Are we still doing this?
That’s right.
Remember that (\infty) has an infinite sum equal to it which is (1 + 1 + 1 + \cdots). This means that if you can’t do arithmetic with infinities (because they are qualities, as you say, and not quantities) then you can’t do arithmetic with infinite sums either. (Which would invalidate Wikipedia proofs.)
Except that (0.9 + 0.09 + 0.009 + \cdots) does not equal (\infty). It equals (0.\dot9), which you can easily do arithmetic with.
Your confusing the infinite sum with the result of the infinite sum. 1 + 1 + 1 + … is not just an infinite sum, it also equals (\infty). (0.9 + 0.09 + 0.009 + \cdots), though it is an finite sum, does not.
What I’m saying is that you can’t do arithmetic with (\infty), but you can do as much arithmetic as you want with an infinite number of terms.
I explained all this right after the snippet you quoted above.
Unless, for some strange reason, you don’t think that infinity can be represented as an infinite sum. In that case, I could simply stop talking about infinities and start talking about. . . infinite sums. There would be no difference with regard to my argument.
Do you agree that infinite sums come in different sizes?
Sure, (0.9 + 0.09 + 0.009 + \cdots) < 1 + 1 + 1 + …
Do you agree that ( (1 + 1 + 1 + \cdots) + 1 > 1 + 1 + 1 + \cdots)?
I don’t know whether to agree or disagree. I don’t know how to make sense of that.
But if you don’t, this means that:
((1 + 1 + 1 + \cdots) + 1 = 1 + 1 + 1 + \cdots)
Do you agree that (1 + 1 + 1 + \cdots = 1 + 1 + 1 + \cdots)?
Remember that one of the Wikipedia proofs claims that (0.9 + 0.09 + 0.009 + \cdots = 0.9 + 0.09 + 0.009 + \cdots). ← Did they really have to do that.
If you don’t agree with this, you also don’t agree with Wikipedia proofs.
If you do agree, let’s subtract (1 + 1 + 1 + \cdots) from the above equation.
What do we get?
We get (1 = 0).
Do you agree with the conclusion?
The problem, Magnus, is that (1 + 1 + 1 + … ) + 1 doesn’t mean anything different than 1 + 1 + 1 + … So yeah, they’re equal.
If you just had a finite sum of 1’s, like this: (1 + 1 + 1) and you added another 1 to it, you’d get this: (1 + 1 + 1) + 1. ← But in that case, why keep the brackets? You can just drop them: 1 + 1 + 1 + 1. ← Now if that was an infinite set of 1’s, you’d write it: 1 + 1 + 1 + … But isn’t that the same as the result above? Just because you have an infinite set of 1s in the brackets doesn’t mean there’s any more reason to keep the brackets. You can drop them for the same reason you can drop them in the finite case (because brackets aren’t needed in sums).
We get (1 = 0).
And that, my friend, is why you don’t do arithemtic with infinity.