User blog comment:Elberos/Help?/@comment-35470197-20181219153612

Even if we assume that there is an inaccessible cardinal \(I\) as an additional axiom, it does not ensures that there are other inaccessible cardinals. But we can consider the case where there are sufficiently many inaccessible cardinals as an additional axiom. In that context, you can use \(I_1\), \(I_{\omega}\), and fixed points.

However, the convention is a little ambiguous. Some authors use \(I_{\omega}\) as the \(\omega\)-th inaccessible, while some authors use \(I_{\omega}\) as the limit of \(I_0,I_1,I_2,\ldots\). (They are different from each other.)