User blog comment:Ecl1psed276/Question about standard notation/@comment-30754445-20180808072912

The inacessible hierarchy are defined in the same way the higher levels (Mahlos, etc.) are defined. For example, Mω is not the limit of Mn. The reason for this, as P進大好きbot said, is that otherwise it becomes a real headache to keep track of which I's are actually inaccesibles and which M's are actually Mahlos. And keeping track of these things is critical if we want our OCF's to work properly.

You can do it the other way, but the overhead becomes more and more cumbersome as we progress. So really, if you want an OCF to be as intuitive and as easy-to-grasp (not to mention easy-to-define) as possible, it is better to define Iω as the ω-th inaccessible cardinal rather than some limit.

IOW it is the Ω's that are the exception. And the reason for this, is that we also need a simple notation to notate "the n-th cardinal" regardless of regularity. You're right that for the sake of creating OCF's, this definition of the Ω's makes things confusing. What we really need here, is a different symbol that would denote "the n-th regular cardinal".

And as P進大好きbot stated, this can be done naturally by extending Deedlit's concept of n-inaccessible cardinals downwards one level. Interestingly, this resembles the way the C-function works. Recall that C(x)=Ωx and C(1,x)=the x-th inaccessible or limit of inaccessibles. We could easily define a similar C-function that skips the limits, so that C(x) would only count regular cardinals, and C(1,x) only counts the inaccessibles.

While this sounds needlessly confusing in the short run, it is actually more intuitive in the long run. And by having two different symbols (Ω and C) we can actually enjoy the benifit of both worlds.