Where you're misleading is in allowing this subjective display positionings of the blues and the yellow to affect the objective position of the purples, which do not overlap or have anything to do with the operation of summing the blues to get the yellow. Likewise for the reds, you could subjectively display them anywhere you like and they'd still objectively sum to the purples. It's only subjectively convenient to put them next to the blues so the visual grouping of the greens is clearer, but even if you displayed them anywhere else that you liked: the first element of the greens would still be the first element of the blues, the second element of the greens would still be the first element of the reds and so on.

If you're saying that my calculations are influenced by my image, you're wrong. The image is nothing but a reflection of my calculations. It's supposed to show how I'm doing my calculations (and it is implied that the way I'm doing them is the way they should be done.)

The first element of the purple rectangle is \(0.9\) and it's constructed from the second element of the green rectangle which is \(0.09\). It is

not constructed from the first element of the green rectangle. That's why the purple rectangle does not start at the same place as the green rectangle.

There is no "product itself" of a divergent infinite product - you never get there.

There is. The fact that the product never ends does not mean it does not exist. (Indeed, I can use Gib's argument against you: infinite sums have no temporal dimension, so they can be considered complete.)

What's true is that there may be no finite number equal to an infinite product. In the case of \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots\), there is no finite number equal to it. It has a limit, which is \(0\), but that's not the same thing as its result (its result being

greater than its limit.)

The only thing you can do is identify some limit that it's tending towards because of the very fact that it never gets there.

Not really.

\(1+\infty\) is almost literally "more boundless" (by the finite quantity of "one").

Something can't be more boundless than boundless, so it's boundless whatever finite thing you do to it along its boundlessness.

Quantities are either finite or they are infinite. There are no degrees. That's where we agree. Where we disagree is that \(\infty + 1\) means "more infinite". It does not. What it means is "larger infinite quantity".

That's why adding one results in boundlessness both before and after you do it

The truth value of the statement \(\infty + 1 = \infty\) depends on its meaning.

If what we mean is "Some infinite quantity X + 1 = Some infinite quantity that is not necessarily equal to X" then the statement is true.

If what we mean is "Some infinite quantity X + 1 = The same infinite quantity X" then it is false.

It's literally impossible to test whether you added, took away or did whatever bounded thing to a boundless length after you've done it and are therefore able to make an equation about it. You can only validly comment on what you did with the finite quantity of 1 apple before it was absorbed into the boundless non-finite mass, you cannot validly comment on that resulting infinity that stays infinite in the same and only way that infinity can be infinite. So with no change in the result, there is therefore no valid equation or statement to make about the result as changed.

It seems to me that like Gib you're not able to distinguish between conceptual and empirical matters.

Gib keeps asking questions such as "How can we empirically determine (specifically, through direct observation) whether any two infinitely long physical objects are equal in length or not?"

The question is irrelevant and it is so precisely because it's empirical and not conceptual.

We're talking about concepts here, and to concepts we should stick. A conceptual matter cannot be resolved empirically.

Your error is to say that since 1=0 is a contradiction, we ought to be able to treat infinites alongside finites as though they were compatible.

That's not what I'm saying.

The contradiction that you speak of arises as a result of not understanding the implications of the concept of infinity that is being employed.

I've previously said that \(\infty + 1 = \infty\) is true. There is no doubt about it. But that's only the case if the symbol \(\infty\) means "some infinite quantity, not necessarily the same as the one represented by the same symbol elsewhere". In such a case, you can't subtract \(\infty\) from both sides and get \(1 = 0\). This is because \(\infty - \infty\) does not equal \(0\) given that what that expression means is "Take some infinity quantity and subtract from it some other infinite quantity that is not necessarily equal to it". If the two symbols do not necessarily represent one and the same quantity, then the difference between them is not necessarily \(0\).

But that's PRECISELY what mathematicians do when they try to prove that \(0.\dot9 = 1\). There is literally no difference between people proving that \(0 = 1\) by subtracting \(\infty\) from both sides of the obviously true equation \(\infty + 1 = \infty\) and various Wikipedia proofs that \(0.\dot9 = 1\) except that it's much easier to see that the former conclusion is nonsense and that the proof must be invalid.

Sure, if saying what something "isn't" gives it a meaning about what it "is".

You don't seem to understand the extent to which "ends" apply when it comes to definition.

Full definitions that include what something "is" as well as what it "isn't" are separating what it "is" from what it "isn't" by a bound (i.e. an "end") that's as clear as possible.

To define some symbol S is to verbally (or non-verbally) describe the meaning of that symbol S. The usual aim is to communicate to others what meaning is assigned to the symbol.

In other words, the meaning of a symbol

precedes its verbal (or non-verbal) representation (which is what definitions are.) They are superficial things, very much in the Freudian "tip of the iceberg" sense.

The meaning of a word does not have to be described in order for it to have a meaning. This means the word "infinity" is a meaningful word so as long there is some kind of meaning assigned to it regardless of what kind and how many descriptions of its meaning exist.

You can describe the meaning of words any way you want, so as long your descriptions do the job they are supposed to do (which is to help others understand what your words mean.) If you can define words using "is not" statements and get the job done, then your definitions are good.

The reason that expressions that include e.g. division by zero are undefined is because you can't clearly bound what it "is" from what it "isn't", since in this case any answer is no more valid than any other:

\(\frac{1}{0}\) has no valid answer (not a single one) since there is no number that gives you \(1\) when you multiply it by \(0\).

As above, limits are at the core of infinite series.

Infinite series and limits is what they taught you at school (and thus what you're familiar with) but they are not relevant to the subject at hand.

See how you're only going for one side of the "undefined" and concluding that the other side therefore doesn't exist?

\(0.\dot01\) never gets to \(0\) also means no gap can ever come into existence between \(0.\dot01\) and \(0\).

That's not true. It simply means the gap cannot be represented using one of those numbers you can understand.

\(0.\dot01\) must attain \(0\) in order for there to be no gap.