User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-30004975-20171217042147/@comment-28606698-20171217074301

An ordinal is inaccessible cardinal if it is both regular cardinal and limit cardinal. So I don't think that $$I_\omega$$ is an inaccessible cardinal since it is not regular cardinal. Regular cardinal has cofinality equal to itself meanwhile the cofinality of $$I_\omega$$ is equal to $$\omega$$. So $$I_\omega$$ is $$\text{sup}\{I_\alpha|\alpha<\omega\}$$