User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-30004975-20171217042147/@comment-28606698-20171217125518

Yes, now I see Deedlit define $$I_\alpha:=$$the $$\alpha$$th inaccessible cardinal and $$I(\alpha, \beta):=$$the $$\beta$$th $$\alpha$$-inaccessible cardinal

We can define as follows:

$$I_\alpha:=\text{min}\{\gamma|\gamma \in R \wedge\aleph_\gamma=\gamma\wedge\forall \beta<\alpha:\gamma>I_\beta\}$$

$$I(\alpha, \beta):=\text{min}\{\gamma|\gamma \in R \wedge(\forall \delta<\beta:\gamma>I(\alpha,\delta))\wedge(\forall \delta<\alpha:I(\delta, \gamma)=\gamma)\}$$

where $$R$$ is the set of all uncountable regular cardinals $$R=\{\alpha|\alpha>\omega\wedge\text{cof}(\alpha)=\alpha\} $$

in this case $$I_\omega$$ is regular and inaccessible.

We also can define as follows:

$$I(\alpha, 0):=\text{min}\{\gamma|\gamma \in R\wedge\forall \delta<\alpha:I(\delta, \gamma)=\gamma\}$$

$$I(\alpha, \beta+1):=\text{min}\{\gamma|\gamma \in R \wedge\gamma>I(\alpha,\beta)\wedge\forall \delta<\alpha:I(\delta, \gamma)=\gamma\}$$

$$I(\alpha, \beta):=\text{sup}\{I(\alpha,\gamma)|\gamma<\beta\}$$

In this case $$I(1, \omega)=\text{sup}\{I(1,n)|n<\omega\}$$ or in other words $$I_\omega=\text{sup}\{I_n|n<\omega\}$$

Anyway the first 1-inaccessible cardinal $$I(1,0)$$ is not the first fixed point of $$\alpha\mapsto\aleph_\alpha$$ since the first aleph-fixed point is not a regular cardinal, its cofinality is equal to $$\omega$$, that is just $$\psi_{I(1,0)}(0)$$

and the first 2-inaccessible cardinal $$I(2,0)$$ is not the first fixed point of $$\alpha\mapsto I(1,\alpha)$$ or in other words $$\alpha\mapsto I_\alpha$$ since the first I-fixed point also is not a regular cardinal, that is just $$\psi_{I(2,0)}(0)$$