User blog comment:Ecl1psed276/Question about standard notation/@comment-35470197-20180808055930

There are (essentially) two common definitions of \(I\). The one is the enumeration of weakly inaccessibles, and the other one is the enumeration of the closures of the classes of weakly inaccessible cardinals. So both conventions are traditionally accepted.

To be more precise, there are several notations using \(I\) depending on the definition of weak \(0\)-inaccessibility. There are also two common definitions of it. One is the regularity of an uncountable cardinal, and the other one is the weak inaccessibility.

On the other hand, I only know the usual convention of \(\Omega_{\alpha}\). If none uses another convention, it is not so reasonable to use one, unless you specify it.

One of the main reason why we need \(I\) in the context where \(I\) enumerates weakly inaccessible cardinals would be the merit that we do not have to mind whether a given value of \(I\) is weakly inaccessible or not. When we define an OCF, we sometimes need to use such a function. (Also, the function \(I\) in this context is commonly denoted by \(\chi\).)

By the way, \(I_0(\beta)\) is just the \(\beta\)-th regular uncountable cardinal in the context where weak \(0\)-inaccessibility is defined as the regularity of an uncountable cardinal and \(I\) is defined as the enumeration of weakly inaccessible cardinals. So you do not have to introduce a new convention for that purpose.

Although it is irrelevant to the question itself, it might be good to mention to the fact that \(I_0(\beta)\) is just the \(\beta\)-th uncountable cardinal \(\Omega_{1+\beta}\) in the context where weak \(0\)-inaccessibility is defined as the regularity of an uncountable cardinal and \(I\) is defined as the enumeration of the closures of the classess of weakly inaccessible cardinals.