And what exactly is the effective difference?
(\infty - \infty) is the same as ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) which is in the same camp as ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)). If you’re going to say that the former is meaningless, what allows you to say that the latter is not? Or vice versa, if you’re going to say the latter is equal to (0), what allows you to say that the former is meaningless?
Meaningless: that which has no meaning. When speaking of symbols (such as words) it refers to symbols that have no meaning assigned to them. By whom? Well, by someone, usually the one using those symbols. In most cases, it’s a way for people to avoid bothering to understand what the other side is trying to communicate, as in “Look, I can’t bother to understand what you’re trying to say, it’s too difficult and/or time-consuming for me, so I’m just gonna conclude that you’re not saying anything, that your statements have no meaning whatsoever, that they mean nothing, that they are meanignless”.
You can do arithmetic with (0.\dot9) but you can’t do arithmetic with (1 + 1 + 1 + \cdots)? Really? You can do arithmetic with convergent series but you can’t do arithmetic with divergent series? W-why? Is it perhaps because you’re REFUSING to do so? As in, you don’t wanna do it because it does not support your present conclusions?
Look closer at what you’re saying.
You’re saying that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0) makes PERFECT SENSE but that ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots) = 0) makes NO SENSE.
Are they infinite sums of non-zero terms or finite sums of non-zero terms? They can’t be both. Their RESULT can be a finite number, sure, just like how the result of (1 \times 1 \times 1 \times \cdots) is a finite number, but they are nonetheless expressions involving an infinite number of terms. The entire point is that WE’RE DOING ARITHMETIC WITH AN INFINITE NUMBER OF NON-ZERO TERMS. And if we can do it in some cases (such as with (0.9 + 0.09 + 0.009 + \cdots)) why can’t we do in other cases (such as with (1 + 1 + 1 + \cdots))?
You think that ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) is not well-defined and at the same think that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)) is well-defined?
How are complex numbers not a category?