User blog comment:Ecl1psed276/Question about standard notation/@comment-30754445-20180808072912/@comment-30754445-20180808203109

"not only does it make things more confusing for learners, but it's also inconsistent and makes the notation uglier"

How is it inconsistent?

And I certainly don't see how it is "more confusing for learners". If I2 is the second inaccessible cardinal, why would you expect the symbol "Iω" to be anything but the ω-th inaccessible cardinal? What could be simpler than saying "Ix is the x-th inaccessible"?

If we do it the other way, then we need to artificially insert all the missing limits into their proper places in the sequence. These limits are not inaccessible themselves, so plucking in a new "Iω" between In and the ω-th inaccessible (which has now moved to position ω+1 in the sequence) is anything but natural. And remember, we need to do this every time a limit is "missing". This gets complicated fairly quickly, and if you are not careful, it will result in an awful mess.

(It can be done properly using what P進大好きbot called closure and keeping careful count of which I's are indeed inaccessible and which I's are the limits. But I'll hardly call the resulting structure "intuitive" or "easy to learn". That single nice property you desire, comes at a very high price.)

BTW P進大好きbot is right, when he says that in general there's no reason to expect the ω-th term of a sequence to be the limit of nth term. Teaching people to expect this is counter-productive in my opinion. Not all functions are continuous.