User blog comment:Ecl1psed276/Question about standard notation/@comment-30754445-20180808072912/@comment-30754445-20180809152851

"If I_w is defined to be the wth inaccessible, then why isn't W_w defined to be the wth regular cardinal?"

Because the Ω's don't count "regular cardinals". Ωn is simply the n-th cardinal.

"W_w is instead defined to be the limit of W_n for finite n"

That's not true.

Ωω is defined as the ω-th cardinal. This is how subscripts are usually used: They represent a simple count of things with a certain property.

Of-course Ωω also happens to be equal to the limit of Ωn for finite n. But that's not how it is defined. It's just a result, due the fact that "the x-th cardinal" happens to be a continuous function.

"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."'''

You're right that at this level, it's not that hard.

But once you start dealing with multiple kinds on inaccessibles and Mahlos and what's not, it becomes a nightmare to manage.

And even at the level we're talking about, I fail to see how your definition is any easier to learn. True, it's not a big deal to add those couple of lines, but it's still an additional couple of lines.

I also don't understand what we gain by this. If we did things your way, then people would just ask "why isn't I_w the w-th inaccessible cardinal?", wouldn't they?

The real problem is that our intuition would like I_w to be two different things which are not compatible with one another. So no matter how we choose, I_w is bound to be a source of confusion.