User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-30004975-20171217042147/@comment-28606698-20171217083506

I still think that $$I_\omega$$ is not regular and not inaccessible, same way as, for example, $$\aleph_\omega$$ is not \omegath regular cardinal since $$\text{cof}(\aleph_\omega)=\omega$$ and only $$\aleph_{\omega+1}$$ is $$\omega$$th regular cardinal. Same way and I_\omega. The cofinality of $$I_\omega$$ is the least cardinality of a set of ordinals less than $$I_\omega$$ that sums to  $$I-\omega: I_1+I_2+...$$