cdots0,\gamma+1)\) be the next weakly Mahlo cardinal \(\pi\) such that \(\pi=M(\alpha_1,\alpha_2,\cdots\alpha_n,\delta,\pi,0,\cdots0,0)\) for all \(\delta<\beta\) after \(M(\alpha_1,\alpha_2,\cdots\alpha_n,\beta,0,0,\cdots0,\gamma)\), and \(M(\alpha_1,\alpha_2,\cdots\alpha_n,\beta)=\sup\{M(\alpha_1,\alpha_2,\cdots\alpha_n,\delta)|\delta<\beta\}\) for limit ordinal \(\beta\). Especially, when all the \(\alpha_1,\alpha_2,\cdots\alpha_n,\beta\) are 0, it results \(M(0,\gamma)=M_{1+\gamma}\). This definition is similar to \(I(\alpha_1,\alpha_2,\cdots\alpha_n,\beta)\), except "uncountable regular cardinal" is substituted into "weakly Mahlo cardinal", so it's called "inaccessibility over weakly Mahlos".

\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)|\gamma_1,\gamma_2\cdots,\gamma_k\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is weakly Mahlo}\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is uncountable regular}\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \text{for weakly Mahlo }\pi,\ \chi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular}\} \\ \text{for uncountable regular }\pi,\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*} And \(\Omega\) is a shorthand for \(\Omega_1\) (first uncountable cardinal), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(M\) is a shorthand for \(M_1=M(0,0)\), \(\chi(\alpha)\) is a shorthand for \(\chi_M(\alpha)\). At this point, weakly Mahlo cardinals can be written as \(M(\alpha_1,\alpha_2,\cdots\alpha_n,0)\) or \(M(\alpha_1,\alpha_2,\cdots\alpha_n,\beta+1)\), uncountable regular cardinals can be written as \(\chi_\pi(\alpha)\) or in the forms of weakly Mahlos.

This time, \(\psi_\pi(0)\) have different values because \(C(0,0)\) contains \(M(0,0)=M\), \(M_M\), \(M_{M_M}\), \(M(M,0)\), \(M(M,M,M)\), etc. \(\psi_\pi(0)=1\) for \(\Omega\le\pi\le M\), \(\psi_\pi(0)=M\omega\) for \(\Omega_{M+1}\le\pi\le M_2\), \(\psi_{M(1,0)}(0)=M_*\). \(\chi_\pi(0)\) also have different values. \(\chi(0)=\Omega\), \(\chi_{M_2}(0)=\Omega_{M+1}\), \(\chi_{M_{M_*+1}}(0)=\Omega_{M_*+1}\), \(\chi_{M_{M_*+2}}(0)=\Omega_{M_{M_*+1}+1}\). Now consider \(\chi_{M(1,0)}(0)\): It's at least \(M_*\), but it must be uncountable regular, so we need at least \(\Omega_{M_*+1}\); \(M_*+1\in C(0,\Omega_{M_*+1})\), then we have larger \(M_{M_*+1}\), \(M_{M_{M_*+1}}\), etc. so we need some point that \(M_\alpha\) cannot cross it - that's the first "weakly inaccessible which is a limit of weakly Mahlos". \(\chi_{M_{M(1,0)+1}}(0)=\Omega_{M(1,0)+1}\) again.

\(\psi_{M(1,0)}(1)\) is the next "weakly Mahlo-fixed-point" after \(\chi_{M(1,0)}(0)\). \(\chi(1)=\Omega_2\), \(\chi_{M_2}(1)=\Omega_{M+2}\), \(\chi_{M_{M_*+1}}(1)=\Omega_{M_*+2}\), \(\chi_{M_{M_*+2}}(1)=\Omega_{M_{M_*+1}+2}\), etc. And \(\chi_{M(1,0)}(1)\) is the next "weakly inaccessible which is a limit of weakly Mahlos" after \(\chi_{M(1,0)}(0)\).

\(\psi_{\chi_{M(1,0)}(0)}(0)=M_*\) and \(\psi_{\chi_{M(1,0)}(0)}(1)\) is the next "weakly Mahlo-fixed-point" after \(M_*\), etc., which suggest that \(\chi_{M(1,0)}(0)\) indeed works as the weakly inaccessible over weakly Mahlos.

Here come comparisons between this OCF and pDAN.

pDAN separator Ordinal
{1{1,,2,,}1{1{1,,2,,}1,,2}2} or {1{1,,2,,}1,,2} \(\psi(\psi_{M(1,0)}(0))=\psi(\psi_{\chi_{M(1,0)}(0)}(0))\)
{1{1 {1{1,,2,,}1{1{1,,2,,}1,,2}2} 2,,}3} \(\psi(\psi_{M(1,0)}(0)+\psi_{\Omega_{M+1}}(\psi_{M(1,0)}(0)))\)
{1 {1,,1,2{1,,2,,}2} 2{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(\psi_{M(1,0)}(0)+\Omega_{M+\omega})\)
{1 {1{1,,2,,}1,2} 2{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(\psi_{M(1,0)}(0)+M_\omega)\)
{1 {1{1,,2,,}1{1{1,,2,,}2}2} 2{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(\psi_{M(1,0)}(0)+M_M)\)
{1 {1{1,,2,,}1{1{1,,2,,}1,,2}2} 2{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(\psi_{M(1,0)}(0)2)\)
{1,,2{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(\Omega_{\psi_{M(1,0)}(0)+1})=\psi(\chi_{M_{\psi_{M(1,0)}(0)+1}}(0))\)
{1,,3{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(\Omega_{\psi_{M(1,0)}(0)+2})=\psi(\chi_{M_{\psi_{M(1,0)}(0)+1}}(1))\)
{1,,1,,2{1,,2,,}1{1{1,,2,,}1,,2}2} \(\psi(M_{\psi_{M(1,0)}(0)+1})\)
{1{1,,2,,}2{1{1,,2,,}1,,2}2} \(\psi(M_{\psi_{M(1,0)}(0)+1}^{M_{\psi_{M(1,0)}(0)+1}})\)
{1,,1,,2{1,,2,,}2{1{1,,2,,}1,,2}2} \(\psi(M_{\psi_{M(1,0)}(0)+2})\)
{1{1,,2,,}1,2{1{1,,2,,}1,,2}2} \(\psi(M_{\psi_{M(1,0)}(0)+\omega})\)
{1{1,,2,,}1 {1{1,,2,,}1{1{1,,2,,}1,,2}2} 2{1{1,,2,,}1,,2}2} \(\psi(M_{\psi_{M(1,0)}(0)2})\)
{1{1,,2,,}1 {1{1,,2,,}1,2{1{1,,2,,}1,,2}2} 2{1{1,,2,,}1,,2}2} \(\psi(M_{M_{\psi_{M(1,0)}(0)+\omega}})\)
{1{1,,2,,}1{1{1,,2,,}1,,2}3} \(\psi(\psi_{\chi_{M(1,0)}(0)}(1))\)
{1{1,,2,,}1{1{1,,2,,}1,,2}4} \(\psi(\psi_{\chi_{M(1,0)}(0)}(2))\)
{1{1,,2,,}1{1{1,,2,,}1,,2}1 {1{1,,2,,}2} 2} \(\psi(\psi_{\chi_{M(1,0)}(0)}(M))\)
{1{1,,2,,}1{1{1,,2,,}1,,2}1 {1{1,,2,,}1{1{1,,2,,}1,,2}2} 2} \(\psi(\psi_{\chi_{M(1,0)}(0)}(\psi_{\chi_{M(1,0)}(0)}(0)))\)
{1{1,,2,,}1{1{1,,2,,}1,,2}1{1{1,,2,,}1,,2}2} \(\psi(\chi_{M(1,0)}(0))\)
{1{1,,2,,}1{1{1,,2,,}1,,2}1{1{1,,2,,}1,,2}3} \(\psi(\chi_{M(1,0)}(0)2)\)
{1{1,,2,,}1 {1{1,,2,,}1,,2}1 {1{1,,2,,}1,,2}1 {1{1,,2,,}1,,2}2} \(\psi(\chi_{M(1,0)}(0)^2)\)
{1{1{1,,2,,}1,,2}2{1,,2,,}1,,2} \(\psi(\chi_{M(1,0)}(0)^{\chi_{M(1,0)}(0)})\)
{1,,2{1,,2,,}1,,2} \(\psi(\Omega_{\chi_{M(1,0)}(0)+1})=\psi(\chi_{M_{\chi_{M(1,0)}(0)+1}}(1))\)

(Note that \(\chi_{M_{\chi_{M(1,0)}(0)+1}}(0)=\chi_{M(1,0)}(0)\), which cannot be generated directly by \(M_\gamma\))

{1,,3{1,,2,,}1,,2} \(\psi(\Omega_{\chi_{M(1,0)}(0)+2})=\psi(\chi_{M_{\chi_{M(1,0)}(0)+1}}(2))\)
{1,,1 {1{1,,2,,}1{1{1,,2,,}1,,2}2} 2{1,,2,,}1,,2} \(\psi(\Omega_{\chi_{M(1,0)}(0)+\psi_{\chi_{M(1,0)}(0)}(0)})\)
{1,,1 {1{1,,2,,}1,,2} 2{1,,2,,}1,,2} \(\psi(\Omega_{\chi_{M(1,0)}(0)2})=\)

\(\psi(\chi_{M_{\chi_{M(1,0)}(0)+1}}(\chi_{M_{\chi_{M(1,0)}(0)+1}}(0)))\)

{1,,1,,2{1,,2,,}1,,2} \(\psi(M_{\chi_{M(1,0)}(0)+1})\)
{1,,1,,3{1,,2,,}1,,2} \(\psi(M_{\chi_{M(1,0)}(0)+1}2)\)
{1{1,,2,,}2,,2} \(\psi(M_{\chi_{M(1,0)}(0)+1}^{M_{\chi_{M(1,0)}(0)+1}})\)
{1,,1,,2{1,,2,,}2,,2} \(\psi(M_{\chi_{M(1,0)}(0)+2})\)
{1{1,,2,,}1,2,,2} \(\psi(M_{\chi_{M(1,0)}(0)+\omega})\)
{1{1,,2,,}1{1{1,,2,,}1,,2}2,,2} \(\psi(M_{\chi_{M(1,0)}(0)2})\)
{1{1,,2,,}1,,3} \(\psi(\psi_{M(1,0)}(1))=\psi(\psi_{\chi_{M(1,0)}(1)}(1))\)

(Again, \(\psi_{\chi_{M(1,0)}(1)}(0)=\psi_{\chi_{M(1,0)}(0)}(0)\). Those are similar to the "\(\psi\)-following pattern" in "Using weakly Mahlo" section)

{1{1,,2,,}1{1{1,,2,,}1,,3}1{1{1,,2,,}1,,3}2,,2} \(\psi(\chi_{M(1,0)}(1))\)
{1,,2{1,,2,,}1,,3} \(\psi(\Omega_{\chi_{M(1,0)}(1)+1})=\psi(\chi_{M_{\chi_{M(1,0)}(1)+1}}(2))\)
{1,,1,,2{1,,2,,}1,,3} \(\psi(M_{\chi_{M(1,0)}(1)+1})\)
{1{1,,2,,}1,2,,3} \(\psi(M_{\chi_{M(1,0)}(1)+\omega})\)
{1{1,,2,,}1,,4} \(\psi(\psi_{M(1,0)}(2))\)
{1{1,,2,,}1{1{1,,2,,}1,,4}1{1{1,,2,,}1,,4}2,,3} \(\psi(\chi_{M(1,0)}(2))\)
{1{1,,2,,}1,,1,2} \(\psi(\psi_{M(1,0)}(\omega))\)
{1{1{1,,2,,}1,,1,2}2{1,,2,,}1,,1,2} \(\psi(\psi_{M(1,0)}(\omega)2)\)
{1,,2{1,,2,,}1,,1,2} \(\psi(\Omega_{\psi_{M(1,0)}(\omega)+1})=\psi(\chi_{M_{\psi_{M(1,0)}(\omega)+1}}(\omega))\)
{1,,3{1,,2,,}1,,1,2} \(\psi(\chi_{M_{\psi_{M(1,0)}(\omega)+1}}(\omega+1))\)
{1,,1,,2{1,,2,,}1,,1,2} \(\psi(M_{\psi_{M(1,0)}(\omega)+1})\)
{1{1,,2,,}1,2,,1,2} \(\psi(M_{\psi_{M(1,0)}(\omega)+\omega})\)
{1{1,,2,,}1{1{1,,2,,}1,,2,2}2,,1,2} or {1{1,,2,,}1,,2,2} \(\psi(\psi_{\chi_{M(1,0)}(\omega)}(\omega+1))\)
{1{1,,2,,}1{1{1,,2,,}1,,2,2}3,,1,2} \(\psi(\psi_{\chi_{M(1,0)}(\omega)}(\omega+2))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2,2}1 {1{1,,2,,}1,,2,2} 2,,1,2} \(\psi(\chi_{M(1,0)}(\omega))\)
{1{1,,2,,}1 {1{1,,2,,}1,,3,2}1 {1{1,,2,,}1,,3,2} 2,,2,2} \(\psi(\chi_{M(1,0)}(\omega+1))\)
{1{1,,2,,}1,,1,3} \(\psi(\psi_{M(1,0)}(\omega2))\)
{1{1,,2,,}1,,1{1{1,,2,,}1,,2}2} \(\psi(\psi_{M(1,0)}(\chi_{M(1,0)}(0)))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2{1{1,,2,,}1,,2}2}1 {1{1,,2,,}1,,2{1{1,,2,,}1,,2}2} 2,,1{1{1,,2,,}1,,2}2} \(\psi(\chi_{M(1,0)}(\chi_{M(1,0)}(0)))\)
{1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} or {1{1,,2,,}1,,1,,2} \(\psi(M(1,0))\)
{1 {1{1,,2,,}1,2} 2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+M_\omega)\)
{1 {1{1,,2,,}1 {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} 2} 2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\psi_{\chi_{M(1,0)}(0)}(M(1,0)))\)
{1 {1{1,,2,,}1,,2} 2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\chi_{M(1,0)}(0))\)
{1 {1{1,,2,,}1,2,,2} 2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+M_{\chi_{M(1,0)}(0)+\omega})\)
{1 {1{1,,2,,}1,,3} 2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\chi_{M(1,0)}(1))\)
{1 {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} 2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\psi_{M(1,0)}(M(1,0)))\)
{1,,2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\Omega_{\psi_{M(1,0)}(M(1,0))+1})=\)

\(\psi(M(1,0)+\chi_{M_{\psi_{M(1,0)}(M(1,0))+1}}(M(1,0)))\)

{1,,1,,2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+M_{\psi_{M(1,0)}(M(1,0))+1})\)
{1{1,,2,,}2,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+M_{\psi_{M(1,0)}(M(1,0))+1}^{M_{\psi_{M(1,0)}(M(1,0))+1}})\)
{1{1,,2,,}1,2,,1{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+M_{\psi_{M(1,0)}(M(1,0))+\omega})\)
{1{1,,2,,}1,,2{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\psi_{\chi_{M(1,0)}(\psi_{M(1,0)}(M(1,0)))}(M(1,0)+1))\)
{1{1,,2,,}1,,1,2{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\psi_{\chi_{M(1,0)}(\psi_{M(1,0)}(M(1,0))+\omega)}(\psi_{M(1,0)}(M(1,0))+\omega))\)
{1{1,,2,,}1,,1 {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} 2{1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\psi_{\chi_{M(1,0)}(\psi_{M(1,0)}(M(1,0))2)}(\psi_{M(1,0)}(M(1,0))2))\)
{1{1,,2,,}1{1{1,,2,,}1,,1,,2}3} \(\psi(M(1,0)+\psi_{\chi_{M(1,0)}(M(1,0))}(M(1,0)+1))\)
{1{1,,2,,}1 {1{1,,2,,}1,,1,,2}1 {1{1,,2,,}1,,1,,2}2} \(\psi(M(1,0)+\chi_{M(1,0)}(M(1,0)))\)
{1,,2{1,,2,,}1,,1,,2} \(\psi(M(1,0)+\Omega_{\chi_{M(1,0)}(M(1,0))+1})=\)

\(\psi(M(1,0)+\chi_{M_{\chi_{M(1,0)}(M(1,0))+1}}(M(1,0)+1))\)

{1,,1,,2{1,,2,,}1,,1,,2} \(\psi(M(1,0)+M_{\chi_{M(1,0)}(M(1,0))+1})\)
{1{1,,2,,}1,2,,1,,2} \(\psi(M(1,0)+M_{\chi_{M(1,0)}(M(1,0))+\omega})\)
{1{1,,2,,}1 {1{1,,2,,}1,,1,,2} 2,,1,,2} \(\psi(M(1,0)+M_{\chi_{M(1,0)}(M(1,0))2})\)
{1{1,,2,,}1,,2,,2} \(\psi(M(1,0)+\psi_{M(1,0)}(M(1,0)+1))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2,,2}1 {1{1,,2,,}1,,2,,2} 2,,1,,2} \(\psi(M(1,0)+\chi_{M(1,0)}(M(1,0)+1))\)
{1{1,,2,,}1,,1,2,,2} \(\psi(M(1,0)+\psi_{M(1,0)}(M(1,0)+\omega))\)
{1,,2{1,,2,,}1,,1 {1{1,,2,,}1,,1,,2} 2,,2} \(\psi(M(1,0)+\chi_{M(1,0)}(M(1,0)+\chi_{M(1,0)}(M(1,0))))\)
{1{1,,2,,}1,,1,,3} \(\psi(M(1,0)2)\)
{1{1,,2,,}1,,1,,4} \(\psi(M(1,0)3)\)
{1{1,,2,,}1,,1,,1,,2} \(\psi(M(1,0)^2)\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} or {1{1,,2,,}1{1,,2,,}2} \(\psi(M(1,0)^{M(1,0)})\)
{1 {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} 2{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)}+\psi_{M(1,0)}(M(1,0)^{M(1,0)}))\)
{1,,2{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)}+\chi_{M_{\psi_{M(1,0)}(M(1,0)^{M(1,0)})+1}}(M(1,0)^{M(1,0)}))\)
{1,,1,,2{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)}+M_{\psi_{M(1,0)}(M(1,0)^{M(1,0)})+1})\)
{1{1,,2,,}1,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)}+M_{\psi_{M(1,0)}(M(1,0)^{M(1,0)})+\omega})\)
{1{1,,2,,}1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)}+\psi_{\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})})}(M(1,0)^{M(1,0)}+1))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2}1 {1{1,,2,,}1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} 2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)}+\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})}))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2,2{1{1{1,,2,,}1{1,,2,,}2}2,,}2}1 {1{1,,2,,}1,,2,2{1{1{1,,2,,}1{1,,2,,}2}2,,}2} 2,,1,2{1{1{1,,2,,}1{1,,2,,}2}2,,}2} \(\psi(M(1,0)^{M(1,0)}+\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})}+\omega))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2{2,,}2{1{1{1,,2,,}1{1,,2,,}2}2,,}2}1 {1{1,,2,,}1,,2{2,,}2{1{1{1,,2,,}1{1,,2,,}2}2,,}2} 2{2,,}2{1{1{1,,2,,}1{1,,2,,}2}2,,}2} \(\psi(M(1,0)^{M(1,0)}+\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})}+M(1,0)^\omega))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2 #}1 {1{1,,2,,}1,,2 #}2 #} where # = "{1 {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} 2,,}2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2" \(\psi(M(1,0)^{M(1,0)}+\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})}2))\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}3} \(\psi(M(1,0)^{M(1,0)}+\psi_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})}(M(1,0)^{M(1,0)}+1))\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1 {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} 2} \(\psi(M(1,0)^{M(1,0)}+\psi_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})}(M(1,0)^{M(1,0)}+\psi_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})}(M(1,0)^{M(1,0)})))\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+\chi_{M(1,0)}(M(1,0)^{M(1,0)}))\)
{1,,2{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+\chi_{M_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})+1}}(M(1,0)^{M(1,0)}+1))\)
{1,,1,,2{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+M_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})+1})\)
{1{1,,2,,}1,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+M_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})+\omega})\)
{1{1,,2,,}1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+\psi_{M(1,0)}(M(1,0)^{M(1,0)}+1))\)
{1{1,,2,,}1,,1,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+\psi_{M(1,0)}(M(1,0)^{M(1,0)}+\omega))\)
{1{1,,2,,}1,,1 {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} 2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+\psi_{M(1,0)}(M(1,0)^{M(1,0)}+\chi_{M(1,0)}(M(1,0)^{M(1,0)})))\)
{1{1,,2,,}1,,1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+M(1,0))\)
{1{1,,2,,}1,,1,,1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2} \(\psi(M(1,0)^{M(1,0)}+M(1,0)^2)\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2,,2} \(\psi(M(1,0)^{M(1,0)}2)\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,1,,2} \(\psi(M(1,0)^{M(1,0)+1})\)
{1{1,,2,,}1{2 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)\omega})\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 3,,}2} \(\psi(M(1,0)^{M(1,0)^2})\)
{1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2}1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} \(\psi(M(1,0)^{M(1,0)^{M(1,0)}})\)
{1 {1{1,,2,,}1{1,,2,,}2} 2{1,,2,,}1{1,,2,,}2} \(\psi(M(1,0)^{M(1,0)^{M(1,0)^{M(1,0)}}})\)
{1,,2{1,,2,,}1{1,,2,,}2} \(\psi(\Omega_{M(1,0)+1})=\psi(\chi_{M_{M(1,0)+1}}(0))\)
{1,,1,,2{1,,2,,}1{1,,2,,}2} \(\psi(M_{M(1,0)+1})\)
{1{1,,2,,}1,2{1,,2,,}2} \(\psi(M_{M(1,0)+\omega})\)
{1{1,,2,,}1,,2{1,,2,,}2} \(\psi(\psi_{M(1,1)}(0))=\psi(\psi_{\chi_{M(1,1)}(0)}(0))\)
{1{1,,2,,}1 {1{1,,2,,}1,,2{1,,2,,}2}1 {1{1,,2,,}1,,2{1,,2,,}2} 2{1,,2,,}2} \(\psi(\chi_{M(1,1)}(0))\)
{1{1,,2,,}1,,3{1,,2,,}2} \(\psi(\psi_{M(1,1)}(1))\)
{1{1,,2,,}1,,1,,2{1,,2,,}2} \(\psi(M(1,1))\)
{1{1,,2,,}1{1,,2,,}3} \(\psi(M(1,1)^{M(1,1)})\)
{1,,2{1,,2,,}1{1,,2,,}3} \(\psi(\chi_{M_{M(1,1)+1}}(0))\)
{1{1,,2,,}1,,1,,2{1,,2,,}3} \(\psi(M(1,2))\)
{1{1,,2,,}1,,1,,2{1,,2,,}4} \(\psi(M(1,3))\)
{1{1,,2,,}1{1,,2,,}1,2} \(\psi(M(1,\omega))\)
{1 {1{1,,2,,}1{1,,2,,}1,2} 2{1,,2,,}1{1,,2,,}1,2} \(\psi(M(1,\omega)2)\)
{1,,2{1,,2,,}1{1,,2,,}1,2} \(\psi(\chi_{M_{M(1,\omega)+1}}(0))\)
{1,,1,,2{1,,2,,}1{1,,2,,}1,2} \(\psi(M_{M(1,\omega)+1})\)
{1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,2} 2{1,,2,,}1,2} \(\psi(M_{M(1,\omega)2})\)
{1{1,,2,,}1,,2{1,,2,,}1,2} \(\psi(\psi_{M(1,\omega+1)}(0))\)
{1{1,,2,,}1,,1,,2{1,,2,,}1,2} \(\psi(M(1,\omega+1))\)
{1{1,,2,,}1,,1,,2{1,,2,,}2,2} \(\psi(M(1,\omega+2))\)
{1{1,,2,,}1{1,,2,,}1,3} \(\psi(M(1,\omega2))\)
{1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,2} 2} \(\psi(M(1,M(1,\omega)))\)
{1{1,,2,,}1{1,,2,,}1,,2} \(\psi(\psi_{M(2,0)}(0))=\psi(\psi_{\chi_{M(2,0)}(0)}(0))\)
{1,,1,,2{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 2} \(\psi(M_{\psi_{M(2,0)}(0)+1})\)
{1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 2} 2{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 2} \(\psi(M_{\psi_{M(2,0)}(0)2})\)
{1{1,,2,,}1,,2{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 2} \(\psi(\psi_{M(1,\psi_{M(2,0)}(0)+1)}(1))\)
{1{1,,2,,}1,,1,,2{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 2} \(\psi(M(1,\psi_{M(2,0)}(0)+1))\)
{1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 2}2 {1{1,,2,,}1{1,,2,,}1,,2} 2} \(\psi(M(1,\psi_{M(2,0)}(0)2))\)
{1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2} 3} \(\psi(\psi_{\chi_{M(2,0)}(0)}(1))\)
{1{1,,2,,}1{1,,2,,}1 {1{1,,2,,}1{1,,2,,}1,,2}1 {1{1,,2,,}1{1,,2,,}1,,2}2} \(\psi(\chi_{M(2,0)}(0))\)
{1,,1,,2{1,,2,,}1{1,,2,,}1,,2} \(\psi(M_{\chi_{M(2,0)}(0)+1})\)
{1{1,,2,,}1,2{1,,2,,}1,,2} \(\psi(M_{\chi_{M(2,0)}(0)+\omega})\)
{1{1,,2,,}1,,1,,2{1,,2,,}1,,2} \(\psi(M(1,\chi_{M(2,0)}(0)+1))\)
{1{1,,2,,}1{1,,2,,}1,2,,2} \(\psi(M(1,\chi_{M(2,0)}(0)+\omega))\)
{1{1,,2,,}1{1,,2,,}1,,3} \(\psi(\psi_{M(2,0)}(1))\)
{1{1,,2,,}1{1,,2,,}1,,1,2} \(\psi(\psi_{M(2,0)}(\omega))\)
{1{1,,2,,}1{1,,2,,}1,,1 {1{1,,2,,}1{1,,2,,}1,,2} 2} \(\psi(\psi_{M(2,0)}(\chi_{M(2,0)}(0)))\)
{1{1,,2,,}1{1,,2,,}1,,1,,2} \(\psi(M(2,0))\)
{1{1,,2,,}1{1,,2,,}1{1,,2,,}2} \(\psi(M(2,0)^{M(2,0)})\)
{1,,1,,2{1,,2,,}1{1,,2,,}1{1,,2,,}2} \(\psi(M_{M(2,0)+1})\)
{1{1,,2,,}1,,1,,2{1,,2,,}1{1,,2,,}2} \(\psi(M(1,M(2,0)+1))\)
{1{1,,2,,}1{1,,2,,}1,,1,,2{1,,2,,}2} \(\psi(M(2,1))\)
{1{1,,2,,}1{1,,2,,}1{1,,2,,}1,2} \(\psi(M(2,\omega))\)
{1{1,,2,,}1{1,,2,,}1{1,,2,,}1,,1,,2} \(\psi(M(3,0))\)
{1{1,,2,,}1 {1,,2,,}1 {1,,2,,}1 {1,,2,,}1,,1,,2} \(\psi(M(4,0))\)
{1{2,,2,,}2} \(\psi(\psi_{M(\omega,0)}(0))=\psi(\psi_{\chi_{M(\omega,0)}(0)}(0))\)
{1{1{2,,2,,}2}2{2,,2,,}2} \(\psi(\psi_{M(\omega,0)}(0)2)\)
{1,,1,,2{2,,2,,}2} \(\psi(M_{\psi_{M(\omega,0)}(0)+1})\)
{1{1,,2,,}1,,1,,2{2,,2,,}2} \(\psi(M(1,\psi_{M(\omega,0)}(0)+1))\)
{1{1,,2,,}1{1,,2,,}1,,1,,2 {2,,2,,}2} \(\psi(M(2,\psi_{M(\omega,0)}(0)+1))\)
{1{2,,2,,}3} \(\psi(\psi_{\chi_{M(\omega,0)}(0)}(1))\)
{1{2,,2,,}1{1{2,,2,,}2}2} \(\psi(\psi_{\chi_{M(\omega,0)}(0)}(\psi_{\chi_{M(\omega,0)}(0)}(0)))\)
{1{2,,2,,}1,,2} \(\psi(\chi_{M(\omega,0)}(0))\)
{1,,1,,2{2,,2,,}1,,2} \(\psi(M_{\chi_{M(\omega,0)}(0)+1})\)
{1{1,,2,,}1,,1,,2{2,,2,,}1,,2} \(\psi(M(1,\chi_{M(\omega,0)}(0)+1))\)
{1{2,,2,,}2,,2} \(\psi(\psi_{M(\omega,0)}(1))=\psi(\psi_{\chi_{M(\omega,0)}(1)}(1))\)
{1{2,,2,,}1,,3} \(\psi(\chi_{M(\omega,0)}(1))\)
{1{2,,2,,}1,,1{1{2,,2,,}1,,2}2} \(\psi(\chi_{M(\omega,0)}(\chi_{M(\omega,0)}(0)))\)
{1{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} or {1{2,,2,,}1,,1,,2} \(\psi(M(\omega,0))\)
{1 {1{2,,2,,}2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(0)}(0))\)
{1 {1{2,,2,,}3} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(0)}(1))\)
{1 {1{2,,2,,}1 {1{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} 2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(0)}(M(\omega,0)))\)
{1 {1{2,,2,,}1,,2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(0))\)
{1 {1{1,,2,,}2{2,,2,,}1,,2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+M_{\chi_{M(\omega,0)}(0)+1})\)
{1 {1{1,,2,,}1{1,,2,,}2{2,,2,,}1,,2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+M(1,\chi_{M(\omega,0)}(0)+1))\)
{1 {1{2,,2,,}2,,2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(1)}(1))\)
{1 {1{2,,2,,}1,,3} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(1))\)
{1 {1{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} 2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\psi_{M(\omega,0)}(M(\omega,0)))\)
{1,,1,,2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+M_{\psi_{M(\omega,0)}(M(\omega,0))+1})\)
{1{1,,2,,}1,,1,,2{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+M(1,\psi_{M(\omega,0)}(M(\omega,0))+1))\)
{1{2,,2,,}2,,1{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(\psi_{M(\omega,0)}(M(\omega,0)))}(M(\omega,0)+1))\)
{1{2,,2,,}1,,2{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(\psi_{M(\omega,0)}(M(\omega,0))))\)
{1{2,,2,,}1,,3{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(\psi_{M(\omega,0)}(M(\omega,0))+1))\)
{1{2,,2,,}1,,1 {1{2,,2,,}1,,2{1{2,,2,,}1,,1,,2}2} 2{1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(\psi_{M(\omega,0)}(M(\omega,0))2))\)
{1{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}3} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(M(\omega,0))}(M(\omega,0)+1))\)
{1{2,,2,,}1,,1 {1{2,,2,,}1,,1,,2}1 {1{2,,2,,}1,,1,,2}2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(M(\omega,0)))\)
{1,,1,,2{2,,2,,}1,,1,,2} \(\psi(M(\omega,0)+M_{\chi_{M(\omega,0)}(M(\omega,0))+1})\)
{1{1,,2,,}1,,1,,2{2,,2,,}1,,1,,2} \(\psi(M(\omega,0)+M(1,\chi_{M(\omega,0)}(M(\omega,0))+1))\)
{1{2,,2,,}2,,1,,2} \(\psi(M(\omega,0)+\psi_{\chi_{M(\omega,0)}(M(\omega,0)+1)}(M(\omega,0)+1))\)
{1{2,,2,,}1,,2,,2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(M(\omega,0)+1))\)
{1{2,,2,,}1,,1{1{2,,2,,}1,,1,,2}2,,2} \(\psi(M(\omega,0)+\chi_{M(\omega,0)}(M(\omega,0)+\chi_{M(\omega,0)}(M(\omega,0))))\)
{1{2,,2,,}1,,1,,3} \(\psi(M(\omega,0)2)\)
{1{2,,2,,}1,,1,,1,,2} \(\psi(M(\omega,0)^2)\)
{1{2,,2,,}1{1,,2,,}2} \(\psi(M(\omega,0)^{M(\omega,0)})\)
{1{2,,2,,}1,,1,,2{1,,2,,}2} \(\psi(M(\omega,1))\)
{1{2,,2,,}1,,1,,2{1,,2,,}3} \(\psi(M(\omega,2))\)
{1{2,,2,,}1{1,,2,,}1,,1,,2} \(\psi(M(\omega+1,0))\)
{1{2,,2,,}1{1,,2,,}1{1,,2,,}1,,1,,2} \(\psi(M(\omega+2,0))\)
{1{2,,2,,}1{2,,2,,}1,,1,,2} \(\psi(M(\omega2,0))\)
{1{3,,2,,}1,,1,,2} \(\psi(M(\omega^2,0))\)
{1{1 {1{1,,2,,}2} 2,,2,,}1,,1,,2} \(\psi(M(M,0))\)
{1{1 {1{1 {1{1,,2,,}2} 2,,2,,}1{1,,2,,}2} 2,,2,,}1,,1,,2} \(\psi(M(M(M,0),0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}2} or {1{1,,3,,}2} \(\psi(\psi_{M(1,0,0)}(0))=\psi(\psi_{\chi_{M(1,0,0)}(0)}(0))\)
{1,,1,,2{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M_{\psi_{M(1,0,0)}(0)+1})\)
{1{1,,2,,}1,,1,,2{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M(1,\psi_{M(1,0,0)}(0)+1))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}2{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(\psi_{M(\psi_{M(1,0,0)}(0),1)}(1))\)
{1,,1,,2{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}2{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M_{\psi_{M(\psi_{M(1,0,0)}(0),1)}(1)+1})\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}3{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(\psi_{\chi_{M(\psi_{M(1,0,0)}(0),1)}(1)}(2))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}1,,2{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(\chi_{M(\psi_{M(1,0,0)}(0),1)}(1))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}1,,1,,2{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M(\psi_{M(1,0,0)}(0),1))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}1,,1,,2{1,,2,,}2 {1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M(\psi_{M(1,0,0)}(0),2))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}1{1,,2,,}1,,1,,2 {1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M(\psi_{M(1,0,0)}(0)+1,0))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 3,,2,,}1,,1,,2 {1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M(\psi_{M(1,0,0)}(0)^2,0))\)
{1{1 {1{1 {1{1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}1{1,,2,,}1{1,,2,,}2 {1 {1{1,,3,,}2} 2,,2,,}2} 2,,2,,}1,,1,,2 {1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(M(M(\psi_{M(1,0,0)}(0)+1,0),0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}3} \(\psi(\psi_{\chi_{M(1,0,0)}(0)}(1))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1,,2} \(\psi(\chi_{M(1,0,0)}(0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}2,,2} \(\psi(\psi_{M(1,0,0)}(1))=\psi(\psi_{\chi_{M(1,0,0)}(1)}(1))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1,,3} \(\psi(\chi_{M(1,0,0)}(1))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1,,1,,2} \(\psi(M(1,0,0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1,,1,,2{1,,2,,}2} \(\psi(M(1,0,1))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1{1,,2,,}1,,1,,2} \(\psi(M(1,1,0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1{1 {1{1 {1{1,,3,,}2} 2,,2,,}1{1,,2,,}2} 2,,2,,}1,,1,,2} \(\psi(M(1,M(1,0,0),0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1{1 {1{1,,3,,}2} 2,,2,,}2} \(\psi(\psi_{M(2,0,0)}(0))\)
{1{1 {1{1,,3,,}2} 2,,2,,}1{1 {1{1,,3,,}2} 2,,2,,}1,,1,,2} \(\psi(M(2,0,0))\)
{1{2 {1{1,,3,,}2} 2,,2,,}1,,1,,2} \(\psi(M(\omega,0,0))\)
{1{1 {1{1 {1{1,,3,,}2} 2,,2,,}1{1,,2,,}2}2 {1{1,,3,,}2} 2,,2,,}1,,1,,2} \(\psi(M(M(1,0,0),0,0))\)
{1{1 {1{1,,3,,}2} 3,,2,,}2} \(\psi(\psi_{M(1,0,0,0)}(0))=\)

\(\psi(\psi_{\chi_{M(1,0,0,0)}(0)}(0))\)

{1{1 {1{1,,3,,}2} 3,,2,,}3} \(\psi(\psi_{\chi_{M(1,0,0,0)}(0)}(1))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1,,2} \(\psi(\chi_{M(1,0,0,0)}(0))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1,,3} \(\psi(\chi_{M(1,0,0,0)}(1))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1,,1,,2} \(\psi(M(1,0,0,0))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1,,1,,2{1,,2,,}2} \(\psi(M(1,0,0,1))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1{1,,2,,}1,,1,,2} \(\psi(M(1,0,1,0))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1{1 {1{1,,3,,}2} 2,,2,,}1,,1,,2} \(\psi(M(1,1,0,0))\)
{1{1 {1{1,,3,,}2} 3,,2,,}1{1 {1{1,,3,,}2} 3,,2,,}1,,1,,2} \(\psi(M(2,0,0,0))\)
{1{1 {1{1,,3,,}2} 4,,2,,}1,,1,,2} \(\psi(M(1,0,0,0,0))\)
{1{1 {1{1,,3,,}2} 5,,2,,}1,,1,,2} \(\psi(M(1,0,0,0,0,0))\)

Here're some approximations, to make the comparisons above more clear:

  • \(\psi_{M(1,0)}(0)=\psi_{\chi_{M(1,0)}(0)}(0)\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1,,2}2}
  • \(\chi_{M_{\psi_{M(1,0)}(0)+1}}(0)\) approximately corresponds to {1,,2{1,,2,,}2{1{1,,2,,}1,,2}2}
  • \(\chi_{M_{\psi_{M(1,0)}(0)+1}}(1)\) approximately corresponds to {1,,3{1,,2,,}2{1{1,,2,,}1,,2}2}
  • \(\chi_{M_{\psi_{M(1,0)}(0)+1}}(M_{\psi_{M(1,0)}(0)+1})\) approximately corresponds to {1,,1,,2{1,,2,,}2{1{1,,2,,}1,,2}2}
  • \(M_{\psi_{M(1,0)}(0)+1}\) approximately corresponds to {1{1,,2,,}2{1{1,,2,,}1,,2}2}
  • \(M_{\psi_{M(1,0)}(0)+2}\) approximately corresponds to {1{1,,2,,}3{1{1,,2,,}1,,2}2}
  • \(M_{\psi_{M(1,0)}(0)2}\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1{1{1,,2,,}1,,2}2}2{1{1,,2,,}1,,2}2}
  • \(\psi_{\chi_{M(1,0)}(0)}(1)\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1,,2}3}
  • \(\psi_{\chi_{M(1,0)}(0)}(\chi_{M(1,0)}(0))\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1,,2}1{1{1,,2,,}1,,2}2}
  • \(\chi_{M(1,0)}(0)\) approximately corresponds to {1{1,,2,,}1,,2}
  • \(\chi_{M_{\chi_{M(1,0)}(0)+1}}(1)\) approximately corresponds to {1,,2{1,,2,,}1,,2}
  • \(\chi_{M_{\chi_{M(1,0)}(0)+1}}(M_{\chi_{M(1,0)}(0)+1})\) approximately corresponds to {1,,1,,2{1,,2,,}1,,2}
  • \(M_{\chi_{M(1,0)}(0)+1}\) approximately corresponds to {1{1,,2,,}2,,2}
  • \(M_{\chi_{M(1,0)}(0)+2}\) approximately corresponds to {1{1,,2,,}3,,2}
  • \(\psi_{M(1,0)}(1)=\psi_{\chi_{M(1,0)}(1)}(1)\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1,,3}2,,2}
  • \(\chi_{M(1,0)}(1)\) approximately corresponds to {1{1,,2,,}1,,3}
  • \(\chi_{M(1,0)}(2)\) approximately corresponds to {1{1,,2,,}1,,4}
  • \(\psi_{M(1,0)}(\omega)=\psi_{\chi_{M(1,0)}(\omega)}(\omega)\) approximately corresponds to {1{1,,2,,}1,,1,2}
  • \(M_{\psi_{\chi_{M(1,0)}(\omega)}(\omega)+1}\) approximately corresponds to {1{1,,2,,}2,,1,2}
  • \(\psi_{\chi_{M(1,0)}(\omega)}(\omega+1)\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1,,2,2}2,,1,2}
  • \(\chi_{M(1,0)}(\omega)\) approximately corresponds to {1{1,,2,,}1,,2,2}
  • \(\psi_{M(1,0)}(M(1,0))=\psi_{\chi_{M(1,0)}(M(1,0))}(M(1,0))\) approximately corresponds to {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2}
  • \(\chi_{M_{\psi_{M(1,0)}(M(1,0))+1}}(M(1,0)+1)\) approximately corresponds to {1,,2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2}
  • \(\chi_{M_{\psi_{M(1,0)}(M(1,0))+1}}(M(1,0)+M_{\psi_{\chi_{M(1,0)}(M(1,0))}(M(1,0))+1})\) approximately corresponds to {1,,1,,2{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2}
  • \(M_{\psi_{M(1,0)}(M(1,0))+1}\) approximately corresponds to {1{1,,2,,}2,,1{1{1,,2,,}1,,1,,2}2}
  • \(\chi_{M(1,0)}(\psi_{M(1,0)}(M(1,0)))\) approximately corresponds to {1{1,,2,,}1,,2{1{1,,2,,}1,,1,,2}2}
  • \(\chi_{M(1,0)}(\psi_{M(1,0)}(M(1,0))+1)\) approximately corresponds to {1{1,,2,,}1,,3{1{1,,2,,}1,,1,,2}2}
  • \(\chi_{M(1,0)}(\psi_{M(1,0)}(M(1,0))2)\) approximately corresponds to {1{1,,2,,}1,,2 {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}2} 2{1{1,,2,,}1,,1,,2}2}
  • \(\psi_{\chi_{M(1,0)}(M(1,0))}(M(1,0)+1)\) approximately corresponds to {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}3}
  • \(\psi_{\chi_{M(1,0)}(M(1,0))}(M(1,0)+\chi_{M(1,0)}(M(1,0)))\) approximately corresponds to {1{1,,2,,}1,,1{1{1,,2,,}1,,1,,2}1{1{1,,2,,}1,,1,,2}2}
  • \(\chi_{M(1,0)}(M(1,0))\) approximately corresponds to {1{1,,2,,}1,,1,,2}
  • \(\chi_{M_{\chi_{M(1,0)}(M(1,0))+1}}(M(1,0)+1)\) approximately corresponds to {1,,2{1,,2,,}1,,1,,2}
  • \(\chi_{M_{\chi_{M(1,0)}(M(1,0))+1}}(M(1,0)+M_{\chi_{M(1,0)}(M(1,0))+1})\) approximately corresponds to {1,,1,,2{1,,2,,}1,,1,,2}
  • \(M_{\chi_{M(1,0)}(M(1,0))+1}\) approximately corresponds to {1{1,,2,,}2,,1,,2}
  • \(\psi_{\chi_{M(1,0)}(M(1,0)+1)}(M(1,0)+1)\) approximately corresponds to {1{1,,2,,}1{1{1,,2,,}1,,2,,2}2,,1,,2}
  • \(\chi_{M(1,0)}(M(1,0)+1)\) approximately corresponds to {1{1,,2,,}1,,2,,2}
  • \(\chi_{M(1,0)}(M(1,0)2)\) approximately corresponds to {1{1,,2,,}1,,1,,3}
  • \(\chi_{M(1,0)}(M(1,0)^2)\) approximately corresponds to {1{1,,2,,}1,,1,,1,,2}
  • \(\psi_{M(1,0)}(M(1,0)^{M(1,0)})=\psi_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})}(M(1,0)^{M(1,0)})\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2}
  • \(M_{\psi_{M(1,0)}(M(1,0)^{M(1,0)})+1}\) approximately corresponds to {1{1,,2,,}2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2}
  • \(\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})})\) approximately corresponds to {1{1,,2,,}1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2}
  • \(\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})}+1)\) approximately corresponds to {1{1,,2,,}1,,3{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2}
  • \(\chi_{M(1,0)}(M(1,0)^{\psi_{M(1,0)}(M(1,0)^{M(1,0)})}2)\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2} 2,,}2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}2}
  • \(\psi_{\chi_{M(1,0)}(M(1,0)^{M(1,0)})}(M(1,0)^{M(1,0)}+1)\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}3}
  • \(\chi_{M(1,0)}(M(1,0)^{M(1,0)})\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2}
  • \(\chi_{M(1,0)}(M(1,0)^{M(1,0)}+1)\) approximately corresponds to {1{1,,2,,}1,,2{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,2}
  • \(\chi_{M(1,0)}(M(1,0)^{M(1,0)}2)\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,3}
  • \(\chi_{M(1,0)}(M(1,0)^{M(1,0)+1})\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}2} 2,,}1,,1,,2}
  • \(M(1,0)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}2}
  • \(\chi_{M_{M(1,0)+1}}(0)\) approximately corresponds to {1,,2{1,,2,,}1{1,,2,,}2}
  • \(M_{M(1,0)+1}\) approximately corresponds to {1{1,,2,,}2{1,,2,,}2}
  • \(\psi_{M(1,1)}(0)=\psi_{\chi_{M(1,1)}(0)}(0)\) approximately corresponds to {1{1,,2,,}1 {1{1,,2,,}1,,2{1,,2,,}2} 2{1,,2,,}2}
  • \(\chi_{M(1,1)}(0)\) approximately corresponds to {1{1,,2,,}1,,2{1,,2,,}2}
  • \(\chi_{M(1,1)}(1)\) approximately corresponds to {1{1,,2,,}1,,3{1,,2,,}2}
  • \(\chi_{M(1,1)}(M(1,1))\) approximately corresponds to {1{1,,2,,}1,,1,,2{1,,2,,}2}
  • \(\chi_{M(1,1)}(M(1,1)^{M(1,1)})\) approximately corresponds to {1{1,,2,,}1{1 {1{1,,2,,}1{1,,2,,}3} 2,,}2{1,,2,,}2}
  • \(M(1,1)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}3}
  • \(M(1,2)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}4}
  • \(M(1,\omega)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1,2}
  • \(M(1,\omega+1)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}2,2}
  • \(M(1,M(1,\omega))\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1{1{1,,2,,}1{1,,2,,}1,2}2}
  • \(\psi_{M(2,0)}(0)=\psi_{\chi_{M(2,0)}(0)}(0)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1{1{1,,2,,}1{1,,2,,}1,,2}2}
  • \(\psi_{\chi_{M(2,0)}(0)}(1)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1{1{1,,2,,}1{1,,2,,}1,,2}3}
  • \(\chi_{M(2,0)}(0)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1,,2}
  • \(\chi_{M(2,0)}(1)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1,,3}
  • \(\chi_{M(2,0)}(M(2,0))\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1,,1,,2}
  • \(M(2,0)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1{1,,2,,}2}
  • \(M(3,0)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}1{1,,2,,}1{1,,2,,}2}
  • \(\psi_{M(\omega,0)}(0)=\psi_{\chi_{M(\omega,0)}(0)}(0)\) approximately corresponds to {1{2,,2,,}2}
  • \(\psi_{\chi_{M(\omega,0)}(0)}(1)\) approximately corresponds to {1{2,,2,,}3}
  • \(\chi_{M(\omega,0)}(0)\) approximately corresponds to {1{2,,2,,}1,,2}
  • \(\chi_{M(\omega,0)}(1)\) approximately corresponds to {1{2,,2,,}1,,3}
  • \(M(\omega,0)\) approximately corresponds to {1{2,,2,,}1{1,,2,,}2}
  • \(M(\omega2,0)\) approximately corresponds to {1{2,,2,,}1{2,,2,,}1{1,,2,,}2}
  • \(M(\omega^2,0)\) approximately corresponds to {1{3,,2,,}1{1,,2,,}2}
  • \(M(M,0)\) approximately corresponds to {1{1{1{1,,2,,}2}2,,2,,}1{1,,2,,}2}
  • \(\psi_{M(1,0,0)}(0)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}2}
  • \(M_{\psi_{M(1,0,0)}(0)+1}\) approximately corresponds to {1{1,,2,,}2{1{1{1,,3,,}2}2,,2,,}2}
  • \(M(1,\psi_{M(1,0,0)}(0)+1)\) approximately corresponds to {1{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}2,,2,,}2}
  • \(\psi_{M(\psi_{M(1,0,0)}(0),1)}(1)=\psi_{\chi_{M(\psi_{M(1,0,0)}(0),1)}(1)}(1)\) approximately corresponds to {1{1 {1{1{1{1,,3,,}2}2,,2,,}2} 2,,2,,}2{1{1{1,,3,,}2}2,,2,,}2}
  • \(\chi_{M(\psi_{M(1,0,0)}(0),1)}(1)\) approximately corresponds to {1{1 {1{1{1{1,,3,,}2}2,,2,,}2} 2,,2,,}1,,2{1{1{1,,3,,}2}2,,2,,}2}
  • \(M(\psi_{M(1,0,0)}(0),1)\) approximately corresponds to {1{1 {1{1{1{1,,3,,}2}2,,2,,}2} 2,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}2,,2,,}2}
  • \(M(\psi_{M(1,0,0)}(0)+1,0)\) approximately corresponds to {1{1 {1{1{1{1,,3,,}2}2,,2,,}2} 2,,2,,}1{1,,2,,}1{1,,2,,}2{1{1{1,,3,,}2}2,,2,,}2}
  • \(\psi_{\chi_{M(1,0,0)}(0)}(1)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}3}
  • \(\chi_{M(1,0,0)}(0)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}1,,2}
  • \(\chi_{M(1,0,0)}(1)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}1,,3}
  • \(M(1,0,0)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}1{1,,2,,}2}
  • \(M(1,0,1)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}1{1,,2,,}3}
  • \(M(1,1,0)\) approximately corresponds to {1{1{1{1,,3,,}2}2,,2,,}1{1,,2,,}1{1,,2,,}2}
  • \(M(2,0,0)\) approximately corresponds to {1 {1{1{1,,3,,}2}2,,2,,}1 {1{1{1,,3,,}2}2,,2,,} 1{1,,2,,}2}
  • \(\psi_{M(1,0,0,0)}(0)\) approximately corresponds to {1{1{1{1,,3,,}2}3,,2,,}2}
  • \(\chi_{M(1,0,0,0)}(0)\) approximately corresponds to {1{1{1{1,,3,,}2}3,,2,,}1,,2}
  • \(M(1,0,0,0)\) approximately corresponds to {1{1{1{1,,3,,}2}3,,2,,}1{1,,2,,}2}
  • \(M(1,0,0,0,0)\) approximately corresponds to {1{1{1{1,,3,,}2}4,,2,,}1{1,,2,,}2}
  • \(M(1,0,0,0,0,0)\) approximately corresponds to {1{1{1{1,,3,,}2}5,,2,,}1{1,,2,,}2}

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