I list significant properties, e.g. typical expansions of standard expressions of ordinals, of several OCFs which are helpful in analysis. I am not good at explicit computation, and hence tables below might contain many errors.
For an ordinal \(\alpha\), I denote by \(\alpha[n]\) a fixed choice of a fundamental sequence of \(\alpha\), i.e. a strictly increasing map \(\textrm{cof}(\alpha) \to \alpha, \ n \mapsto \alpha[n]\) above whose image \(\alpha\) is the least ordinal.
Buchholz's OCF[]
References:
- W. Buchholz, A new system of proof-theoretic ordinal functions, Annals of Pure and Applied Logic, Volume 32 (1986), pp. 195--207.
- W. Buchholz, Relating ordinals to proofs in a prespicious way, unpublished article.
- Buchholz's function, wiki article.
I list characters in standard expressions with respect to Buchholz's \(\psi\).
| character | property |
|---|---|
| \(0\) | the constant given as the least ordinal |
| \(+ \colon (\alpha,\beta) \mapsto \alpha + \beta\) | the associative \(2\)-ary function given as the addition |
| \(\psi_{\nu} \colon \alpha \mapsto \psi_{\nu}(\alpha)\) | a \(1\)-ary function indexed by a \(\nu \leq \omega\) |
I list standard expressions of specific ordinals with respect to Buchholz's \(\psi\), which will be used in order to define the conventional notion of a semi-standard expression.
| ordinal | standard expression |
|---|---|
| \(\Omega_0 := 1\) | \(\psi_0(0)\) |
| \(\omega\) | \(\psi_0(\psi_0(0))\) |
I call an expression of an ordinal semi-standard if the expression given by replacing all the occurence of \(1\) and \(\omega\) which are not indices of \(\psi\) in it by \(\psi_0(0)\) and \(\psi_0(\psi_0(0))\) respectively is standard with respect to Buchholz's \(\psi\). I list semi-standard expressions of specific ordinals with respect to Buchholz's \(\psi\).
| ordinal | semi-standard expression | restriction |
|---|---|---|
| \(2\) | \(1+1\) | |
| \(3\) | \(1+1+1\) | |
| \(\omega^{\alpha}\) | \(\psi_0(\alpha)\) | \(\alpha < \varepsilon_0\) |
| \(\varepsilon_0\) | \(\psi_0(\psi_1(0))\) | |
| \(\varepsilon_1\) | \(\psi_0(\psi_1(0)+\psi_1(0))\) | |
| \(\varepsilon_2\) | \(\psi_0(\psi_1(0)+\psi_1(0)+\psi_1(0))\) | |
| \(\varepsilon_{\omega^{\alpha}}\) | \(\psi_0(\psi_1(\alpha))\) | \(\alpha \in C_1(\alpha) \land \alpha < \psi_1(\alpha)\) |
| \(\Omega_{\nu} := \aleph_{\nu}\) | \(\psi_{\nu}(0)\) | \(1 \leq \nu \leq \omega\). |
I list common mistakes of values of Buchholz's \(\psi\).
| wrong property | correct property |
|---|---|
| Buchholz's \(\psi\) is a computable function. | Buchholz's \(\psi\) is not a computable function. |
| \(\psi_0(\varepsilon_0+1) = \varepsilon_0 \times \omega\) | \(\psi_0(\varepsilon_0+1) = \varepsilon_0\) |
| \(\psi_0(\epsilon_1) = \varepsilon_1\) | \(\psi_0(\epsilon_1) = \varepsilon_0\) |
| \(\psi_0(\zeta_0) = \zeta_0\) | \(\psi_0(\zeta_0) = \varepsilon_0\) |
| \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\Omega_3)\) | \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\Omega_2)\) |
| \(\psi_0(\psi_1(\psi_2(\psi_3(\cdots)))) = \psi_0(\Omega_{\omega})\) | \(\psi_0(\psi_1(\psi_2(\psi_3(\cdots)))) = \psi_0(\Omega_2)\) |
| \(\psi_0(\Omega_{\omega+1}+1) = \psi_0(\Omega_{\omega+1}) \times \omega\) | \(\psi_0(\Omega_{\omega+1}+1) = \psi_0(\Omega_{\omega+1}) = \psi_0(\psi_{\omega}(\psi_{\omega}(\cdots \psi_{\omega}(0) \cdots)))\) |
| \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}) > \psi_0(\Omega_{\Omega})\) | \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}) = \psi_0(\Omega_{\Omega}) = \psi_0(\psi_{\omega}(\psi_{\omega}(\cdots \psi_{\omega}(0)\cdots)))\) |
| \(\psi_1(0) = \psi_0(\psi_0(\cdots \psi_0(0) \cdots))\) | \(\psi_1(0) = \Omega_1\) |
| \(\psi_2(0) = \psi_1(\psi_1(\cdots \psi_1(0)\cdots))\) | \(\psi_2(0) = \Omega_2\) |
| \(\psi_{\omega}(\psi_{\omega}(\cdots \psi_{\omega}(0)\cdots)) = \Omega_{\omega+1}\) | \(\psi_{\omega}(\psi_{\omega}(\cdots \psi_{\omega}(0)\cdots)) = \varepsilon_{\Omega_{\omega}+1}\) |
| \(\psi_I(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega}}}}}\) | The function \(\psi_I\) is undefined. |
I list nest-free expansions of semi-standard expressions of ordinals with respect to Buchholz's \(\psi\).
| ordinal \(\alpha\) | expansion \(\alpha[n]\) | restriction |
|---|---|---|
| \(0\) | ||
| \(\beta+\psi_{\nu}(\gamma)\) | \(\beta\) | \(\psi_{\nu}(\delta)[n] = 0\). |
| \(\beta+\psi_{\nu}(\delta)[n]\) | \(\psi_{\nu}(\delta)[n] \neq 0\). | |
| \(\psi_{\nu}(\beta)\) | \(0\) | \(\beta = 0\) and \(\nu = 0\). |
| \(n\) | \(\beta = 0\) and \(\nu \notin \{0,\omega\}\). | |
| \(\psi_0(0)\) \(\psi_1(0)\) \(\psi_2(0)\) |
\(\beta = 0\) and \(\nu = \omega\). | |
| \(\psi_{\nu}(\beta[0])\) \(\psi_{\nu}(\beta[0])+\psi_{\nu}(\beta[0])\) \(\psi_{\nu}(\beta[0])+\psi_{\nu}(\beta[0])+\psi_{\nu}(\beta[0])\) |
\(\textrm{cof}(\beta) = 1\) | |
| \(\psi_{\nu}(\beta[n])\) | \(\omega \leq \textrm{cof}(\beta) < \Omega_{\nu}\). |
I list nesting expansions of standard expressions of countable ordinals with respect to Buchholz's \(\psi\).
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_0(\psi_1(0))\) | \(\psi_0(\psi_0(0))\) \(\psi_0(\psi_0(\psi_0(0)))\) \(\psi_0(\psi_0(\psi_0(\psi_0(0))))\) |
| \(\psi_0(\psi_1(0)+\psi_0(\psi_1(0)))\) | \(\psi_0(\psi_1(0)+\psi_0(\psi_0(0)))\) \(\psi_0(\psi_1(0)+\psi_0(\psi_0(\psi_0(0))))\) \(\psi_0(\psi_1(0)+\psi_0(\psi_0(\psi_0(\psi_0(0)))))\) |
| \(\psi_0(\psi_1(0)+\psi_1(0))\) | \(\psi_0(\psi_1(0)+\psi_0(0))\) \(\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(0)))\) \(\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(0))))\) |
| \(\psi_0(\psi_1(\psi_1(0)))\) | \(\psi_0(\psi_1(\psi_0(0)))\) \(\psi_0(\psi_1(\psi_0(\psi_1(\psi_0(0)))))\) \(\psi_0(\psi_1(\psi_0(\psi_1(\psi_0(\psi_1(\psi_0(0)))))))\) |
| \(\psi_0(\psi_1(\psi_1(0)+\psi_1(0)))\) | \(\psi_0(\psi_1(\psi_1(0)+\psi_0(0)))\) \(\psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)+\psi_0(0)))))\) \(\psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)+\psi_0(0)))))))\) |
| \(\psi_0(\psi_1(\psi_1(\psi_1(0))))\) | \(\psi_0(\psi_1(\psi_1(\psi_0(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_0(0)))))))\) \(\psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_0(0)))))))))\) |
| \(\psi_0(\psi_2(0))\) | \(\psi_0(\psi_1(0))\) \(\psi_0(\psi_1(\psi_1(0)))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0))))\) |
| \(\psi_0(\psi_2(0)+\psi_1(0))\) | \(\psi_0(\psi_2(0)+\psi_0(0))\) \(\psi_0(\psi_2(0)+\psi_0(\psi_2(0)+\psi_0(0)))\) \(\psi_0(\psi_2(0)+\psi_0(\psi_2(0)+\psi_0(\psi_2(0)+\psi_0(0))))\) |
| \(\psi_0(\psi_2(0)+\psi_1(\psi_2(0)))\) | \(\psi_0(\psi_2(0)+\psi_1(\psi_1(0)))\) \(\psi_0(\psi_2(0)+\psi_1(\psi_1(\psi_1(0))))\) \(\psi_0(\psi_2(0)+\psi_1(\psi_1(\psi_1(\psi_1(0)))))\) |
| \(\psi_0(\psi_2(0)+\psi_2(0))\) | \(\psi_0(\psi_2(0)+\psi_1(0))\) \(\psi_0(\psi_2(0)+\psi_1(\psi_2(0)+\psi_1(0)))\) \(\psi_0(\psi_2(0)+\psi_1(\psi_2(0)+\psi_1(\psi_2(0)+\psi_1(0))))\) |
| \(\psi_0(\psi_2(\psi_0(\psi_1(0))))\) | \(\psi_0(\psi_2(\psi_0(\psi_0(0))))\) \(\psi_0(\psi_2(\psi_0(\psi_0(\psi_0(0)))))\) \(\psi_0(\psi_2(\psi_0(\psi_0(\psi_0(\psi_0(0)))))\) |
| \(\psi_0(\psi_2(\psi_0(\psi_2(0))))\) | \(\psi_0(\psi_2(\psi_0(\psi_1(0))))\) \(\psi_0(\psi_2(\psi_0(\psi_1(\psi_1(0)))))\) \(\psi_0(\psi_2(\psi_0(\psi_1(\psi_1(\psi_1(0)))))\) |
| \(\psi_0(\psi_2(\psi_1(0)))\) | \(\psi_0(\psi_2(\psi_0(0)))\) \(\psi_0(\psi_2(\psi_0(\psi_2(\psi_0(0)))))\) \(\psi_0(\psi_2(\psi_0(\psi_2(\psi_0(\psi_2(\psi_0(0)))))))\) |
| \(\psi_0(\psi_2(\psi_1(0)+\psi_1(0)))\) | \(\psi_0(\psi_2(\psi_1(0)+\psi_0(0)))\) \(\psi_0(\psi_2(\psi_1(0)+\psi_0(\psi_2(\psi_1(0)+\psi_0(0)))))\) \(\psi_0(\psi_2(\psi_1(0)+\psi_0(\psi_2(\psi_1(0)+\psi_0(\psi_2(\psi_1(0)+\psi_0(0)))))))\) |
| \(\psi_0(\psi_2(\psi_1(\psi_1(0))))\) | \(\psi_0(\psi_2(\psi_1(\psi_0(0))))\) \(\psi_0(\psi_2(\psi_1(\psi_0(\psi_2(\psi_1(\psi_0(0)))))))\) \(\psi_0(\psi_2(\psi_1(\psi_0(\psi_2(\psi_1(\psi_0(\psi_2(\psi_1(\psi_0(0)))))))))\) |
| \(\psi_0(\psi_2(\psi_1(\psi_2(0))))\) | \(\psi_0(\psi_2(\psi_1(\psi_1(0))))\) \(\psi_0(\psi_2(\psi_1(\psi_1(\psi_1(0)))))\) \(\psi_0(\psi_2(\psi_1(\psi_1(\psi_1(\psi_1(0)))))\) |
| \(\psi_0(\psi_2(\psi_2(0)))\) | \(\psi_0(\psi_2(\psi_1(0)))\) \(\psi_0(\psi_2(\psi_1(\psi_2(\psi_1(0)))))\) \(\psi_0(\psi_2(\psi_1(\psi_2(\psi_1(\psi_2(\psi_1(0)))))))\) |
| \(\psi_0(\psi_2(\psi_2(0)+\psi_1(0)))\) | \(\psi_0(\psi_2(\psi_2(0)+\psi_0(0)))\) \(\psi_0(\psi_2(\psi_2(0)+\psi_0(\psi_2(\psi_2(0)+\psi_0(0)))))\) \(\psi_0(\psi_2(\psi_2(0)+\psi_0(\psi_2(\psi_2(0)+\psi_0(\psi_2(\psi_2(0)+\psi_0(0)))))))\) |
| \(\psi_0(\psi_2(\psi_2(\psi_1(0))))\) | \(\psi_0(\psi_2(\psi_2(\psi_0(0))))\) \(\psi_0(\psi_2(\psi_2(\psi_0(\psi_2(\psi_2(\psi_0(0)))))))\) \(\psi_0(\psi_2(\psi_2(\psi_0(\psi_2(\psi_2(\psi_0(\psi_2(\psi_2(\psi_0(0)))))))))\) |
| \(\psi_0(\psi_2(\psi_2(\psi_2(0))))\) | \(\psi_0(\psi_2(\psi_2(\psi_1(0))))\) \(\psi_0(\psi_2(\psi_2(\psi_1(\psi_2(\psi_2(\psi_1(0)))))))\) \(\psi_0(\psi_2(\psi_2(\psi_1(\psi_2(\psi_2(\psi_1(\psi_2(\psi_2(\psi_1(0)))))))))\) |
| \(\psi_0(\psi_3(0))\) | \(\psi_0(\psi_2(0))\) \(\psi_0(\psi_2(\psi_2(0)))\) \(\psi_0(\psi_2(\psi_2(\psi_2(0))))\) |
| \(\psi_0(\psi_{\omega}(0)+\psi_1(0))\) | \(\psi_0(\psi_{\omega}(0)+\psi_0(0))\) \(\psi_0(\psi_{\omega}(0)+\psi_0(\psi_{\omega}(0)+\psi_0(0)))\) \(\psi_0(\psi_{\omega}(0)+\psi_0(\psi_{\omega}(0)+\psi_0(\psi_{\omega}(0)+\psi_0(0))))\) |
| \(\psi_0(\psi_{\omega}(0)+\psi_2(0))\) | \(\psi_0(\psi_{\omega}(0)+\psi_1(0))\) \(\psi_0(\psi_{\omega}(0)+\psi_1(\psi_{\omega}(0)+\psi_1(0)))\) \(\psi_0(\psi_{\omega}(0)+\psi_1(\psi_{\omega}(0)+\psi_1(\psi_{\omega}(0)+\psi_1(0))))\) |
| \(\psi_0(\psi_{\omega}(\psi_1(0)))\) | \(\psi_0(\psi_{\omega}(\psi_0(0)))\) \(\psi_0(\psi_{\omega}(\psi_0(\psi_{\omega}(\psi_0(0)))))\) \(\psi_0(\psi_{\omega}(\psi_0(\psi_{\omega}(\psi_0(\psi_{\omega}(\psi_0(0)))))))\) |
| \(\psi_0(\psi_{\omega}(\psi_2(0)))\) | \(\psi_0(\psi_{\omega}(\psi_1(0)))\) \(\psi_0(\psi_{\omega}(\psi_1(\psi_{\omega}(\psi_1(0)))))\) \(\psi_0(\psi_{\omega}(\psi_1(\psi_{\omega}(\psi_1(\psi_{\omega}(\psi_1(0)))))))\) |
| Limit | \(\psi_0(\psi_{\omega}(0))\) \(\psi_0(\psi_{\omega}(\psi_{\omega}(0)))\) \(\psi_0(\psi_{\omega}(\psi_{\omega}(\psi_{\omega}(0))))\) |
Extended Buchholz's OCF[]
References:
- D. Maksudov, The extension of Buchholz's function, Traveling To The Infinity.
- Buchholz's function#Extension, wiki article.
I list characters in standard expressions with respect to extended Buchholz' \(\psi\).
| character | property |
|---|---|
| \(0\) | the constant given as the least ordinal |
| \(+ \colon (\alpha,\beta) \mapsto \alpha + \beta\) | the associative \(2\)-ary function given as the addition |
| \(\psi \colon (\alpha,\beta) \mapsto \psi_{\alpha}(\beta)\) | a \(2\)-ary function |
I list standard expressions of specific ordinals with respect to extended Buchholz's \(\psi\), which will be used in order to define the conventional notion of a semi-standard expression.
| ordinal | standard expression |
|---|---|
| \(1\) | \(\psi_0(0)\) |
| \(\omega\) | \(\psi_0(\psi_0(0))\) |
I call an expression of an ordinal semi-standard if the expression given by replacing all the occurence of \(1\) and \(\omega\) in it by \(\psi_0(0)\) and \(\psi_0(\psi_0(0))\) respectively is standard with respect to extended Buchholz's \(\psi\). I list semi-standard expressions of specific ordinals with respect to extended Buchholz's \(\psi\).
| ordinal | semi-standard expression | restriction |
|---|---|---|
| \(2\) | \(1+1\) | |
| \(3\) | \(1+1+1\) | |
| \(\omega^{\alpha}\) | \(\psi_0(\alpha)\) | \(\alpha < \varepsilon_0\) |
| \(\varepsilon_0\) | \(\psi_0(\psi_1(0))\) | |
| \(\varepsilon_1\) | \(\psi_0(\psi_1(0)+\psi_1(0))\) | |
| \(\varepsilon_2\) | \(\psi_0(\psi_1(0)+\psi_1(0)+\psi_1(0))\) | |
| \(\varepsilon_{\omega^{\alpha}}\) | \(\psi_0(\psi_1(\alpha))\) | \(\alpha \in C_1(\alpha) \land \alpha < \psi_1(\alpha)\) |
| \(\Omega_{\nu}\) | \(\psi_{\nu}(0)\) | \(\nu \geq 1\). |
I list common mistakes of values of Extended Buchholz's \(\psi\).
| wrong property | correct property |
|---|---|
| Extended Buchholz's \(\psi\) is a computable function. | Extended Buchholz's \(\psi\) is not a computable function. |
| Values of Extended Buchholz's \(\psi\) coincides with the corresponding values of Buchholz's \(\psi\) as far as both are defined. | Values of Extended Buchholz's \(\psi_0\) coincides with the corresponding values of Buchholz's \(\psi_0\) when we restrict them to ordinals below \(\varepsilon_{\Omega_{\omega}+1}\). |
| \(\psi_0(\varepsilon_0+1) = \varepsilon_0 \times \omega\) | \(\psi_0(\varepsilon_0+1) = \varepsilon_0\) |
| \(\psi_0(\epsilon_1) = \varepsilon_1\) | \(\psi_0(\epsilon_1) = \varepsilon_0\) |
| \(\psi_0(\zeta_0) = \zeta_0\) | \(\psi_0(\zeta_0) = \varepsilon_0\) |
| \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\Omega_3)\) | \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\Omega_2)\) |
| \(\psi_0(\psi_1(\psi_2(\psi_3(\cdots)))) = \psi_0(\Omega_{\omega})\) | \(\psi_0(\psi_1(\psi_2(\psi_3(\cdots)))) = \psi_0(\Omega_2)\) |
| \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_1(0)+1}(\psi_{\psi_1(0)+2}(0)))) = \psi_0(\psi_{\psi_1(0)+2}(0))\) | \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_1(0)+1}(\psi_{\psi_1(0)+2}(0)))) = \psi_0(\psi_{\psi_1(0)+1}(0))\) |
| \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_2(0)}(\psi_{\psi_3(0)}(0)))) = \psi_0(\psi_{\psi_3(0)}(0))\) | \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_2(0)}(\psi_{\psi_3(0)}(0)))) = \psi_0(\psi_{\psi_1(0)+1}(0))\) |
| \(\psi_1(0) = \psi_0(\psi_0(\cdots \psi_0(0) \cdots))\) | \(\psi_1(0) = \Omega_1\) |
| \(\psi_2(0) = \psi_1(\psi_1(\cdots \psi_1(0)\cdots))\) | \(\psi_2(0) = \Omega_2\) |
| \(\psi_{\omega}(\psi_{\omega}(\cdots \psi_{\omega}(0)\cdots)) = \Omega_{\omega+1}\) | \(\psi_{\omega}(\psi_{\omega}(\cdots \psi_{\omega}(0)\cdots)) = \varepsilon_{\Omega_{\omega}+1}\) |
| \(\psi_{\Omega}(0) = 1\) | \(\psi_{\Omega}(0) = \Omega_{\Omega}\) |
| \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_3}(\cdots))) = \psi_{\Omega}(\psi_{\Omega_{\omega}}(0))\) | \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_3}(\cdots))) = \psi_{\Omega}(\psi_{\Omega_2+1}(0))\) |
| \(\psi_I(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega}}}}}\) | \(\psi_I(0) = I\) |
I list nest-free expansions of semi-standard expressions of ordinals with respect to extended Buchholz's \(\psi\).
| ordinal \(\alpha\) | expansion \(\alpha[n]\) | restriction |
|---|---|---|
| \(0\) | ||
| \(\beta+\psi_{\nu}(\gamma)\) | \(\beta\) | \(\psi_{\nu}(\delta)[n] = 0\). |
| \(\beta+\psi_{\nu}(\delta)[n]\) | \(\psi_{\nu}(\delta)[n] \neq 0\). | |
| \(\psi_{\nu}(\beta)\) | \(0\) | \(\beta = 0\) and \(\nu = 0\). |
| \(n\) | \(\beta = 0\) and \(\textrm{cof}(\nu) = 1\). | |
| \(\psi_{\nu[n]}(0)\) | \(\beta = 0\) and \(\textrm{cof}(\nu) \geq \omega\). | |
| \(\psi_{\nu}(\beta[0])\) \(\psi_{\nu}(\beta[0])+\psi_{\nu}(\beta[0])\) \(\psi_{\nu}(\beta[0])+\psi_{\nu}(\beta[0])+\psi_{\nu}(\beta[0])\) |
\(\textrm{cof}(\beta) = 1\). | |
| \(\psi_{\nu}(\beta[n])\) | \(\omega \leq \textrm{cof}(\beta) < \Omega_{\nu}\). |
I list nesting expansions of semi-standard expressions of countable ordinals with respect to extended Buchholz's \(\psi\). Since extended Buchholz's \(\psi_0\) restricted to \(\varepsilon_{\Omega_{\omega}+1}\) coincides with Buchholz's \(\psi_0\), I only list expansions of ordinals greater than or equal to \(\psi_0(\varepsilon_{\Omega_{\omega}+1}) = \psi_0(\psi_{\omega+1}(0))\) in extended Buchholz's \(\psi_0\).
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_0(\psi_{\omega+1}(0))\) | \(\psi_0(\psi_{\omega}(0))\) \(\psi_0(\psi_{\omega}(\psi_{\omega}(0)))\) \(\psi_0(\psi_{\omega}(\psi_{\omega}(\psi_{\omega}(0))))\) |
| \(\psi_0(\psi_{\omega+1}(0)+\psi_1(0))\) | \(\psi_0(\psi_{\omega+1}(0)+\psi_0(0))\) \(\psi_0(\psi_{\omega+1}(0)+\psi_0(\psi_{\omega+1}(0)+\psi_0(0)))\) \(\psi_0(\psi_{\omega+1}(0)+\psi_0(\psi_{\omega+1}(0)+\psi_0(\psi_{\omega+1}(0)+\psi_0(0))))\) |
| \(\psi_0(\psi_{\omega+1}(0)+\psi_2(0))\) | \(\psi_0(\psi_{\omega+1}(0)+\psi_1(0))\) \(\psi_0(\psi_{\omega+1}(0)+\psi_1(\psi_{\omega+1}(0)+\psi_1(0)))\) \(\psi_0(\psi_{\omega+1}(0)+\psi_1(\psi_{\omega+1}(0)+\psi_1(\psi_{\omega+1}(0)+\psi_1(0))))\) |
| \(\psi_0(\psi_{\omega+1}(0)+\psi_{\omega+1}(0))\) | \(\psi_0(\psi_{\omega+1}(0)+\psi_{\omega}(0))\) \(\psi_0(\psi_{\omega+1}(0)+\psi_{\omega}(\psi_{\omega+1}(0)+\psi_{\omega}(0)))\) \(\psi_0(\psi_{\omega+1}(0)+\psi_{\omega}(\psi_{\omega+1}(0)+\psi_{\omega}(\psi_{\omega+1}(0)+\psi_{\omega}(0))))\) |
| \(\psi_0(\psi_{\omega+1}(\psi_1(0)))\) | \(\psi_0(\psi_{\omega+1}(\psi_0(0)))\) \(\psi_0(\psi_{\omega+1}(\psi_0(\psi_{\omega+1}(\psi_0(0)))))\) \(\psi_0(\psi_{\omega+1}(\psi_0(\psi_{\omega+1}(\psi_0(\psi_{\omega+1}(\psi_0(0)))))))\) |
| \(\psi_0(\psi_{\omega+1}(\psi_2(0)))\) | \(\psi_0(\psi_{\omega+1}(\psi_1(0)))\) \(\psi_0(\psi_{\omega+1}(\psi_1(\psi_{\omega+1}(\psi_1(0)))))\) \(\psi_0(\psi_{\omega+1}(\psi_1(\psi_{\omega+1}(\psi_1(\psi_{\omega+1}(\psi_1(0)))))))\) |
| \(\psi_0(\psi_{\omega+1}(\psi_{\omega+1}(0)))\) | \(\psi_0(\psi_{\omega+1}(\psi_{\omega}(0)))\) \(\psi_0(\psi_{\omega+1}(\psi_{\omega}(\psi_{\omega+1}(\psi_{\omega}(0)))))\) \(\psi_0(\psi_{\omega+1}(\psi_{\omega}(\psi_{\omega+1}(\psi_{\omega}(\psi_{\omega+1}(\psi_{\omega}(0)))))))\) |
| \(\psi_0(\psi_{\omega+2}(0))\) | \(\psi_0(\psi_{\omega+1}(0))\) \(\psi_0(\psi_{\omega+1}(\psi_{\omega+1}(0)))\) \(\psi_0(\psi_{\omega+1}(\psi_{\omega+1}(\psi_{\omega+1}(0))))\) |
| \(\psi_0(\psi_{\omega+\omega+1}(0))\) | \(\psi_0(\psi_{\omega+\omega}(0))\) \(\psi_0(\psi_{\omega+\omega}(\psi_{\omega+\omega}(0)))\) \(\psi_0(\psi_{\omega+\omega}(\psi_{\omega+\omega}(\psi_{\omega+\omega}(0))))\) |
| \(\psi_0(\psi_{\psi_0(2)+1}(0))\) | \(\psi_0(\psi_{\psi_0(2)}(0))\) \(\psi_0(\psi_{\psi_0(2)}(\psi_{\psi_0(2)}(0)))\) \(\psi_0(\psi_{\psi_0(2)}(\psi_{\psi_0(2)}(\psi_{\psi_0(2)}(0))))\) |
| \(\psi_0(\psi_{\psi_0(\psi_1(0))}(0))\) | \(\psi_0(\psi_{\psi_0(\psi_0(0))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_0(\psi_0(0)))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_0(\psi_0(\psi_0(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_0(\psi_1(0)+\psi_1(0))}(0))\) | \(\psi_0(\psi_{\psi_0(\psi_1(0)+\psi_0(0))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(0)))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_0(\psi_2(0))}(0))\) | \(\psi_0(\psi_{\psi_0(\psi_1(0))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_1(\psi_1(0)))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_1(\psi_1(\psi_1(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_0(\psi_{\omega+1}(0))}(0))\) | \(\psi_0(\psi_{\psi_0(\psi_{\omega}(0))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_{\omega}(\psi_{\omega}(0)))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_{\omega}(\psi_{\omega}(\psi_{\omega}(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_0(\psi_{\psi_0(2)+1}(0))}(0))\) | \(\psi_0(\psi_{\psi_0(\psi_{\psi_0(2)}(0))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_{\psi_0(2)}(\psi_{\psi_0(2)}(0)))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_{\psi_0(2)}(\psi_{\psi_0(2)}(\psi_{\psi_0(2)}(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_1(0)}(0))\) | \(\psi_0(\psi_{\psi_0(0)}(0))\) \(\psi_0(\psi_{\psi_0(\psi_{\psi_0(0)}(0))}(0))\) \(\psi_0(\psi_{\psi_0(\psi_{\psi_0(\psi_{\psi_0(0)}(0))}(0))}(0))\) |
| \(\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_1(0)}(0))\) | \(\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_0(0)}(0))\) \(\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_0(0)}(0))}(0))\) \(\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_0(\psi_{\psi_1(0)}(0)+\psi_{\psi_0(0)}(0))}(0))}(0))\) |
| \(\psi_0(\psi_{\psi_1(0)}(\psi_1(0)))\) | \(\psi_0(\psi_{\psi_1(0)}(\psi_0(0)))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_0(\psi_{\psi_1(0)}(\psi_0(0)))))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_0(\psi_{\psi_1(0)}(\psi_0(\psi_{\psi_1(0)}(\psi_0(0)))))))\) |
| \(\psi_0(\psi_{\psi_1(0)}(\psi_2(0)))\) | \(\psi_0(\psi_{\psi_1(0)}(\psi_1(0)))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_1(\psi_{\psi_1(0)}(\psi_1(0)))))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_1(\psi_{\psi_1(0)}(\psi_1(\psi_{\psi_1(0)}(\psi_1(0))))))\) |
| \(\psi_0(\psi_{\psi_1(0)}(\psi_{\omega+1}(0)))\) | \(\psi_0(\psi_{\psi_1(0)}(\psi_{\omega}(0)))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_{\omega}(\psi_{\psi_1(0)}(\psi_{\omega}(0))))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_{\omega}(\psi_{\psi_1(0)}(\psi_{\omega}(\psi_{\psi_1(0)}(\psi_{\omega}(0)))))\) |
| \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_1(0)}(0)))\) | \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_0(0)}(0)))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_0(0)}(0)))}(0)))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_0(0)}(0)))}(0)))}(0)))\) |
| \(\psi_0(\psi_{\psi_1(0)+1}(0))\) | \(\psi_0(\psi_{\psi_1(0)}(0))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_1(0)}(0)))\) \(\psi_0(\psi_{\psi_1(0)}(\psi_{\psi_1(0)}(\psi_{\psi_1(0)}(0))))\) |
| \(\psi_0(\psi_{\psi_1(0)+\psi_1(0)}(0))\) | \(\psi_0(\psi_{\psi_1(0)+\psi_0(0)}(0))\) \(\psi_0(\psi_{\psi_1(0)+\psi_0(\psi_{\psi_1(0)+\psi_0(0)}(0))}(0))\) \(\psi_0(\psi_{\psi_1(0)+\psi_0(\psi_{\psi_1(0)+\psi_0(\psi_{\psi_1(0)+\psi_0(0)}(0))}(0))}(0))\) |
| \(\psi_0(\psi_{\psi_1(\psi_1(0))}(0))\) | \(\psi_0(\psi_{\psi_1(\psi_0(0))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_0(\psi_{\psi_1(\psi_0(0))}(0)))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_0(\psi_{\psi_1(\psi_0(\psi_{\psi_1(\psi_0(0))}(0)))}(0)))}(0))\) |
| \(\psi_0(\psi_{\psi_1(\psi_2(0))}(0))\) | \(\psi_0(\psi_{\psi_1(\psi_1(0))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_1(\psi_1(0)))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_1(\psi_1(\psi_1(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_1(\psi_{\omega+1}(0))}(0))\) | \(\psi_0(\psi_{\psi_1(\psi_{\omega}(0))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_{\omega}(\psi_{\omega}(0)))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_{\omega}(\psi_{\omega}(\psi_{\omega}(0))))}(0))\) |
| \(\psi_0(\psi_{\psi_1(\psi_{\psi_1(0)}(0))}(0))\) | \(\psi_0(\psi_{\psi_1(\psi_{\psi_0(0)}(0))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_{\psi_0(\psi_{\psi_1(\psi_{\psi_0(0)}(0))}(0))}(0))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_{\psi_0(\psi_{\psi_1(\psi_{\psi_0(\psi_{\psi_1(\psi_{\psi_0(0)}(0))}(0))}(0))}(0))}(0))}(0))\) |
| \(\psi_0(\psi_{\psi_2(0)}(0))\) | \(\psi_0(\psi_{\psi_1(0)}(0))\) \(\psi_0(\psi_{\psi_1(\psi_{\psi_1(0)}(0))}(0))\) \(\psi_0(\psi_{\psi_1(\psi_{\psi_1(\psi_{\psi_1(0)}(0))}(0))}(0))\) |
| \(\psi_0(\psi_{\psi_{\psi_1(0)}(0)}(0))\) | \(\psi_0(\psi_{\psi_{\psi_0(0)}(0)}(0))\) \(\psi_0(\psi_{\psi_{\psi_0(\psi_{\psi_{\psi_0(0)}(0)}(0))}(0)}(0))\) \(\psi_0(\psi_{\psi_{\psi_0(\psi_{\psi_{\psi_0(\psi_{\psi_{\psi_0(0)}(0)}(0))}(0)}(0))}(0)}(0))\) |
| Limit | \(\psi_0(\psi_{\psi_0(0)}(0))\) \(\psi_0(\psi_{\psi_{\psi_0(0)}(0)}(0))\) \(\psi_0(\psi_{\psi_{\psi_{\psi_0(0)}(0)}(0)}(0))\) |
Rathjen's OCF Based on a Weakly Mahlo Cardinal[]
References:
- M. Rathjen, Ordinal Notations Based on a Weakly Mahlo Cardinal, Archive for Mathematical Logic, Volume 29, Issue 4 (1990), pp. 249--263.
- Ordinal notation#Rathjen's ψ, wiki article.
I deal with Rathjen's weakly Mahlo \(\psi\), i.e. Rathjen's \(\psi\) based on the least weakly Mahlo cardinal \(M\) with a specific ordinal notation system introduced in the first reference above. I note that it does not coincide with the simplified OCF introduced in The Realm of Ordinal Analysis without a specific associated ordinal notation system, and is not a formal symbol in the ordinal notation system introduced in Proof-Theoretic Analysis of KPM. Be careful that many googologists compfound them, because they sometimes talk about Rathjen's \(\psi\) even though many of them have not even take a brief look at the precise definition.
I list characters in standard expressions with respect to Rathjen's weakly Mahlo \(\psi\).
| character | property | restriction |
|---|---|---|
| \(0\) | the constant given as the least ordinal | |
| \(M\) | the constant given as the least weakly Mahlo cardinal | |
| \(+ \colon (\alpha,\beta) \mapsto \alpha + \beta\) | the associative \(2\)-ary function given as the addition | |
| \(\varphi \colon (\alpha,\beta) \mapsto \varphi_{\alpha}(\beta)\) | the \(2\)-ary function given as Veblen's function | |
| \(\Phi \colon (\alpha,\beta) \mapsto \Phi_{\alpha}(\beta)\) | the \(2\)-ary function | |
| \(\chi \colon (\alpha,\beta) \mapsto \chi_{\alpha}(\beta)\) | a \(2\)-ary function | \(\alpha < \Gamma_{M+1}\) and \(\beta < M\) |
| \(\psi \colon (\pi,\alpha) \mapsto \psi_{\pi}(\alpha)\) | a \(2\)-ary function | \(\alpha < \Gamma_{M+1}\) and \(\pi = \chi_{\gamma}(\delta)\) for some pair \((\gamma,\delta)\) of ordinals satisfying \(\gamma < \Gamma_{M+1}\), \(\delta < M\), and \(\textrm{cof}(\delta) \leq 1\). |
I list standard expressions of specific ordinals with respect to Rathjen's weakly Mahlo \(\psi\), which will be used in order to define the conventional notion of a semi-standard expression.
| ordinal | standard expression |
|---|---|
| \(1\) | \(\varphi_{0}(0)\) |
| \(2\) | \(\varphi_{0}(0)+\varphi_{0}(0)\) |
| \(3\) | \(\varphi_{0}(0)+\varphi_{0}(0)+\varphi_{0}(0)\) |
| \(\omega\) | \(\varphi_{0}(\varphi_{0}(0))\) |
| \(\Omega\) | \(\chi_{0}(0)\) |
| \(\Omega_2\) | \(\chi_{0}(\varphi_{0}(0))\) |
| \(\Omega_3\) | \(\chi_{0}(\varphi_{0}(0)+\varphi_{0}(0))\) |
| the least weakly inaccessible cardinal \(I\), which is also denoted by \(I_0\) or \(I_1\) depending on the convention | \(\chi_{\varphi_{0}(0)}(0)\) |
| the second weakly inaccessible cardinal \(I_2\), which is also denoted by \(I_1\) depending on the convention | \(\chi_{\varphi_{0}(0)}(\varphi_{0}(0))\) |
| the third weakly inaccessible cardinal \(I_3\), which is also denoted by \(I_2\) depending on the convention | \(\chi_{\varphi_{0}(0)}(\varphi_{0}(0)+\varphi_{0}(0))\) |
I call an expression of an ordinal semi-standard if the expression given by replacing all the occurence of \(1\), \(2\), \(3\), \(\omega\), \(\Omega\), \(\Omega_2\), \(\Omega_3\), \(I\), \(I_2\), and \(I_3\) by \(\varphi_{0}(0)\), \(\varphi_{0}(0)+\varphi_{0}(0)\), \(\varphi_{0}(0)+\varphi_{0}(0)+\varphi_{0}(0)\), \(\varphi_{0}(\varphi_{0}(0))\), \(\chi_{0}(0)\), \(\chi_{0}(\varphi_{0}(0))\), \(\chi_{0}(\varphi_{0}(0)+\varphi_{0}(0))\), \(\chi_{\varphi_{0}(0)}(0)\), \(\chi_{\varphi_{0}(0)}(\varphi_{0}(0))\), and \(\chi_{\varphi_{0}(0)}(\varphi_{0}(0)+\varphi_{0}(0))\) respectively is standard. I list semi-standard expressions of specific ordinals with respect to Rathjen's weakly Mahlo \(\psi\).
| ordinal | semi-standard expression | restriction |
|---|---|---|
| \(\Gamma_{0} = \varphi(1,0,0)\) | \(\psi_{\Omega}(0)\) | |
| \(\Gamma_{\alpha} = \varphi(1,0,\alpha)\) | \(\psi_{\Omega}(\alpha)\) | \(\alpha < \varphi(1,1,0)\) |
| \(\Omega_{1+\alpha}\) | \(\chi_0(\alpha)\) | \(\alpha\) is not an omega fixed point. |
| the least omega fixed point \(\Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega}}}}}\) | \(\Phi_{1}(0)\) | |
| the \((1+\alpha)\)-th omega fixed point | \(\Phi_{1}(\alpha)\) | \(\alpha\) is not a fixed point of \(\Phi_{1}\). |
| the \((1+\alpha)\)-th cardinal which is a fixed point of \(\Phi_{\gamma}\) for any \(\gamma < \beta\) | \(\Phi_{\beta}(\alpha)\) | \(\alpha\) is not a fixed point of \(\Phi_{\beta}\). |
| the \((1+\alpha)\)-th ordinal above \(\beta\) which is a weakly \(\beta\)-inaccessible cardinal or a limit of weakly \(\beta\)-inaccessible cardinals | \(\chi_{\beta}(\alpha)\) | \(\alpha\) is not a weakly \((\beta+1)\)-inaccessible cardinal, \(\alpha < M\), and \(\beta < \psi_{\chi_{M}(0)}(0)\). |
| the \((1+\alpha)\)-th ordinal which is a weakly \((1,0)\)-inaccessible, i.e. hyper-inaccessible, cardinal or a limit of weakly \((1,0)\)-inaccessible cardinals | \(\chi_{M}(\alpha)\) | \(\alpha\) is not a weakly \((1,1)\)-inaccessible cardinal and \(\alpha < M\) |
| the \((1+\alpha)\)-th ordinal greater than or equal to \(\sup_{\gamma < \beta} \chi_{M}(\gamma)\) closed under the map \(x \mapsto \chi_{x}(0)\) | \(\psi_{\chi_{M}(\beta)}(\alpha)\) | \(\alpha \in C_{\chi_{M}(\beta)}(\alpha)\), \(\beta\) is not a weakly \((1,1)\)-inaccessible cardinal, and \(\beta < M\). |
| the \((2+\alpha)\)-th ordinal which is a weakly \(\chi_{M}(0)\)-inaccessible cardinal or a limit of weakly \(\chi_{M}(0)\)-inaccessible cardinals | \(\chi_{\chi_{M}(0)}(\alpha)\) | \(\alpha\) is not a weakly \((\chi_{M}(0)+1)\)-inaccessible cardinal and \(\alpha < M\). |
| the \((1+\alpha)\)-th ordinal which is a weakly \((1,\beta)\)-inaccessible cardinal or a limit of weakly \((1,\beta)\)-inaccessible cardinals | \(\chi_{M+\beta}(\alpha)\) | \(\alpha\) is not a weakly \((1,\beta+1)\)-inaccessible cardinal, \(\alpha < M\), and \(\beta < \psi_{\chi_{M+M}(0)}(0)\). |
| the \((1+\alpha)\)-th ordinal which is a weakly \((2,0)\)-inaccessible cardinal or a limit of weakly \((2,0)\)-inaccessible cardinals | \(\chi_{M+M}(\alpha)\) | \(\alpha\) is not a weakly \((2,1)\)-inaccessible cardinal and \(\alpha < M\). |
| the \((1+\alpha)\)-th ordinal greater than or equal to \(\sup_{\gamma < \beta} \chi_{M+M}(\gamma)\) closed under the map \(x \mapsto \chi_{M+x}(0)\) | \(\psi_{\chi_{M+M}(\beta)}(\alpha)\) | \(\alpha \in C_{\chi_{M+M}(\beta)}(\alpha)\), \(\beta\) is not a weakly \((2,1)\)-inaccessible cardinal, and \(\beta < M\). |
| the \((2+\alpha)\)-th ordinal which is a weakly \(\chi_{M+M}(0)\)-inaccessible cardinal or a limit of weakly \(\chi_{M+M}(0)\)-inaccessible cardinals | \(\chi_{M+\chi_{M+M}(0)}(\alpha)\) | \(\alpha\) is not a weakly \((\chi_{M+M}(0)+1)\)-inaccessible cardinal and \(\alpha < M\). |
I list expressions of specific ordinals using multiplications and powers, which are not characters which are allowed to appear in semi-standard expressions with respect to Rathjen's weakly Mahlo \(\psi\). Since the definition is complicated, the table might contain descriptions which lack necessary restrictions.
| ordinal | expression | restriction |
|---|---|---|
| the \((1+\alpha)\)-th ordinal above \((M^n \times a_n + \cdots M \times a_1 + a_0)^{-}\) which is a weakly \((a_n,\ldots,a_1,a_0)\)-inaccessible cardinal or a limit of weakly \((a_n,\ldots,a_1,a_0)\)-inaccessible cardinal | \(\chi_{M^n \times a_n + \cdots M \times a_1 + a_0}(\alpha)\) | \(n \geq 1\), \(a_n \geq 1\), \(\alpha,a_n,\ldots,a_1,a_0 < M\), and \(\alpha\) is not a weakly \((a_n,\ldots,a_1,a_0+1)\)-inaccessible cardinal. |
| the \((1+\alpha)\)-th ordinal which is a weakly \((1,0,Z,0)\)-inaccessible cardinal for any finite string \(Z\) consisting of \(0\)'s separated by commas | \(\chi_{M^{\omega}}(\alpha)\) | The subset of \(\alpha\) consisting of ordinals satisfying the same property is bounded and \(\alpha < M\). |
I list common mistakes of values of Rathjen's weakly Mahlo \(\psi\).
| wrong property | correct property |
|---|---|
| Rathjen's weakly Mahlo \(\psi\) is a computable function. | Rathjen's weakly Mahlo \(\psi\) is not a computable function. |
| \(\psi_{\Omega}(0) = 1\) | \(\psi_{\Omega}(0) = \Gamma_0\) |
| \(\psi_{\Omega}(1) = \omega\) | \(\psi_{\Omega}(1) = \Gamma_1\) |
| \(\psi_{\Omega}(2) = \omega^2\) | \(\psi_{\Omega}(2) = \Gamma_2\) |
| \(\psi_{\Omega}(\Omega) = \varepsilon_0\) | \(\psi_{\Omega}(\Omega) = \varphi(1,1,0)\) |
| \(\psi_{\Omega}(\varphi(1,1,0)+1) = \varphi(1,1,1)\) | \(\psi_{\Omega}(\varphi(1,1,0)+1) = \varphi(1,1,0)\) |
| \(\psi_{\Omega}(\Omega_2) = \psi_{\Omega}(\psi_{\Omega}(\cdots \psi_{\Omega}(0)))\) | \(\psi_{\Omega}(\Omega_2) = \psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(\cdots \psi_{\Omega_2}(0))))\) |
| \(\psi_{\Omega_2}(0) = \psi_{\Omega}(\psi_{\Omega}(\cdots \psi_{\Omega}(0) \cdots))\) | \(\psi_{\Omega_2}(0) = \Gamma_{\Omega+1}\) |
| \(\psi_{\Omega_3}(0) = \psi_{\Omega_2}(\psi_{\Omega_2}(\cdots \psi_{\Omega_2}(0)\cdots))\) | \(\psi_{\Omega_3}(0) = \Gamma_{\Omega_2+1}\) |
| \(\psi_{\Omega_{\omega}}(0) = \Gamma_{\Omega_{\omega}+1}\) | The function \(\psi_{\Omega_{\omega}}\) is undefined. |
| \(\chi_1(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega}}}}}\) | \(\chi_1(0) = I\) |
| The \(\omega\)-th weakly inaccessible cardinal is \(\chi_1(\omega)\). | The \(\omega\)-th weakly inaccessible cardinal is \(\chi_1(\omega+1)\). |
| The limit of the sequence \(\chi_{0}(0), \chi_{\chi_{0}(0)}(0), \chi_{\chi_{\chi_{0}(0)}(0)}(0), \ldots\) is the least weakly hyper-inaccessible cardinal. | The limit of the sequence \(\chi_{0}(0), \chi_{\chi_{0}(0)}(0), \chi_{\chi_{\chi_{0}(0)}(0)}(0), \ldots\) is of cofinality \(\omega\). |
| \(\chi_M(0) > \chi_{\chi_M(0)}(0)\) | \(\chi_M(0) < \chi_{\chi_M(0)}(0)\) |
I list nest-free expansions of semi-standard expressions of ordinals with respect to Rathjen's weakly Mahlo \(\psi\).
| ordinal \(\alpha\) | expansion \(\alpha[n]\) | restriction |
|---|---|---|
| \(0\) | ||
| \(\beta+\gamma\) | \(\beta\) | \(\gamma\) is an additive principal number, and \(\gamma[n] = 0\). |
| \(\beta+\gamma[n]\) | \(\gamma\) is an additive principal number, and \(\gamma[n] \neq 0\). | |
| \(\varphi_{\beta}(\gamma)\) | \(0\) | \(\gamma = 0\) and \(\beta = 0\) |
| \(\varphi_{\beta[0]}(0)\) \(\varphi_{\beta[0]}(\varphi_{\beta[0]}(0))\) \(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\varphi_{\beta[0]}(0)))\) |
\(\gamma = 0\) and \(\textrm{cof}(\beta) = 1\). | |
| \(\varphi_{\beta[n]}(0)\) | \(\gamma = 0\) and \(\textrm{cof}(\beta) \geq \omega\). | |
| \(\varphi_{0}(\gamma[0])\) \(\varphi_{0}(\gamma[0])+\varphi_{0}(\gamma[0])\) \(\varphi_{0}(\gamma[0])+\varphi_{0}(\gamma[0])+\varphi_{0}(\gamma[0]))\) |
\(\textrm{cof}(\gamma) = 1\), \(\beta = 0\), and \(\gamma[0]\) is not an epsilon number. | |
| \(\gamma[0]\) \(\gamma[0]+\gamma[0]\) \(\gamma[0]+\gamma[0]+\gamma[0]\) |
\(\textrm{cof}(\gamma) = 1\), \(\beta = 0\), and \(\gamma[0]\) is an epsilon number. | |
| \(\varphi_{\beta[0]}(\varphi_{\beta}(\gamma[0])+1)\) \(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\varphi_{\beta}(\gamma[0])+1))\) \(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\varphi_{\beta}(\gamma[0])+1)))\) |
\(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) = 1\), and \(\gamma[0]\) is not a fixed point of \(\varphi_{\beta}\). | |
| \(\varphi_{\beta[0]}(\gamma[0]+1)\) \(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\gamma[0]+1))\) \(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\varphi_{\beta[0]}(\gamma[0]+1)))\) |
\(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) = 1\), and \(\gamma[0]\) is a fixed point of \(\varphi_{\beta}\). | |
| \(\varphi_{\beta[n]}(\varphi_{\beta}(\gamma[0])+1)\) | \(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) \geq \omega\), and \(\gamma[0]\) is not a fixed point of \(\varphi_{\beta}\). | |
| \(\varphi_{\beta[n]}(\gamma[0]+1)\) | \(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) \geq \omega\), and \(\gamma[0]\) is a fixed point of \(\varphi_{\beta}\). | |
| \(\varphi_{\beta}(\gamma[n])\) | \(\textrm{cof}(\gamma) \geq \omega\) and \(\gamma[n]\) is not a fixed point of \(\varphi_{\beta}\). | |
| \(\gamma[n]\) | \(\textrm{cof}(\gamma) \geq \omega\) and \(\gamma[n]\) is a fixed point of \(\varphi_{\beta}\). | |
| \(\Phi_{\beta}(\gamma)\) | ||
| \(\chi_{0}(0)\) \(\chi_{0}(\chi_{0}(0))\) \(\chi_{0}(\chi_{0}(\chi_{0}(0)))\) |
\(\gamma = 0\) and \(\beta = 1\). | |
| \(\Phi_{\beta[0]}(0)\) \(\Phi_{\beta[0]}(\Phi_{\beta[0]}(0))\) \(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\Phi_{\beta[0]}(0)))\) |
\(\gamma = 0\), \(\textrm{cof}(\beta) = 1\), and \(\beta \neq 1\). | |
| \(\Phi_{\beta[n]}(0)\) | \(\gamma = 0\), and \(\textrm{cof}(\beta) \geq \omega\). | |
| \(\chi_{0}(\Phi_{1}(\gamma[0])+1)\) \(\chi_{0}(\chi_{0}(\Phi_{1}(\gamma[0])+1))\) \(\chi_{0}(\chi_{0}(\chi_{0}(\Phi_{1}(\gamma[0])+1)))\) |
\(\textrm{cof}(\gamma) = 1\), \(\beta = 1\), and \(\gamma[0]\) is not an omega fixed point. | |
| \(\chi_{0}(\gamma[0]+1)\) \(\chi_{0}(\chi_{0}(\gamma[0]+1))\) \(\chi_{0}(\chi_{0}(\chi_{0}(\gamma[0]+1)))\) |
\(\textrm{cof}(\gamma) = 1\), \(\beta = 1\), and \(\gamma[0]\) is an omega fixed point. | |
| \(\Phi_{\beta[0]}(\Phi_{\beta}(\gamma[0])+1\)) \(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\Phi_{\beta}(\gamma[0])+1))\) \(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\Phi_{\beta}(\gamma[0])+1)))\) |
\(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) = 1\), \(\beta \neq 1\), and \(\gamma[0]\) is not a fixed point of \(\Phi_{\beta}\). | |
| \(\Phi_{\beta[0]}(\gamma[0]+1)\) \(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\gamma[0]+1))\) \(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\Phi_{\beta[0]}(\gamma[0]+1)))\) |
\(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) = 1\), \(\beta \neq 1\), and \(\gamma[0]\) is a fixed point of \(\Phi_{\beta}\). | |
| \(\Phi_{\beta[n]}(\Phi_{\beta}(\gamma[0])+1)\) | \(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) \geq \omega\), and \(\gamma[0]\) is not a fixed point of \(\Phi_{\beta}\). | |
| \(\Phi_{\beta[n]}(\gamma[0]+1)\) | \(\textrm{cof}(\gamma) = 1\), \(\textrm{cof}(\beta) \geq \omega\), and \(\gamma[0]\) is a fixed point of \(\Phi_{\beta}\). | |
| \(\Phi_{\beta}(\gamma[n])\) | \(\textrm{cof}(\gamma) \geq \omega\) and \(\gamma[n]\) is not a fixed point of \(\Phi_{\beta}\). | |
| \(\gamma[n]\) | \(\textrm{cof}(\gamma) \geq \omega\) and \(\gamma[n]\) is a fixed point of \(\Phi_{\beta}\). | |
| \(\chi_{\beta}(\gamma)\) | \(n\) | \(\textrm{cof}(\gamma) \leq 1\) |
| \(\chi_{\beta}(\gamma[n])\) | \(\textrm{cof}(\gamma) \geq \omega\) and \(\gamma[n]\) is not a fixed point of \(\chi_{\beta}\). | |
| \(\gamma[n]\) | \(\textrm{cof}(\gamma) \geq \omega\) and \(\gamma[n]\) is a fixed point of \(\chi_{\beta}\). | |
| \(\psi_{\chi_{\beta}(\gamma)}(\delta)\) | \(\varphi_{0}(0)\) \(\varphi_{\varphi_{0}(0)}(0)\) \(\varphi_{\varphi_{\varphi_{0}(0)}(0)}(0)\) |
\(\delta = 0\), \(\beta = 0\), and \(\gamma = 0\). |
| \(\varphi_{\chi_{0}(\gamma[0])+1}(0)\) \(\varphi_{\varphi_{\chi_{0}(\gamma[0])+1}(0)}(0)\) \(\varphi_{\varphi_{\varphi_{\chi_{0}(\gamma[0])+1}(0)}(0)}(0)\) |
\(\delta = 0\), \(\beta = 0\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is not an omega fixed point. | |
| \(\varphi_{\gamma[0]+1}(0)\) \(\varphi_{\varphi_{\gamma[0]+1}(0)}(0)\) \(\varphi_{\varphi_{\varphi_{\gamma[0]+1}(0)}(0)}(0)\) |
\(\delta = 0\), \(\beta = 0\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is an omega fixed point. | |
| \(\Phi_{1}(0)\) \(\Phi_{\Phi_{1}(0)}(0)\) \(\Phi_{\Phi_{\Phi_{1}(0)}(0)}(0)\) |
\(\delta = 0\), \(\beta = 1\), and \(\gamma = 0\). | |
| \(\Phi_{\chi_{1}(\gamma[0])+1}(0)\) \(\Phi_{\Phi_{\chi_{1}(\gamma[0])+1}(0)}(0)\) \(\Phi_{\Phi_{\Phi_{\chi_{1}(\gamma[0])+1}(0)}(0)}(0)\) |
\(\delta = 0\), \(\beta = 1\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is not a weakly \(2\)-inaccessible cardinal. | |
| \(\Phi_{\gamma[0]+1}(0)\) \(\Phi_{\Phi_{\gamma[0]+1}(0)}(0)\) \(\Phi_{\Phi_{\Phi_{\gamma[0]+1}(0)}(0)}(0)\) |
\(\delta = 0\), \(\beta = 1\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is a weakly \(2\)-inaccessible cardinal. | |
| \(\chi_{\beta[0]}(0)\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(0))\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\chi_{\beta[0]}(0)))\) |
\(\delta = 0\), \(\textrm{cof}(\beta) = 1\), \(\beta \neq 1\), and \(\gamma = 0\). | |
| \(\chi_{\beta[0]}(\chi_{\beta}(\gamma[0])+1)\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\chi_{\beta}(\gamma[0])+1))\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\chi_{\beta[0]}(\chi_{\beta}(\gamma[0])+1)))\) |
\(\delta = 0\), \(\textrm{cof}(\beta) = 1\), \(\beta \neq 1\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is not a weakly \((\beta+1)\)-inaccessible cardinal. | |
| \(\chi_{\beta[0]}(\gamma[0]+1)\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\gamma[0]+1))\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\chi_{\beta[0]}(\gamma[0]+1)))\) |
\(\delta = 0\), \(\textrm{cof}(\beta) = 1\), \(\beta \neq 1\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is a weakly \((\beta+1)\)-inaccessible cardinal. | |
| \(\chi_{\beta[n]}(0)\) | \(\delta = 0\), \(\omega \leq \textrm{cof}(\beta) < \psi_{\chi_M(0)}(0)\), and \(\gamma = 0\). | |
| \(\chi_{\beta[n]}(\chi_{\beta}(\gamma[0])+1)\) | \(\delta = 0\), \(\omega \leq \textrm{cof}(\beta) < \psi_{\chi_M(0)}(0)\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is not a fixed point of \(\chi_{\beta}\). | |
| \(\chi_{\beta[n]}(\gamma[0]+1)\) | \(\delta = 0\), \(\omega \leq \textrm{cof}(\beta) < \psi_{\chi_M(0)}(0)\), \(\textrm{cof}(\gamma) = 1\), and \(\gamma[0]\) is a fixed point of \(\chi_{\beta}\). | |
| \(\varphi_{\psi_{\chi_{0}(\gamma)}(\delta[0])+1}(0)\) \(\varphi_{\varphi_{\psi_{\chi_{0}(\gamma)}(\delta[0])+1}(0)}(0)\) \(\varphi_{\varphi_{\varphi_{\psi_{\chi_{0}(\gamma)}(\delta[0])+1}(0)}(0)}(0)\) |
\(\textrm{cof}(\delta) = 1\), and \(\beta = 0\). | |
| \(\chi_{\beta[0]}(\psi_{\chi_{\beta}(\gamma)}(\delta[0])+1)\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\psi_{\chi_{\beta}(\gamma)}(\delta[0])+1))\) \(\chi_{\beta[0]}(\chi_{\beta[0]}(\chi_{\beta[0]}(\psi_{\chi_{\beta}(\gamma)}(\delta[0])+1)))\) |
\(\textrm{cof}(\delta) = 1\), and \(\textrm{cof}(\beta) = 1\). | |
| \(\chi_{\beta[n]}(\psi_{\chi_{\beta}(\gamma)}(\delta[0])+1)\) | \(\textrm{cof}(\delta) = 1\), and \(\omega \leq \textrm{cof}(\beta) < M\). | |
| \(\psi_{\chi_{\beta}(\gamma)}(\delta[n])\) | \(\omega \leq \textrm{cof}(\delta) < \chi_{\beta}(\gamma)\). |
I list nesting expansions of semi-standard expressions of countable ordinals with respect to Rathjen's weakly Mahlo \(\psi\).
- Below \(\psi_{\Omega}(\psi_{\Omega_2}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\Omega)\) | \(\psi_{\Omega}(\psi_{\Omega}(0))\) \(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\Omega+\psi_{\Omega}(\Omega))\) | \(\psi_{\Omega}(\Omega+\psi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega+\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\Omega+\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0)))))\) |
| \(\psi_{\Omega}(\Omega+\Omega)\) | \(\psi_{\Omega}(\Omega+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\Omega+\psi_{\Omega}(\Omega+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega+\psi_{\Omega}(\Omega+\psi_{\Omega}(\Omega+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\varphi_{0}(\Omega+1)+\Omega)\) | \(\psi_{\Omega}(\varphi_{0}(\Omega+1)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\varphi_{0}(\Omega+1)+\psi_{\Omega}(\varphi_{0}(\Omega+1)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{0}(\Omega+1)+\psi_{\Omega}(\varphi_{0}(\Omega+1)+\psi_{\Omega}(\varphi_{0}(\Omega+1)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\varphi_{0}(\Omega+\Omega))\) | \(\psi_{\Omega}(\varphi_{0}(\Omega+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\varphi_{0}(\Omega+\psi_{\Omega}(\varphi_{0}(\Omega+\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\varphi_{0}(\Omega+\psi_{\Omega}(\varphi_{0}(\Omega+\psi_{\Omega}(\varphi_{0}(\Omega+\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{1}(\Omega+1)+\Omega)\) | \(\psi_{\Omega}(\varphi_{1}(\Omega+1)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\varphi_{1}(\Omega+1)+\psi_{\Omega}(\varphi_{1}(\Omega+1)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{1}(\Omega+1)+\psi_{\Omega}(\varphi_{1}(\Omega+1)+\psi_{\Omega}(\varphi_{1}(\Omega+1)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\varphi_{1}(\Omega+\Omega))\) | \(\psi_{\Omega}(\varphi_{1}(\Omega+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{1}(\Omega+\psi_{\Omega}(\varphi_{1}(\Omega+\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{1}(\Omega+\psi_{\Omega}(\varphi_{1}(\Omega+\psi_{\Omega}(\varphi_{1}(\Omega+\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\Omega)\) | \(\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\psi_{\Omega}(\varphi_{\omega}(\Omega+1)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\varphi_{\omega}(\Omega+\Omega))\) | \(\psi_{\Omega}(\varphi_{\omega}(\Omega+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{\omega}(\Omega+\psi_{\Omega}(\Omega+\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\varphi_{\omega}(\Omega+\psi_{\Omega}(\varphi_{\omega}(\Omega+\psi_{\Omega}(\Omega+\psi_{\Omega}(0))))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega}(1))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\Omega+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\Omega+1))}(\Omega+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\Omega+1))}(\Omega+1))}(\Omega+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega}(2))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\varphi_{\Omega}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\varphi_{\Omega}(1)+1)}(\varphi_{\Omega}(1)+1)\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\varphi_{\Omega}(1)+1)}(\varphi_{\Omega}(1)+1)}(\varphi_{\Omega}(1)+1)\) |
| \(\psi_{\Omega}(\varphi_{\Omega}(\omega+1))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\varphi_{\Omega}(\omega)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\varphi_{\Omega}(\omega)+1))}(\varphi_{\Omega}(\omega)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(\varphi_{\psi_{\Omega}(0)}(\varphi_{\Omega}(\omega)+1))}(\varphi_{\Omega}(\omega)+1))}(\varphi_{\Omega}(\omega)+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega}(\Omega))\) | \(\psi_{\Omega}(\varphi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega}(\psi_{\Omega}(\varphi_{\Omega}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega}(\psi_{\Omega}(\varphi_{\Omega}(\psi_{\Omega}(\varphi_{\Omega}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega}(\Omega+\Omega))\) | \(\psi_{\Omega}(\varphi_{\Omega}(\Omega+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega}(\Omega+\psi_{\Omega}(\varphi_{\Omega}(\Omega+\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega}(\Omega+\psi_{\Omega}(\varphi_{\Omega}(\Omega+\psi_{\Omega}(\varphi_{\Omega}(\Omega+\psi_{\Omega}(0))))))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega+\Omega}(0))\) | \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(0)}(0))\) \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(0)}(0))}(0))\) \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\varphi_{\Omega+\Omega}(1))\) | \(\psi_{\Omega}(\varphi_{\Omega}(\varphi_{\Omega+\Omega}(0)+1))\) \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega}(\varphi_{\Omega+\Omega}(0)+1))}(\varphi_{\Omega+\Omega}(0)+1))\) \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega}(\varphi_{\Omega+\Omega}(0)+1))}(\varphi_{\Omega+\Omega}(0)+1))}(\varphi_{\Omega+\Omega}(0)+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega+\Omega}(2))\) | \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(0)}(\varphi_{\Omega+\Omega}(1)+1))\) \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(0)}(\varphi_{\Omega+\Omega}(1)+1))}(\varphi_{\Omega+\Omega}(1)+1))\) \(\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(\varphi_{\Omega+\psi_{\Omega}(0)}(\varphi_{\Omega+\Omega}(1)+1))}(\varphi_{\Omega+\Omega}(1)+1))}(\varphi_{\Omega+\Omega}(1)+1))\) |
- Below \(\psi_{\Omega}(\Phi_{1}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\psi_{\Omega_2}(\Omega))\) | \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Omega_2)\) | \(\psi_{\Omega}(\psi_{\Omega_2}(0))\) \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0))))\) |
| \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\Omega))\) | \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0)))))\) |
| \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\Omega_2))\) | \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(0))))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0)))))\) |
| \(\psi_{\Omega}(\Omega_2+\Omega)\) | \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\Omega_2+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\Omega_2+\psi_{\Omega}(\Omega_2+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\Omega))\) | \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\Omega_2))\) | \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0))))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0)))))\) |
| \(\psi_{\Omega}(\Omega_2+\Omega_2)\) | \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(0))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(0))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_2}(1))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega_2}(0)}(\Omega_2+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_2}(\varphi_{\psi_{\Omega_2}(0)}(\Omega_2+1))}(\Omega_2+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_2}(\varphi_{\psi_{\Omega_2}(\varphi_{\psi_{\Omega_2}(0)}(\Omega_2+1))}(\Omega_2+1))}(\Omega_2+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_2}(2))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega_2}(0)}(\varphi_{\Omega_2}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_2}(\varphi_{\psi_{\Omega_2}(0)}(\varphi_{\Omega_2}(1)+1))}(\varphi_{\Omega_2}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_2}(\varphi_{\psi_{\Omega_2}(\varphi_{\psi_{\Omega_2}(0)}(\varphi_{\Omega_2}(1)+1))}(\varphi_{\Omega_2}(1)+1))}(\varphi_{\Omega_2}(1)+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_2}(\Omega))\) | \(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_2}(\Omega_2))\) | \(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega_2}(\varphi_{\Omega_2}(\psi_{\Omega_2}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega_2}(\psi_{\Omega_2}(\varphi_{\Omega_2}(\psi_{\Omega_2}(\varphi_{\Omega_2}(\psi_{\Omega_2}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\Omega_3}(\Omega))\) | \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\Omega_3}(\Omega_2))\) | \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_2}(\psi_{\Omega_3}(\psi_{\Omega_2}(0)))))\) \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_2}(\psi_{\Omega_3}(\psi_{\Omega_2}(\psi_{\Omega_3}(\psi_{\Omega_2}(0)))))))\) |
| \(\psi_{\Omega}(\Omega_3)\) | \(\psi_{\Omega}(\psi_{\Omega_3}(0))\) \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_3}(0)))\) \(\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_3}(\psi_{\Omega_3}(0))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\Omega))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0)))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\Omega_2))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(0))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0)))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\Omega_3))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega_3}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_3}(0))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\psi_{\Omega_3}(\psi_{\Omega_3}(\psi_{\Omega_3}(0)))))\) |
| \(\psi_{\Omega}(\Omega_3+\Omega)\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\Omega_3+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\Omega_3+\psi_{\Omega}(\Omega_3+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\Omega))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\Omega_2))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(\psi_{\Omega_2}(0)))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\Omega_3))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega_3}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega_3}(\psi_{\Omega_3}(0))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\psi_{\Omega_3}(\psi_{\Omega_3}(\psi_{\Omega_3}(0)))))\) |
| \(\psi_{\Omega}(\Omega_3+\Omega_2)\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(0))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_2}(0))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\Omega))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\Omega_2))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_2}(0)))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_2}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_2}(0)))))))\) |
| \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\Omega_3))\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_3}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_3}(\psi_{\Omega_3}(0))))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\psi_{\Omega_3}(\psi_{\Omega_3}(\psi_{\Omega_3}(0)))))\) |
| \(\psi_{\Omega}(\Omega_3+\Omega_3)\) | \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(0))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(0)))\) \(\psi_{\Omega}(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(\Omega_3+\psi_{\Omega_3}(0))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_3}(1))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega_3}(0)}(\Omega_3+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_3}(\varphi_{\psi_{\Omega_3}(0)}(\Omega_3+1))}(\Omega_3+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_3}(\varphi_{\psi_{\Omega_3}(\varphi_{\psi_{\Omega_3}(0)}(\Omega_3+1))}(\Omega_3+1))}(\Omega_3+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_3}(2))\) | \(\psi_{\Omega}(\varphi_{\psi_{\Omega_3}(0)}(\varphi_{\Omega_3}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_3}(\varphi_{\psi_{\Omega_3}(0)}(\varphi_{\Omega_3}(1)+1))}(\varphi_{\Omega_3}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\Omega_3}(\varphi_{\psi_{\Omega_3}(\varphi_{\psi_{\Omega_3}(0)}(\varphi_{\Omega_3}(1)+1))}(\varphi_{\Omega_3}(1)+1))}(\varphi_{\Omega_3}(1)+1))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_3}(\Omega))\) | \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_3}(\Omega_2))\) | \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega_2}(\varphi_{\Omega_3}(\psi_{\Omega_2}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega_2}(\varphi_{\Omega_3}(\psi_{\Omega_2}(\varphi_{\Omega_3}(\psi_{\Omega_2}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{\Omega_3}(\Omega_3))\) | \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega_3}(0)))\) \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega_3}(\varphi_{\Omega_3}(\psi_{\Omega_3}(0)))))\) \(\psi_{\Omega}(\varphi_{\Omega_3}(\psi_{\Omega_3}(\varphi_{\Omega_3}(\psi_{\Omega_3}(\varphi_{\Omega_3}(\psi_{\Omega_3}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{0}(\omega+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\omega+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\omega+1)}(\psi_{\chi_{0}(\omega+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\omega+1)}(\psi_{\chi_{0}(\omega+1)}(\psi_{\chi_{0}(\omega+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Omega))\) | \(\psi_{\Omega}(\chi_{0}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\chi_{0}(\psi_{\Omega}(\chi_{0}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\chi_{0}(\psi_{\Omega}(\chi_{0}(\psi_{\Omega}(\chi_{0}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{0}(\chi_{0}(\Omega)))\) | \(\psi_{\Omega}(\chi_{0}(\chi_{0}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\chi_{0}(\chi_{0}(\psi_{\Omega}(\chi_{0}(\chi_{0}(\psi_{\Omega}(0)))))))\) \(\psi_{\Omega}(\chi_{0}(\chi_{0}(\psi_{\Omega}(\chi_{0}(\chi_{0}(\psi_{\Omega}(\chi_{0}(\chi_{0}(\psi_{\Omega}(0))))))))))\) |
- Below \(\psi_{\Omega}(\psi_{I}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\Phi_{1}(0)+\Omega)\) | \(\psi_{\Omega}(\Phi_{1}(0)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\Phi_{1}(0)+\psi_{\Omega}(\Phi_{1}(0)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{1}(0)+\psi_{\Omega}(\Phi_{1}(0)+\psi_{\Omega}(\Phi_{1}(0)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\Omega))\) | \(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\psi_{\Omega}(\Phi_{1}(0)+\chi_{0}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{1}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{1}(1)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(1)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(1)+1)}(\psi_{\chi_{0}(\Phi_{1}(1)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(1)+1)}(\psi_{\chi_{0}(\Phi_{1}(1)+1)}(\psi_{\chi_{0}(\Phi_{1}(1)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{1}(\omega)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(\omega)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(\omega)+1)}(\psi_{\chi_{0}(\Phi_{1}(\omega)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{1}(\omega)+1)}(\psi_{\chi_{0}(\Phi_{1}(\omega)+1)}(\psi_{\chi_{0}(\Phi_{1}(\omega)+1)}(0))))\) |
| \(\psi_{\Omega}(\Phi_{1}(\Omega))\) | \(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{1}(\Omega_2))\) | \(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega_2}(\Phi_{1}(\psi_{\Omega_2}(0)))))\) \(\psi_{\Omega}(\Phi_{1}(\psi_{\Omega_2}(\Phi_{1}(\psi_{\Omega_2}(\Phi_{1}(\psi_{\Omega_2}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\Omega)))\) | \(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\psi_{\Omega}(0)))))))\) \(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\psi_{\Omega}(\Phi_{1}(\Phi_{1}(\psi_{\Omega}(0))))))))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{2}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(0)+1)}(0)\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(0)+1)}(\psi_{\chi_{0}(\Phi_{2}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(0)+1)}(\psi_{\chi_{0}(\Phi_{2}(0)+1)}(\psi_{\chi_{0}(\Phi_{2}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{2}(1)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(1)+1)}(0)\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(1)+1)}(\psi_{\chi_{0}(\Phi_{2}(1)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(1)+1)}(\psi_{\chi_{0}(\Phi_{2}(1)+1)}(\psi_{\chi_{0}(\Phi_{2}(1)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{2}(\omega)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(\omega)+1)}(0)\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(\omega)+1)}(\psi_{\chi_{0}(\Phi_{2}(\omega)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{2}(\omega)+1)}(\psi_{\chi_{0}(\Phi_{2}(\omega)+1)}(\psi_{\chi_{0}(\Phi_{2}(\omega)+1)}(0))))\) |
| \(\psi_{\Omega}(\Phi_{2}(\Omega))\) | \(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{2}(\Omega_2))\) | \(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega_2}(0)))\) \(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega_2}(\Phi_{2}(\psi_{\Omega_2}(0)))))\) \(\psi_{\Omega}(\Phi_{2}(\psi_{\Omega_2}(\Phi_{2}(\psi_{\Omega_2}(\Phi_{2}(\psi_{\Omega_2}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{2}(\chi_{0}(\Phi_{1}(0)+1)))\) | \(\psi_{\Omega}(\Phi_{2}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(0)))\) \(\psi_{\Omega}(\Phi_{2}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(\Phi_{2}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(0)))))\) \(\psi_{\Omega}(\Phi_{2}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(\Phi_{2}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(\Phi_{2}(\psi_{\chi_{0}(\Phi_{1}(0)+1)}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\Omega)))\) | \(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\psi_{\Omega}(0)))))))\) \(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\psi_{\Omega}(\Phi_{2}(\Phi_{1}(\psi_{\Omega}(0))))))))))\) |
| \(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\Omega)))\) | \(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\psi_{\Omega}(0)))))))\) \(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\psi_{\Omega}(\Phi_{2}(\Phi_{2}(\psi_{\Omega}(0))))))))))\) |
| \(\psi_{\Omega}(\chi_{0}(\Phi_{\omega}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{\omega}(0)+1)}(0)\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{\omega}(0)+1)}(\psi_{\chi_{0}(\Phi_{\omega}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\Phi_{\omega}(0)+1)}(\psi_{\chi_{0}(\Phi_{\omega}(0)+1)}(\psi_{\chi_{0}(\Phi_{\omega}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\Phi_{\omega}(\Omega))\) | \(\psi_{\Omega}(\Phi_{\omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{\omega}(\psi_{\Omega}(\Phi_{\omega}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{\omega}(\psi_{\Omega}(\Phi_{\omega}(\psi_{\Omega}(\Phi_{\omega}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{\Omega}(0))\) | \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(0))\) \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(0))}(0))\) \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\Phi_{\Omega}(1))\) | \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(0)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(0)+1))}(\Phi_{\Omega}(0)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(0)+1))}(\Phi_{\Omega}(0)+1))}(\Phi_{\Omega}(0)+1))\) |
| \(\psi_{\Omega}(\Phi_{\Omega}(2))\) | \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(1)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(1)+1))}(\Phi_{\Omega}(1)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(1)+1))}(\Phi_{\Omega}(1)+1))}(\Phi_{\Omega}(1)+1))\) |
| \(\psi_{\Omega}(\Phi_{\Omega}(\Omega))\) | \(\psi_{\Omega}(\Phi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{\Omega}(\psi_{\Omega}(\Phi_{\Omega}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{\Omega}(\psi_{\Omega}(\Phi_{\Omega}(\psi_{\Omega}(\Phi_{\Omega}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\Omega)\) | \(\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\psi_{\Omega}(\Phi_{\Phi_{1}(0)}(0)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\Phi_{\Phi_{\Omega}(0)}(0))\) | \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{\Omega}(0)}(0)}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{\Omega}(\Phi_{\Phi_{\psi_{\Omega}(0)}(0)}(0))}(0)}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{\Omega}(\Phi_{\Phi_{\psi_{\Omega}(\Phi_{\Phi_{\psi_{\Omega}(0)}(0)}(0))}(0)}(0))}(0)}(0))\) |
- Below \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\psi_{I}(0)+\Omega)\) | \(\psi_{\Omega}(\psi_{I}(0)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\psi_{I}(0)+\psi_{\Omega}(\psi_{I}(0)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{I}(0)+\psi_{\Omega}(\psi_{I}(0)+\psi_{\Omega}(\psi_{I}(0)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\chi_{0}(\psi_{I}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\psi_{I}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\psi_{I}(0)+1)}(\psi_{\chi_{0}(\psi_{I}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\psi_{I}(0)+1)}(\psi_{\chi_{0}(\psi_{I}(0)+1)}(\psi_{\chi_{0}(\psi_{I}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(I)\) | \(\psi_{\Omega}(\psi_{I}(0))\) \(\psi_{\Omega}(\psi_{I}(\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{I}(\psi_{I}(\psi_{I}(0))))\) |
| \(\psi_{\Omega}(I+\Omega)\) | \(\psi_{\Omega}(I+\psi_{\Omega}(0))\) \(\psi_{\Omega}(I+\psi_{\Omega}(I+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(I+\psi_{\Omega}(I+\psi_{\Omega}(I+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(I+\psi_{I}(I))\) | \(\psi_{\Omega}(I+\psi_{I}(\psi_{I}(0)))\) \(\psi_{\Omega}(I+\psi_{I}(\psi_{I}(\psi_{I}(0))))\) \(\psi_{\Omega}(I+\psi_{I}(\psi_{I}(\psi_{I}(\psi_{I}(0)))))\) |
| \(\psi_{\Omega}(I+I)\) | \(\psi_{\Omega}(I+\psi_{I}(0))\) \(\psi_{\Omega}(I+\psi_{I}(I+\psi_{I}(0)))\) \(\psi_{\Omega}(I+\psi_{I}(I+\psi_{I}(I+\psi_{I}(0))))\) |
| \(\psi_{\Omega}(\varphi_{I}(1))\) | \(\psi_{\Omega}(\varphi_{\psi_{I}(0)}(I+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I}(\varphi_{\psi_{I}(0)}(I+1))}(I+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I}(\varphi_{\psi_{I}(\varphi_{\psi_{I}(0)}(I+1))}(I+1))}(I+1))\) |
| \(\psi_{\Omega}(\varphi_{I}(2))\) | \(\psi_{\Omega}(\varphi_{\psi_{I}(0)}(\varphi_{I}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I}(\varphi_{\psi_{I}(0)}(\varphi_{I}(1)+1))}(\varphi_{I}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I}(\varphi_{\psi_{I}(\varphi_{\psi_{I}(0)}(\varphi_{I}(1)+1))}(\varphi_{I}(1)+1))}(\varphi_{I}(1)+1))\) |
| \(\psi_{\Omega}(\varphi_{I}(\Omega))\) | \(\psi_{\Omega}(\varphi_{I}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{I}(\psi_{\Omega}(\varphi_{I}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{I}(\psi_{\Omega}(\varphi_{I}(\psi_{\Omega}(\varphi_{I}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{I}(I))\) | \(\psi_{\Omega}(\varphi_{I}(\psi_{I}(0)))\) \(\psi_{\Omega}(\varphi_{I}(\psi_{I}(\varphi_{I}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\varphi_{I}(\psi_{I}(\varphi_{I}(\psi_{I}(\varphi_{I}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{0}(I+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(I+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(I+1)}(\psi_{\chi_{0}(I+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(I+1)}(\psi_{\chi_{0}(I+1)}(\psi_{\chi_{0}(I+1)}(0))))\) |
| \(\psi_{\Omega}(\Phi_{I}(1))\) | \(\psi_{\Omega}(\Phi_{\psi_{I}(0)}(I+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I}(\Phi_{\psi_{I}(0)}(I+1))}(I+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I}(\Phi_{\psi_{I}(\Phi_{\psi_{I}(0)}(I+1))}(I+1))}(I+1))\) |
| \(\psi_{\Omega}(\Phi_{I}(2))\) | \(\psi_{\Omega}(\Phi_{\psi_{I}(0)}(\Phi_{I}(1)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I}(\Phi_{\psi_{I}(0)}(\Phi_{I}(1)+1))}(\Phi_{I}(1)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I}(\Phi_{\psi_{I}(\Phi_{\psi_{I}(0)}(\Phi_{I}(1)+1))}(\Phi_{I}(1)+1))}(\Phi_{I}(1)+1))\) |
| \(\psi_{\Omega}(\Phi_{I}(\Omega))\) | \(\psi_{\Omega}(\Phi_{I}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{I}(\psi_{\Omega}(\Phi_{I}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{I}(\psi_{\Omega}(\Phi_{I}(\psi_{\Omega}(\Phi_{I}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{I}(I))\) | \(\psi_{\Omega}(\Phi_{I}(\psi_{I}(0)))\) \(\psi_{\Omega}(\Phi_{I}(\psi_{I}(\Phi_{I}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\Phi_{I}(\psi_{I}(\Phi_{I}(\psi_{I}(\Phi_{I}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{\Phi_{I}(1)}(0))\) | \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{I}(0)}(I)}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{I}(\Phi_{\Phi_{\psi_{I}(0)}(I)}(0))}(I)}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{I}(\Phi_{\Phi_{\psi_{I}(0)}(I)}(0))}(I)}(0))\) |
| \(\psi_{\Omega}(\psi_{I_2}(0)+I)\) | \(\psi_{\Omega}(\psi_{I_2}(0)+\psi_{I}(0))\) \(\psi_{\Omega}(\psi_{I_2}(0)+\psi_{I}(\psi_{I_2}(0)+\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{I_2}(0)+\psi_{I}(\psi_{I_2}(0)+\psi_{I}(\psi_{I_2}(0)+\psi_{I}(0))))\) |
| \(\psi_{\Omega}(\psi_{I_2}(\Omega))\) | \(\psi_{\Omega}(\psi_{I_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{I_2}(\psi_{\Omega}(\psi_{I_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\psi_{I_2}(\psi_{\Omega}(\psi_{I_2}(\psi_{\Omega}(\psi_{I_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{I_2}(I))\) | \(\psi_{\Omega}(\psi_{I_2}(\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{I_2}(\psi_{I}(\psi_{I_2}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\psi_{I_2}(\psi_{I}(\psi_{I_2}(\psi_{I}(\psi_{I_2}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(I_2)\) | \(\psi_{\Omega}(\psi_{I_2}(0))\) \(\psi_{\Omega}(\psi_{I_2}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(0))))\) |
| \(\psi_{\Omega}(I_2+\psi_{I}(\Omega))\) | \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{\Omega}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{\Omega}(\psi_{\Omega}(\psi_{\Omega}(0)))))\) |
| \(\psi_{\Omega}(I_2+\psi_{I}(I))\) | \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{I}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{I}(\psi_{I}(0))))\) \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{I}(\psi_{I}(\psi_{I}(0)))))\) |
| \(\psi_{\Omega}(I_2+\psi_{I}(I_2))\) | \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{I_2}(\psi_{I_2}(0))))\) \(\psi_{\Omega}(I_2+\psi_{I}(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(0)))))\) |
| \(\psi_{\Omega}(I_2+I)\) | \(\psi_{\Omega}(I_2+\psi_{I}(0))\) \(\psi_{\Omega}(I_2+\psi_{I}(I_2+\psi_{I}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I}(I_2+\psi_{I}(I_2+\psi_{I}(0))))\) |
| \(\psi_{\Omega}(I_2+\psi_{I_2}(\Omega))\) | \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(I_2+\psi_{I_2}(I))\) | \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{I}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{I}(I_2+\psi_{I_2}(\psi_{I}(0)))))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{I}(I_2+\psi_{I_2}(\psi_{I}(I_2+\psi_{I_2}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(I_2+\psi_{I_2}(I_2))\) | \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(0))))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(0)))))\) |
| \(\psi_{\Omega}(I_2+I_2)\) | \(\psi_{\Omega}(I_2+\psi_{I_2}(0))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(I_2+\psi_{I_2}(0)))\) \(\psi_{\Omega}(I_2+\psi_{I_2}(I_2+\psi_{I_2}(I_2+\psi_{I_2}(0))))\) |
| \(\psi_{\Omega}(\varphi_{I_2}(1))\) | \(\psi_{\Omega}(\varphi_{\psi_{I_2}(0)}(I_2+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I_2}(\varphi_{\psi_{I_2}(0)}(I_2+1))}(I_2+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I_2}(\varphi_{\psi_{I_2}(\varphi_{\psi_{I_2}(0)}(I_2+1))}(I_2+1))}(I_2+1))\) |
| \(\psi_{\Omega}(\varphi_{I_2}(2))\) | \(\psi_{\Omega}(\varphi_{\psi_{I_2}(0)}(\varphi_{I_2}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I_2}(\varphi_{\psi_{I_2}(0)}(\varphi_{I_2}(1)+1))}(\varphi_{I_2}(1)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{I_2}(\varphi_{\psi_{I_2}(\varphi_{\psi_{I_2}(0)}(\varphi_{I_2}(1)+1))}(\varphi_{I_2}(1)+1))}(\varphi_{I_2}(1)+1))\) |
| \(\psi_{\Omega}(\varphi_{I_2}(\Omega))\) | \(\psi_{\Omega}(\varphi_{I_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\varphi_{I_2}(\psi_{\Omega}(\varphi_{I_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\varphi_{I_2}(\psi_{\Omega}(\varphi_{I_2}(\psi_{\Omega}(\varphi_{I_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{I_2}(I))\) | \(\psi_{\Omega}(\varphi_{I_2}(\psi_{I}(0)))\) \(\psi_{\Omega}(\varphi_{I_2}(\psi_{I}(\varphi_{I_2}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\varphi_{I_2}(\psi_{I}(\varphi_{I_2}(\psi_{I}(\varphi_{I_2}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\varphi_{I_2}(I_2))\) | \(\psi_{\Omega}(\varphi_{I_2}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(\varphi_{I_2}(\psi_{I_2}(\varphi_{I_2}(\psi_{I_2}(0)))))\) \(\psi_{\Omega}(\varphi_{I_2}(\psi_{I_2}(\varphi_{I_2}(\psi_{I_2}(\varphi_{I_2}(\psi_{I_2}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{0}(I_2+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(I_2+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(I_2+1)}(\psi_{\chi_{0}(I_2+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(I_2+1)}(\psi_{\chi_{0}(I_2+1)}(\psi_{\chi_{0}(I_2+1)}(0))))\) |
| \(\psi_{\Omega}(\Phi_{I_2}(1))\) | \(\psi_{\Omega}(\Phi_{\psi_{I_2}(0)}(I_2+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I_2}(\Phi_{\psi_{I_2}(0)}(I_2+1))}(I_2+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I_2}(\Phi_{\psi_{I_2}(\Phi_{\psi_{I_2}(0)}(I_2+1))}(I_2+1))}(I_2+1))\) |
| \(\psi_{\Omega}(\Phi_{I_2}(2))\) | \(\psi_{\Omega}(\Phi_{\psi_{I_2}(0)}(\Phi_{I_2}(1)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I_2}(\Phi_{\psi_{I_2}(0)}(\Phi_{I_2}(1)+1))}(\Phi_{I_2}(1)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{I_2}(\Phi_{\psi_{I_2}(\Phi_{\psi_{I_2}(0)}(\Phi_{I_2}(1)+1))}(\Phi_{I_2}(1)+1))}(\Phi_{I_2}(1)+1))\) |
| \(\psi_{\Omega}(\Phi_{I_2}(\Omega))\) | \(\psi_{\Omega}(\Phi_{I_2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\Phi_{I_2}(\psi_{\Omega}(\Phi_{I_2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\Phi_{I_2}(\psi_{\Omega}(\Phi_{I_2}(\psi_{\Omega}(\Phi_{I_2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{I_2}(I))\) | \(\psi_{\Omega}(\Phi_{I_2}(\psi_{I}(0)))\) \(\psi_{\Omega}(\Phi_{I_2}(\psi_{I}(\Phi_{I_2}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\Phi_{I_2}(\psi_{I}(\Phi_{I_2}(\psi_{I}(\Phi_{I_2}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{I_2}(I_2))\) | \(\psi_{\Omega}(\Phi_{I_2}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(\Phi_{I_2}(\psi_{I_2}(\Phi_{I_2}(\psi_{I_2}(0)))))\) \(\psi_{\Omega}(\Phi_{I_2}(\psi_{I_2}(\Phi_{I_2}(\psi_{I_2}(\Phi_{I_2}(\psi_{I_2}(0)))))))\) |
| \(\psi_{\Omega}(\Phi_{\Phi_{I_2}(1)}(0))\) | \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{I_2}(0)}(I_2)}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{I_2}(\Phi_{\Phi_{\psi_{I_2}(0)}(I_2)}(0))}(I_2)}(0))\) \(\psi_{\Omega}(\Phi_{\Phi_{\psi_{I_2}(\Phi_{\Phi_{\psi_{I_2}(0)}(I_2)}(0))}(I_2)}(0))\) |
| \(\psi_{\Omega}(I_3)\) | \(\psi_{\Omega}(\psi_{I_3}(0))\) \(\psi_{\Omega}(\psi_{I_3}(\psi_{I_3}(0)))\) \(\psi_{\Omega}(\psi_{I_3}(\psi_{I_3}(\psi_{I_3}(0))))\) |
| \(\psi_{\Omega}(\chi_{1}(\omega+1))\) | \(\psi_{\Omega}(\psi_{\chi_{1}(\omega+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{1}(\omega+1)}(\psi_{\chi_{1}(\omega+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{1}(\omega+1)}(\psi_{\chi_{1}(\omega+1)}(\psi_{\chi_{1}(\omega+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{1}(\Omega))\) | \(\psi_{\Omega}(\chi_{1}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\chi_{1}(\psi_{\Omega}(\chi_{1}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\chi_{1}(\psi_{\Omega}(\chi_{1}(\psi_{\Omega}(\chi_{1}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{1}(\varphi_{\Omega}(1)))\) | \(\psi_{\Omega}(\chi_{1}(\varphi_{\psi_{\Omega}(0)}(\Omega+1)))\) \(\psi_{\Omega}(\chi_{1}(\varphi_{\psi_{\Omega}(\chi_{1}(\varphi_{\psi_{\Omega}(0)}(\Omega+1)))}(\Omega+1)))\) \(\psi_{\Omega}(\chi_{1}(\varphi_{\psi_{\Omega}(\chi_{1}(\varphi_{\psi_{\Omega}(\chi_{1}(\varphi_{\psi_{\Omega}(0)}(\Omega+1)))}(\Omega+1)))}(\Omega+1)))\) |
| \(\psi_{\Omega}(\chi_{1}(\Phi_{\Omega}(0)))\) | \(\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(0)}(0)))\) \(\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(0)}(0)))}(0)))\) \(\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(0)}(0)))}(0)))}(0)))\) |
| \(\psi_{\Omega}(\chi_{1}(\Phi_{\Omega}(1)))\) | \(\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(0)+1)))\) \(\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(0)+1)))}(\Phi_{\Omega}(0)+1)))\) \(\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(\chi_{1}(\Phi_{\psi_{\Omega}(0)}(\Phi_{\Omega}(0)+1)))}(\Phi_{\Omega}(0)+1)))}(\Phi_{\Omega}(0)+1)))\) |
| \(\psi_{\Omega}(\chi_{1}(I))\) | \(\psi_{\Omega}(\chi_{1}(\psi_{I}(0)))\) \(\psi_{\Omega}(\chi_{1}(\psi_{I}(\chi_{1}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\chi_{1}(\psi_{I}(\chi_{1}(\psi_{I}(\chi_{1}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{1}(\chi_{1}(I)))\) | \(\psi_{\Omega}(\chi_{1}(\chi_{1}(\psi_{I}(0))))\) \(\psi_{\Omega}(\chi_{1}(\chi_{1}(\psi_{I}(\chi_{1}(\chi_{1}(\psi_{I}(0)))))))\) \(\psi_{\Omega}(\chi_{1}(\chi_{1}(\psi_{I}(\chi_{1}(\chi_{1}(\psi_{I}(\chi_{1}(\chi_{1}(\psi_{I}(0))))))))))\) |
- Below \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\Omega)\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+I)\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{I}(0))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{I}(\psi_{\chi_{2}(0)}(0)+\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{I}(\psi_{\chi_{2}(0)}(0)+\psi_{I}(\psi_{\chi_{2}(0)}(0)+\psi_{I}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+I_2)\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{I_2}(0))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{I_2}(\psi_{\chi_{2}(0)}(0)+\psi_{I_2}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\psi_{I_2}(\psi_{\chi_{2}(0)}(0)+\psi_{I_2}(\psi_{\chi_{2}(0)}(0)+\psi_{I_2}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\Omega))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(I))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{I}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{I}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{I}(\psi_{\chi_{2}(0)}(0)+\chi_{1}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\Omega))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(I))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{I}(\psi_{\chi_{2}(0)}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{I}(\psi_{\chi_{2}(0)}(\psi_{I}(\psi_{\chi_{2}(0)}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(I_2))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{I_2}(\psi_{\chi_{2}(0)}(\psi_{I_2}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{I_2}(\psi_{\chi_{2}(0)}(\psi_{I_2}(\psi_{\chi_{2}(0)}(\psi_{I_2}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{2}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\chi_{2}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(0)}(\psi_{\chi_{2}(0)}(\psi_{\chi_{2}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{2}(0)+\chi_{2}(0))\) | \(\psi_{\Omega}(\chi_{2}(0)+\psi_{\chi_{2}(0)}(0))\) \(\psi_{\Omega}(\chi_{2}(0)+\psi_{\chi_{2}(0)}(\chi_{2}(0)+\psi_{\chi_{2}(0)}(0)))\) \(\psi_{\Omega}(\chi_{2}(0)+\psi_{\chi_{2}(0)}(\chi_{2}(0)+\psi_{\chi_{2}(0)}(\chi_{2}(0)+\psi_{\chi_{2}(0)}(0))))\) |
| \(\psi_{\Omega}(\varphi_{\chi_{2}(0)}(1))\) | \(\psi_{\Omega}(\varphi_{\psi_{\chi_{2}(0)}(0)}(\chi_{2}(0)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\chi_{2}(0)}(\varphi_{\psi_{\chi_{2}(0)}(0)}(\chi_{2}(0)+1))}(\chi_{2}(0)+1))\) \(\psi_{\Omega}(\varphi_{\psi_{\chi_{2}(0)}(\varphi_{\psi_{\chi_{2}(0)}(\varphi_{\psi_{\chi_{2}(0)}(0)}(\chi_{2}(0)+1))}(\chi_{2}(0)+1))}(\chi_{2}(0)+1))\) |
| \(\psi_{\Omega}(\Phi_{\chi_{2}(0)}(1))\) | \(\psi_{\Omega}(\Phi_{\psi_{\chi_{2}(0)}(0)}(\chi_{2}(0)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{\chi_{2}(0)}(\Phi_{\psi_{\chi_{2}(0)}(0)}(\chi_{2}(0)+1))}(\chi_{2}(0)+1))\) \(\psi_{\Omega}(\Phi_{\psi_{\chi_{2}(0)}(\Phi_{\psi_{\chi_{2}(0)}(\Phi_{\psi_{\chi_{2}(0)}(0)}(\chi_{2}(0)+1))}(\chi_{2}(0)+1))}(\chi_{2}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{1}(\chi_{2}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{1}(\chi_{2}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{1}(\chi_{2}(0)+1)}(\psi_{\chi_{1}(\chi_{2}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{1}(\chi_{2}(0)+1)}(\psi_{\chi_{1}(\chi_{2}(0)+1)}(\psi_{\chi_{1}(\chi_{2}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{2}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{2}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{2}(1)}(\psi_{\chi_{2}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{2}(1)}(\psi_{\chi_{2}(1)}(\psi_{\chi_{2}(1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{2}(\Omega))\) | \(\psi_{\Omega}(\chi_{2}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\chi_{2}(\psi_{\Omega}(\chi_{2}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\chi_{2}(\psi_{\Omega}(\chi_{2}(\psi_{\Omega}(\chi_{2}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{2}(I))\) | \(\psi_{\Omega}(\chi_{2}(\psi_{I}(0)))\) \(\psi_{\Omega}(\chi_{2}(\psi_{I}(\chi_{2}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\chi_{2}(\psi_{I}(\chi_{2}(\psi_{I}(\chi_{2}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{2}(I_2))\) | \(\psi_{\Omega}(\chi_{2}(\psi_{I_2}(0)))\) \(\psi_{\Omega}(\chi_{2}(\psi_{I_2}(\chi_{2}(\psi_{I_2}(0)))))\) \(\psi_{\Omega}(\chi_{2}(\psi_{I_2}(\chi_{2}(\psi_{I_2}(\chi_{2}(\psi_{I_2}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\Omega)))\) | \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{\Omega}(0)))))))\) \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{\Omega}(0))))))))))\) |
| \(\psi_{\Omega}(\chi_{2}(\chi_{1}(I)))\) | \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{I}(0))))\) \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{I}(\chi_{2}(\chi_{1}(\psi_{I}(0)))))))\) \(\psi_{\Omega}(\chi_{2}(\chi_{1}(\psi_{I}(\chi_{2}(\chi_{1}(\psi_{I}(\chi_{2}(\chi_{1}(\psi_{I}(0))))))))))\) |
| \(\psi_{\Omega}(\chi_{2}(\chi_{2}(0)))\) | \(\psi_{\Omega}(\chi_{2}(\psi_{\chi_{2}(0)}(0)))\) \(\psi_{\Omega}(\chi_{2}(\psi_{\chi_{2}(0)}(\chi_{2}(\psi_{\chi_{2}(0)}(0)))))\) \(\psi_{\Omega}(\chi_{2}(\psi_{\chi_{2}(0)}(\chi_{2}(\psi_{\chi_{2}(0)}(\chi_{2}(\psi_{\chi_{2}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\omega}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\omega}(0)}(\psi_{\chi_{\omega}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\omega}(0)}(\psi_{\chi_{\omega}(0)}(\psi_{\chi_{\omega}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{\omega}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\omega}(1)}(\psi_{\chi_{\omega}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\omega}(1)}(\psi_{\chi_{\omega}(1)}(\psi_{\chi_{\omega}(1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(\Omega))\) | \(\psi_{\Omega}(\chi_{\omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{\Omega}(\chi_{\omega}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{\Omega}(\chi_{\omega}(\psi_{\Omega}(\chi_{\omega}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(I))\) | \(\psi_{\Omega}(\chi_{\omega}(\psi_{I}(0)))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{I}(\chi_{\omega}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{I}(\chi_{\omega}(\psi_{I}(\chi_{\omega}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\Omega)))\) | \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{\Omega}(0))))\) \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{\Omega}(0)))))))\) \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{\Omega}(0))))))))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(I)))\) | \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{I}(0))))\) \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{I}(\chi_{\omega}(\chi_{1}(\psi_{I}(0)))))))\) \(\psi_{\Omega}(\chi_{\omega}(\chi_{1}(\psi_{I}(\chi_{\omega}(\chi_{1}(\psi_{I}(\chi_{\omega}(\chi_{1}(\psi_{I}(0))))))))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(\chi_{2}(0)))\) | \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{2}(0)}(0)))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{2}(0)}(\chi_{\omega}(\psi_{\chi_{2}(0)}(0)))))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{2}(0)}(\chi_{\omega}(\psi_{\chi_{2}(0)}(\chi_{\omega}(\psi_{\chi_{2}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(\chi_{\omega}(0)))\) | \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{\omega}(0)}(0)))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{\omega}(0)}(\chi_{\omega}(\psi_{\chi_{\omega}(0)}(0)))))\) \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{\omega}(0)}(\chi_{\omega}(\psi_{\chi_{\omega}(0)}(\chi_{\omega}(\psi_{\chi_{\omega}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\Omega}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\Omega}(0)}(0))\) \(\psi_{\Omega}(\chi_{\psi_{\Omega}(\chi_{\psi_{\Omega}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{\psi_{\Omega}(\chi_{\psi_{\Omega}(\chi_{\psi_{\Omega}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\Omega}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\Omega}(0)}(\psi_{\chi_{\Omega}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\Omega}(0)}(\psi_{\chi_{\Omega}(0)}(\psi_{\chi_{\Omega}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\Omega}(1)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\Omega}(0)}(\chi_{\Omega}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\Omega}(\chi_{\psi_{\Omega}(0)}(\chi_{\Omega}(0)+1))}(\chi_{\Omega}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\Omega}(\chi_{\psi_{\Omega}(\chi_{\psi_{\Omega}(0)}(\chi_{\Omega}(0)+1))}(\chi_{\Omega}(0)+1))}(\chi_{\Omega}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{\Omega}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\Omega}(1)}(\psi_{\chi_{\Omega}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\Omega}(1)}(\psi_{\chi_{\Omega}(1)}(\psi_{\chi_{\Omega}(1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(\Omega))\) | \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\Omega}(\chi_{\Omega}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\Omega}(\chi_{\Omega}(\psi_{\Omega}(\chi_{\Omega}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(I))\) | \(\psi_{\Omega}(\chi_{\Omega}(\psi_{I}(0)))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{I}(\chi_{\Omega}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{I}(\chi_{\Omega}(\psi_{I}(\chi_{\Omega}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(\chi_{2}(0)))\) | \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{2}(0)}(0)))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{2}(0)}(\chi_{\Omega}(\psi_{\chi_{2}(0)}(0)))))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{2}(0)}(\chi_{\Omega}(\psi_{\chi_{2}(0)}(\chi_{\Omega}(\psi_{\chi_{2}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(\chi_{\omega}(0)))\) | \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{\omega}(0)}(0)))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{\omega}(0)}(\chi_{\Omega}(\psi_{\chi_{\omega}(0)}(0)))))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{\omega}(0)}(\chi_{\Omega}(\psi_{\chi_{\omega}(0)}(\chi_{\Omega}(\psi_{\chi_{\omega}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(\chi_{\Omega}(0)))\) | \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{\Omega}(0)}(0)))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{\Omega}(0)}(\chi_{\Omega}(\psi_{\chi_{\Omega}(0)}(0)))))\) \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{\Omega}(0)}(\chi_{\Omega}(\psi_{\chi_{\Omega}(0)}(\chi_{\Omega}(\psi_{\chi_{\Omega}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{I}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{I}(0)}(0))\) \(\psi_{\Omega}(\chi_{\psi_{I}(\chi_{\psi_{I}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{\psi_{I}(\chi_{\psi_{I}(\chi_{\psi_{I}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\chi_{I}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{I}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{I}(0)}(\psi_{\chi_{I}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{I}(0)}(\psi_{\chi_{I}(0)}(\psi_{\chi_{I}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{I}(1)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{I}(0)}(\chi_{I}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{I}(\chi_{\psi_{I}(0)}(\chi_{I}(0)+1))}(\chi_{I}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{I}(\chi_{\psi_{I}(\chi_{\psi_{I}(0)}(\chi_{I}(0)+1))}(\chi_{I}(0)+1))}(\chi_{I}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{I}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{I}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{I}(1)}(\psi_{\chi_{I}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{I}(1)}(\psi_{\chi_{I}(1)}(\psi_{\chi_{I}(1)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{I}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{I}(0)}(\chi_{\psi_{\chi_{I}(0)}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{I}(0)}(\chi_{\psi_{\chi_{I}(0)}(\chi_{\psi_{\chi_{I}(0)}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\chi_{\chi_{I}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(0))))\) |
- Below \(\psi_{\Omega}(\chi_{M}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0))\) | \(\psi_{\Omega}(\chi_{0}(0))\) \(\psi_{\Omega}(\chi_{\chi_{0}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{0}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\Omega)\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\Omega}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+I)\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{I}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{I}(\psi_{\chi_{M}(0)}(0)+\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{I}(\psi_{\chi_{M}(0)}(0)+\psi_{I}(\psi_{\chi_{M}(0)}(0)+\psi_{I}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{\omega}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\omega}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\omega}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\omega}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\omega}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\omega}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\omega}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{\Omega}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\Omega}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\Omega}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\Omega}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\Omega}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\Omega}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\Omega}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{I}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{I}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{I}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{I}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{\chi_{I}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\chi_{I}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\chi_{I}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{\chi_{I}(0)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{M}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{0}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{\chi_{0}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0)+\chi_{\chi_{\chi_{0}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\varphi_{\psi_{\chi_{M}(0)}(0)}(1))\) | \(\psi_{\Omega}(\varphi_{\chi_{0}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\varphi_{\chi_{\chi_{0}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\varphi_{\chi_{\chi_{\chi_{0}(0)}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\varphi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{M}(0)}(0)))\) | \(\psi_{\Omega}(\varphi_{\psi_{\chi_{M}(0)}(0)}(\chi_{0}(0)))\) \(\psi_{\Omega}(\varphi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{0}(0)}(0)))\) \(\psi_{\Omega}(\varphi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{\chi_{0}(0)}(0)}(0)))\) |
| \(\psi_{\Omega}(\chi_{0}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{0}(\psi_{\chi_{M}(0)}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{0}(\psi_{\chi_{M}(0)}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{0}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{0}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{0}(\psi_{\chi_{M}(0)}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\Phi_{\psi_{\chi_{M}(0)}(0)}(1))\) | \(\psi_{\Omega}(\Phi_{\chi_{0}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\Phi_{\chi_{\chi_{0}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\Phi_{\chi_{\chi_{\chi_{0}(0)}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\Phi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{M}(0)}(0)))\) | \(\psi_{\Omega}(\Phi_{\psi_{\chi_{M}(0)}(0)}(\chi_{0}(0)))\) \(\psi_{\Omega}(\Phi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{0}(0)}(0)))\) \(\psi_{\Omega}(\Phi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{\chi_{0}(0)}(0)}(0)))\) |
| \(\psi_{\Omega}(\chi_{1}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{1}(\psi_{\chi_{M}(0)}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{1}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{1}(\psi_{\chi_{M}(0)}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{1}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{1}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{1}(\psi_{\chi_{M}(0)}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\omega}(\psi_{\chi_{M}(0)}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\Omega}(\psi_{\chi_{M}(0)}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{I}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{I}(\psi_{\chi_{M}(0)}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{I}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{I}(\psi_{\chi_{M}(0)}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{I}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{I}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{I}(\psi_{\chi_{M}(0)}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1)}(\psi_{\chi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(0)+1)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_0(\psi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\chi_{0}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{0}(0)}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(1))\) | \(\psi_{\Omega}(\chi_0(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\chi_{0}(0)}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{0}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(0)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(2))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(1)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(1)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(1)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\omega+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\omega)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\omega)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(\omega)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\Omega))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\Omega}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\Omega}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\Omega+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\Omega)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\Omega)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(\Omega)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(I))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{I}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{I}(\psi_{\chi_{M}(0)}(\psi_{I}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{I}(\psi_{\chi_{M}(0)}(\psi_{I}(\psi_{\chi_{M}(0)}(\psi_{I}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(I+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(I)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(I)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(I)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{2}(0)))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{2}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{2}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{2}(0)}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{2}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{2}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{2}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{2}(0)+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{2}(0))}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{2}(0))}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{2}(0))}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{I}(0)))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{I}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{I}(0)}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{I}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{I}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{I}(0)+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{I}(0))}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{I}(0))}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{I}(0))}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{\chi_{I}(0)}(0)))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(0)))))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{\chi_{I}(0)}(0)}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{\chi_{I}(0)}(0)+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\chi_{I}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{\chi_{I}(0)}(0))}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{\chi_{I}(0)}(0))}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0)))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{0}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{\chi_{0}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\chi_{\chi_{\chi_{0}(0)}(0)}(0)))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0)+1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0))}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0))}(0)}(0)}(0))\) |
- Below \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\chi_{M}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+1))}(\chi_{M}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+1))}(\chi_{M}(0)+1))}(\chi_{M}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(0)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(0)}(2))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(1)+1))}(\psi_{\chi_{\chi_{M}(0)}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(1)+1))}(\psi_{\chi_{\chi_{M}(0)}(0)}(1)+1))}(\psi_{\chi_{\chi_{M}(0)}(0)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(1)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{M}(0)}(0)+1))}(\chi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{M}(0)}(0)+1))}(\chi_{\chi_{M}(0)}(0)+1))}(\chi_{\chi_{M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(1)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(1)}(\psi_{\chi_{\chi_{M}(0)}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(1)}(\psi_{\chi_{\chi_{M}(0)}(1)}(\psi_{\chi_{\chi_{M}(0)}(1)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(2)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{M}(0)}(1)+1))}(\chi_{\chi_{M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{\chi_{M}(0)}(1)+1))}(\chi_{\chi_{M}(0)}(1)+1))}(\chi_{\chi_{M}(0)}(1)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(2)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(2))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(2)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(2)}(\psi_{\chi_{\chi_{M}(0)}(2)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)}(2)}(\psi_{\chi_{\chi_{M}(0)}(2)}(\psi_{\chi_{\chi_{M}(0)}(2)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)}(0)))\) \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(0)+1}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(\psi_{\chi_{\chi_{M}(0)+1}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+\chi_{M}(0)+1))\)
\(\psi_{\Omega}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+\chi_{M}(0)+1))}(\chi_{M}(0)+\chi_{M}(0)+1))\) \(\psi_{\Omega}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+\chi_{M}(0)+1))}(\chi_{M}(0)+\chi_{M}(0)+1))}(\chi_{M}(0)+\chi_{M}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(1))\) | \(\psi_{\Omega}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)+1))\)
\(\psi_{\Omega}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(\chi_{\chi_{M}(0)+\psi_{\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(0)+\chi_{M}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(\psi_{\chi_{\chi_{M}(0)+\chi_{M}(0)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+1)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+1))}(\chi_{M}(0)+1)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(\chi_{\psi_{\chi_{M}(0)}(0)}(\chi_{M}(0)+1))}(\chi_{M}(0)+1))}(\chi_{M}(0)+1)}(0))\) |
| \(\psi_{\Omega}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(\psi_{\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(\chi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(\chi_{\chi_{M}(0)}(0)+1))}(\chi_{\chi_{M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(\chi_{\psi_{\chi_{\chi_{M}(0)}(0)}(0)}(\chi_{\chi_{M}(0)}(0)+1))}(\chi_{\chi_{M}(0)}(0)+1))}(\chi_{\chi_{M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\chi_{\chi_{\chi_{M}(0)}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(\psi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(0))\) | \(\psi_{\Omega}(\chi_{\chi_{M}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{M}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\chi_{M}(0)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(1)}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(1)}(0)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(2))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(1)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(1)}(1)}(0)}(0))\) |
| \(\psi_{\Omega}(\chi_{M}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(\psi_{\chi_{M}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(1)}(\psi_{\chi_{M}(1)}(\psi_{\chi_{M}(1)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(1)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(1)}(0)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)+1))}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(1)}(0)}(2))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(1)+1))}(\psi_{\chi_{\chi_{M}(1)}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(\chi_{\psi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(1)+1))}(\psi_{\chi_{\chi_{M}(1)}(0)}(1)+1))}(\psi_{\chi_{\chi_{M}(1)}(0)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(1)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(1)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(\psi_{\chi_{\chi_{M}(1)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(0))\) | \(\psi_{\Omega}(\chi_{\chi_{M}(1)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{M}(1)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\chi_{M}(1)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(1))\) | \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(2)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(2)}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{\psi_{\chi_{M}(2)}(0)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(2))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(1))\) \(\psi_{\Omega}(\chi_{\psi_{\chi_{M}(2)}(1)}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\psi_{\chi_{M}(2)}(1)}(0)}(0))\) |
| \(\psi_{\Omega}(\chi_{M}(2))\) | \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(\psi_{\chi_{M}(2)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M}(2)}(\psi_{\chi_{M}(2)}(\psi_{\chi_{M}(2)}(0))))\) |
| \(\psi_{\Omega}(\chi_{M}(\Omega))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\Omega}(0))\) \(\psi_{\Omega}(\chi_{M}(\psi_{\Omega}(\chi_{M}(\psi_{\Omega}(0)))\) \(\psi_{\Omega}(\chi_{M}(\psi_{\Omega}(\chi_{M}(\psi_{\Omega}(\chi_{M}(\psi_{\Omega}(0))))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(\Omega)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(\Omega)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(\Omega)}(0)}(\psi_{\chi_{\chi_{M}(\Omega)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(\Omega)}(0)}(\psi_{\chi_{\chi_{M}(\Omega)}(0)}(\psi_{\chi_{\chi_{M}(\Omega)}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{M}(I))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{I}(0))\) \(\psi_{\Omega}(\chi_{M}(\psi_{I}(\chi_{M}(\psi_{I}(0)))\) \(\psi_{\Omega}(\chi_{M}(\psi_{I}(\chi_{M}(\psi_{I}(\chi_{M}(\psi_{I}(0))))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(I)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(I)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(I)}(0)}(\psi_{\chi_{\chi_{M}(I)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(I)}(0)}(\psi_{\chi_{\chi_{M}(I)}(0)}(\psi_{\chi_{\chi_{M}(I)}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(0)}(0)))\) | \(\psi_{\Omega}(\chi_{M}(\chi_{0}(0))\) \(\psi_{\Omega}(\chi_{M}(\chi_{\chi_{0}(0)}(0)))\) \(\psi_{\Omega}(\chi_{M}(\chi_{\chi_{\chi_{0}(0)}(0)}(0)))\) |
| \(\psi_{\Omega}(\chi_{M}(\chi_{M}(0)))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(0)}(0))\) \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(0)}(\chi_{M}(\psi_{\chi_{M}(0)}(0)))\) \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(0)}(\chi_{M}(\psi_{\chi_{M}(0)}(\chi_{M}(\psi_{\chi_{M}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\chi_{M}(\chi_{M}(0))}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(\psi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(\psi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(\psi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M}(\chi_{M}(0)+1)}(0))\) | \(\psi_{\Omega}(\chi_{M}(\chi_{M}(0)))\) \(\psi_{\Omega}(\chi_{\chi_{M}(\chi_{M}(0))}(0))\) \(\psi_{\Omega}(\chi_{\chi_{\chi_{M}(\chi_{M}(0))}(0)}(0))\) |
| \(\psi_{\Omega}(\chi_{M}(\chi_{M}(1)))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(1)}(0)))\) \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(1)}(\chi_{M}(\psi_{\chi_{M}(1)}(0)))))\) \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M}(1)}(\chi_{M}(\psi_{\chi_{M}(1)}(\chi_{M}(\psi_{\chi_{M}(1)}(0)))))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+1}(0)}(0))\) | \(\psi_{\Omega}(\chi_M(0))\) \(\psi_{\Omega}(\chi_M(\chi_M(0)))\) \(\psi_{\Omega}(\chi_M(\chi_M(\chi_M(0))))\) |
| \(\psi_{\Omega}(\chi_{M+1}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+1}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+1}(0)}(\psi_{\chi_{M+1}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+1}(0)}(\psi_{\chi_{M+1}(0)}(\psi_{\chi_{M+1}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{M+2}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+2}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+2}(0)}(\psi_{\chi_{M+2}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+2}(0)}(\psi_{\chi_{M+2}(0)}(\psi_{\chi_{M+2}(0)}(0))))\) |
| \(\psi_{\Omega}(\chi_{M+\omega}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+\omega}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+\omega}(0)}(\psi_{\chi_{M+\omega}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+\omega}(0)}(\psi_{\chi_{M+\omega}(0)}(\psi_{\chi_{M+\omega}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+\Omega}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M+\psi_{\Omega}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{\Omega}(\chi_{M+\psi_{\Omega}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{\Omega}(\chi_{M+\psi_{\Omega}(\chi_{M+\psi_{\Omega}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\chi_{M+\Omega}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+\Omega}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+\Omega}(0)}(\psi_{\chi_{M+\Omega}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+\Omega}(0)}(\psi_{\chi_{M+\Omega}(0)}(\psi_{\chi_{M+\Omega}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+I}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M+\psi_{I}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{I}(\chi_{M+\psi_{I}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{I}(\chi_{M+\psi_{I}(\chi_{M+\psi_{I}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+I}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M+\psi_{I}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{I}(\chi_{M+\psi_{I}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{I}(\chi_{M+\psi_{I}(\chi_{M+\psi_{I}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\chi_{M+I}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+I}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+I}(0)}(\psi_{\chi_{M+I}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+I}(0)}(\psi_{\chi_{M+I}(0)}(\psi_{\chi_{M+I}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M+\varphi_{\psi_{I}(0)}(I+1)}(0))\) \(\psi_{\Omega}(\chi_{M+\varphi_{\psi_{I}(\chi_{M+\varphi_{\psi_{I}(0)}(I+1)}(0))}(I+1)}(0))\) \(\psi_{\Omega}(\chi_{M+\varphi_{\psi_{I}(\chi_{M+\varphi_{\psi_{I}(\chi_{M+\varphi_{\psi_{I}(0)}(I+1)}(0))}(I+1)}(0))}(I+1)}(0))\) |
| \(\psi_{\Omega}(\chi_{M+\varphi_{I}(1)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(\psi_{\chi_{M+\varphi_{I}(1)}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+\psi_{\chi_{M}(0)}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M+\chi_{0}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\chi_{\chi_{0}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\chi_{\chi_{\chi_{0}(0)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M+\psi_{\chi_{M}(0)}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{\chi_{M}(0)}(\chi_{M+\psi_{\chi_{M}(0)}(0)}(0))}(0))\) \(\psi_{\Omega}(\chi_{M+\psi_{\chi_{M}(0)}(\chi_{M+\psi_{\chi_{M}(0)}(\chi_{M+\psi_{\chi_{M}(0)}(0)}(0))}(0))}(0))\) |
| \(\psi_{\Omega}(\chi_{M+\chi_{M}(0)}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(\psi_{\chi_{M+\chi_{M}(0)}(0)}(0))))\) |
- Below \(\psi_{\Omega}(\psi_{\chi_{\varphi_{1}(M+1)}(0)}(0))\)
| ordinal \(\alpha\) | expansion \(\alpha[n]\) |
|---|---|
| \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(0))\) | \(\psi_{\Omega}(\chi_{M}(0))\) \(\psi_{\Omega}(\chi_{M+\chi_{M}(0)}(0))\) \(\psi_{\Omega}(\chi_{M+\chi_{M+\chi_{M}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(1))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M+M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(0)}(0)+1)}(\psi_{\chi_{M+M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(0)}(0)+1)}(\psi_{\chi_{M+M}(0)}(0)+1)}(\psi_{\chi_{M+M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(2))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M+M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(0)}(1)+1)}(\psi_{\chi_{M+M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(0)}(1)+1)}(\psi_{\chi_{M+M}(0)}(1)+1)}(\psi_{\chi_{M+M}(0)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{M+M}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(0)\) \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(\psi_{\chi_{M+M}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+M}(0)}(\psi_{\chi_{M+M}(0)}(\psi_{\chi_{M+M}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+M}(1)}(0))\) | \(\psi_{\Omega}(\chi_{M}(\chi_{M+M}(0)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M}(\chi_{M+M}(0)+1)}(\chi_{M+M}(0)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M+\chi_{M}(\chi_{M+M}(0)+1)}(\chi_{M+M}(0)+1)}(\chi_{M+M}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+M}(1)}(1))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M+M}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(1)}(0)+1)}(\psi_{\chi_{M+M}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(1)}(0)+1)}(\psi_{\chi_{M+M}(1)}(0)+1)}(\psi_{\chi_{M+M}(1)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{M+M}(1)}(2))\) | \(\psi_{\Omega}(\chi_{M}(\psi_{\chi_{M+M}(1)}(1)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(1)}(1)+1)}(\psi_{\chi_{M+M}(1)}(1)+1))\) \(\psi_{\Omega}(\chi_{M+\chi_{M+\chi_{M}(\psi_{\chi_{M+M}(1)}(1)+1)}(\psi_{\chi_{M+M}(1)}(1)+1)}(\psi_{\chi_{M+M}(1)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{M+M}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{M+M}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{M+M}(1)}(\psi_{\chi_{M+M}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{M+M}(1)}(\psi_{\chi_{M+M}(1)}(\psi_{\chi_{M+M}(1)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)}(0))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(0)}(0))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(2))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+M}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)}(\chi_{\varphi_{0}(M+1)+M}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\chi_{\varphi_{0}(M+1)+M}(0)+1)}(\chi_{\varphi_{0}(M+1)+M}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\chi_{\varphi_{0}(M+1)+M}(0)+1)}(\chi_{\varphi_{0}(M+1)+M}(0)+1)}(\chi_{\varphi_{0}(M+1)+M}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(2))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)+\chi_{\varphi_{0}(M+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1)+1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{\varphi_{0}(M+1)+M}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(\psi_{\chi_{\varphi_{0}(M+1)+M}(1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(0))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(0)}(0)}(0))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(2))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)}(0))))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)+1)}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)+1)}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)+1)}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)+1))\) |
| \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(2))\) | \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1)+1))\) \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))+\chi_{\varphi_{0}(\varphi_{0}(M+1))}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1)+1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(1)+1))\) |
| \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1))\) | \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(M+1))+M}(1)}(0))))\) |
| \(\psi_{\Omega}(\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0))\) | \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0)}(0))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0)}(0)))\) \(\psi_{\Omega}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0)}(\psi_{\chi_{\varphi_{0}(\varphi_{0}(\varphi_{0}(M+1)))}(0)}(0))))\) |
I note that the countable limit of Rathjen's weakly Mahlo \(\psi\) itself is \(\sup \{\psi_{\Omega}(\alpha) \mid \alpha < \Gamma_{M+1}\}\), which is much greater than \(\psi_{\Omega}(\psi_{\chi_{\varphi_{1}(M+1)}(0)}(0))\). On the other hand, the ordinal notation system \(T(M)\) associated to Rathjen's weakly Mahlo \(\psi\) satisfies \(T(M) \cap M = C_{\Omega}(\psi_{\chi_{\varphi_{1}(M+1)}(0)}(0)) \cap \psi_{\chi_{\varphi_{1}(M+1)}(0)}(0)\) by Theorem 6.3 in the first reference. It means that the recursive interpretation of the \(\in\)-relation on ordinals associated to Rathjen's weakly Mahlo \(\psi\) might not be applicable to countable ordinals above \(\psi_{\Omega}(\psi_{\chi_{\varphi_{1}(M+1)}(0)}(0))\). Namely, any "analysis" on Rathjen's weakly Mahlo \(\psi\) above \(\psi_{\Omega}(\psi_{\chi_{\varphi_{1}(M+1)}(0)}(0))\) is not based on an actual algorithm to determine fundamental sequences, and hence is not reproducible. Honestly, I do not even know whether the \(\in\)-relation is decidable or not in the whole system of countable ordinals describable by Rathjen's weakly Mahlo \(\psi\) with respect to the obvious extension of the coding given in Definition 6.1 in the first reference. Therefore it is quite reasonable to stop here.
Rathjen's OCF Based on a Weakly Compact Cardinal[]
References:
- M. Rathjen, Proof Theory of Reflection, Annals of Pure and Applied Logic, Volume 68, Issue 2 (1994), pp. 181--224.
I deal with Rathjen's weakly compact \(\Psi\), i.e. Rathjen's \(\Psi\) based on the least weakly compact cardinal \(K\) with a specific ordinal notation system introduced in the reference above.
I list characters in standard expressions with respect to Rathjen's weakly compact \(\Psi\).
| character | property | restriction |
|---|---|---|
| \(0\) | the constant given as the least ordinal | |
| \(K\) | the constant given as the least weakly compact cardinal | |
| \(+ \colon (\alpha,\beta) \mapsto \alpha + \beta\) | the associative \(2\)-ary function given as the addition | |
| \(\varphi \colon (\alpha,\beta) \mapsto \varphi_{\alpha}(\beta)\) | the \(2\)-ary function given as the Veblen function | |
| \(\Omega \colon \alpha \mapsto \Omega_{\alpha}\) | the \(1\)-ary function which assigns \(0\) to \(0\) and \(\aleph_{\alpha}\) to \(\alpha > 0\) | |
| \(\Xi \colon \xi \mapsto \Xi(\xi)\) | a \(1\)-ary function | |
| \(\Psi \colon (\xi,\pi,\alpha) \mapsto \Psi^{\xi}_{\pi}(\alpha)\) | a \(3\)-ary function | \(\xi \leq \alpha \land \pi \in \textrm{Reg} \cap K\) |
I list expansions of ordinals in Rathjen's weakly compact \(\Psi\).
WIP
Arai's OCF[]
References:
- T. Arai, A Simplified Ordinal Analysis of First-Order Reflection, preprint in arXiv.
I list expansions of ordinals in Arai's \(\psi_{\Omega_1}\) based on the least \(\Pi^1_{N-2}\)-indescribable cardinal \(\mathbb{N}\) for a fixed \(N \in \mathbb{N}\) greater than \(2\).
WIP