User blog comment:Edwin Shade/In Which I Try To Derive Values of Madore's OCF on My Own, and See How Far I Get/@comment-32213734-20171203100352/@comment-28606698-20171208150200

Countable limit ordinals (ordinals less than first uncountable ordinal) always has cofinality $$\omega$$. If cofinality of an ordinal $$\alpha$$ is equal to $$\omega$$ then its fundamental sequence has length omega and and $$\eta$$ for $$\alpha[\eta]$$ is always a natural number.

For assignation of fundamental sequences for this hierarchy we should allow fundamental sequences with length greater than $$\omega$$ and to define them as follows:

The fundamental sequence for an ordinal number $$\alpha$$ with cofinality $$\text{cof}(\alpha)=\beta$$ is a strictly increasing sequence $$(\alpha[\eta])_{\eta<\beta}$$ with length $$\beta$$ and with limit $$\alpha$$, where $$\alpha[\eta]$$ is the $$\eta$$-th element of this sequence.

If $$\alpha$$ is a successor ordinal then $$\text{cof}(\alpha)=1$$ and the fundamental sequence has only one element $$\alpha[0]=\alpha-1$$. If $$\alpha$$ is a limit ordinal then $$\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}$$.

Cofinality of any uncountable regular cardinal (as \Omega_1, \Omega_2,...) is equal to this cardinal. So $$\text{cof}(\Omega_1)=\Omega_1$$, $$\text{cof}(\Omega_2)=\Omega_2$$ and so on. If cofinality of an ordinal $$\alpha$$ is equal to an uncountable cardinal $$\Omega_{\beta+1}$$ then its fundamental sequence has length equal to $$\Omega_{\beta+1}$$ and  $$\eta$$ for $$\alpha[\eta]$$ is ane ordinal less than $$\Omega_{\beta+1}$$.

So for example

$$\psi_0(\Omega)[3]$$

Rule 6 If If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta)\in\{\Omega_{\mu+1}|\mu\geq\nu\}$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])$$ with $$\gamma[0]=\Omega_\mu$$ and $$\gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])$$

$$\text{cof}(\beta)=\Omega_{0+1}$$

$$\gamma[0]=\Omega_0=0$$

$$\gamma[1]=\psi_0(\Omega[\gamma[0]])=\psi[\Omega[0]]=\psi(0)=\varepsilon_0$$

$$\gamma[2]=\psi_0(\Omega[\gamma[1]])=\psi[\Omega[\varepsilon_0]]=\psi(\varepsilon)=\varepsilon_{\varepsilon_0}$$

$$\gamma[3]=\psi_0(\Omega[\gamma[2]])=\psi[\Omega[\varepsilon_{\varepsilon_0}]]=\psi(\varepsilon_{\varepsilon_0})=\varepsilon_{\varepsilon_{\varepsilon_0}}$$