It is an edge case, and so when it’s put this way it seems very ad hoc. But this is kind of an ad hoc way of putting things.
I have a related question: is (0.\dot9) rational or irrational? If it’s rational, what’s the ratio?
I can think of a few. The biggest is that “an infinite number” is different from “the infinite number ‘(\infty)’”: the former implies multiple infinities.
Another is that numbers have properties and operations, and I don’t think any standard operations have an obvious meaning on infinite numbers, and I’m not sure what properties apply or what we could use to infer those properties.
Why? Why not the same as ((2+2+2+…) - (3+3+3+…))? Especially considering that you probably don’t think ((1+1+1+⋯) = (2+2+2+…)), why should we pick one or the other as a stand-in for (\infty)?
So I guess my answer to your question is that I don’t think (\infty - \infty) means the same thing as ((1+1+1+⋯)-(1+1+1+⋯)).
Though I later acknowledge that “meaningless” is perhaps too strong, here I mean it as saying that just because you can string some symbols together doesn’t mean that they express a coherent concept. A “square circle” is meaningless in the sense that, even though the words that compose the phrase are perfectly meaningful, the phrase doesn’t point to a coherent concept.
2 points:
- (.9) can be expressed as a convergent series, but it isn’t a convergent series.
- A divergent series is undefined in the limit. So we can do arithmetic with the limit of a convergent series and not with the limit of a divergent series, because we can do arithmetic with things that are defined and not with things that are undefined.
They are the finite limits of infinite sums. There is no tension at all there. And again, they can be expressed infinite sums, but they are numbers. I’m about 95% sure that every number can be expressed as multiple infinite sums.
Sorry, typo, should have said “‘complex numbers’ is a category”. I was replying to your rejection of identifying what kind of number (L) is, and you offered the argument that (i) does not fit into any pre-complex-numbers categories. But we agree that complex numbers are a category of numbers. So (L) too should fit into some category. So you should be able to say that it’s in some category, or it’s some new number like (i) was and it needs a brand new category of numbers, and you have some rigorous definitions and properties of numbers like (L) (or even some vague, hand-wavey, there-should-be-something-(L)-shaped-in-roughly-this-vicinity type answer).