Carleas wrote: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 \(\infty\)?

You can pick any.

So I guess my answer to your question is that I don't think \(\infty - \infty\) means the same thing as \((1+1+1+...)-(1+1+1+...)\).

So you don't think that \(1 + 1 + 1 + \cdots = \infty\)?

Obviously, you refuse to understand the meaning that your interlocutors assign to the symbol \(\infty\) so maybe I should just stop using it, instead substituting it with an infinite series.

What does \((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)\) equal to? Is it \(0\) or is it \(1 + 1 + 1 + \cdots\)?

If you say it's \(0\) then the second part of

this post is irrelevant to you and so there is no need for you to respond to it.

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.

"Square circle" is a logical contradiction. Are you saying that \(\infty - \infty = \infty\) is a logical contradiction? I don't see why. Remember that I am not the one claiming that \(\infty - \infty = \infty\). On this forum, that would be Phyllo. Outside of the forum, that would be mathematicians.

\(0.\dot9\) can be expressed as a convergent series, but it isn't a convergent series.

That's like saying \(2 + 2\) can be expressed as \(4\) but it's not \(4\). In a way, it's true. They are two different expressions. They aren't the same expression. But also, they are expressions representing the same thing. They are two different expressions representing the same idea. In that sense, \(0.\dot9\) and \(0.9 + 0.09 + 0.009 + \cdots\) are two different expressions representing the same idea.

2) 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.

The Wikipedia proof I constantly quote does not do arithmetic with the limit of convergent series. It does arithmetic with convergent series.

They start with:

\(x = 0.999\dotso\)

Then they multiply both sides by \(10\).

This gives them:

\(10x = 10 \times 0.999\dotso\)

Then they substitute \(0.999\dotso\) with \(0.9 + 0.09 + 0.009 + \dotso\) telling us that the number of terms in the sum is \(\infty\).

This leads to:

\(10x = 10 \times (0.9 + 0.09 + 0.009 + \dotso)\)

How do they calculate the result of the right side of the expression?

Certainly not by asking "What's the limit of \(0.9 + 0.09 + 0.009 + \dotso\)?"

What they do is they multiply every term by \(10\), like so:

\(10x = 10 \times 0.9 + 10 \times 0.09 + 10 \times 0.009 + \dotso\)

This leads to:

\(10x = 9 + 0.9 + 0.09 + 0.009 + \dotso\)

Note that the number of terms in the resulting sum is the same as before: \(\infty\).

Then we "split off" the integer part:

\(10x = 9 + 0.999\dotso\)

The number of terms in the resulting \(0.999\dots\) is \(\infty - 1\) since the resulting sum is produced by removing one term from the previous sum. \(\infty - 1\), they tell us, equals to \(\infty\), which makes the number of terms the same as before. However, saying that \(\infty - 1 = \infty\) means that \(\infty - \infty \neq 0\) making \(\infty\) a non-specific number.

Thus, we can substitute the resulting \(0.999\dotso\) with \(x\), like so:

\(10x = 9 + x\)

Then we subtract \(x\) from both sides:

\(10x - x = 9 + x - x\)

And this is where the problem lies. We don't get what they tell us we get.

We don't get \(9x = 9\) because \(x - x\) is not \(0\) since \(x\) represents a non-specific number.

AN INFINITE NUMBER OF NON-ZERO TERMS MINUS AN INFINITE NUMBER OF NON-ZERO TERMS IS NOT ZERO NON-ZERO TERMS IF WE ALREADY ACCEPTED THAT INFINITY MINUS INFINITY IS NOT ZERO.

THAT'S THE ENTIRE POINT.