User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-30004975-20171217042147/@comment-1605058-20171217092348

Psi, Denis, you are both kind of right: Psi is right that given the definition in this post, \(I_\omega\) is inaccessible and strictly larger than the limit of \(I_1,I_2,\dots\), however it seems that Denis "should" be right, at least according to Rathjen's original paper: he uses slightly different notation, but there \(I_1(\alpha)\) is the function we are interested in - it is the enumeration function of not the set of inaccessible cardinals, but of its closure, i.e. of the set inaccessible cardinals and their limits. Hence \(I_1(\omega)\) is the \(\omega\)th inaccessible or limit of inaccessibles - hence it's the limit of \(I_1(0),I_1(1),I_0(2),\dots\).

Going back to Deedlit's notation, \(\alpha\mapsto I_\alpha\) is a continuous function with this definition, which is very helpful since in particular it does make the limit of \(I,I_I,I_{I_I},\dots\) the first fixed point of this function. Let me also comment that if we were to use the definition ignoring the limits of inaccessibles, then the first fixed point of \(\alpha\mapsto I_\alpha\) would be the first 1-inaccessible cardinal, which probably doesn't help making the OCF strong.