User blog comment:Ecl1psed276/Question about standard notation/@comment-30754445-20180808072912/@comment-35870936-20180809145734

@PsiCubed2

Consider this: If I_w is defined to be the wth inaccessible, then why isn't W_w defined to be the wth regular cardinal? W_w is instead defined to be the limit of W_n for finite n, which doesn't match the way I_w is defined. The wth regular cardinal is actually W_(w+1).

And you seem to think that it will be too complicated to have a system where I_w = limit of I_n. I don't get that. You could define I_a as follows:

I_1 = the first inaccessible

I_(a+1) = the next inaccessible after I_a

I_a = limit of I_n for n<a, if a is a limit.

See? Not that hard.