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. 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.)
Not really.
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”.
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 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.
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.
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.
(\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).
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.
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.
I suppose what you want to say is: no gap between numbers = no difference between numbers = equal numbers. If that’s what you’re trying to say, then I agree.
That’s not the point of our disagreement. What we’re disputing is your claim that there is no gap between (0.\dot01) and (0).
This is the problematic statement. It’s a non-sequitir.
A non-existent gap would be a gap that has zero size. The size of an infinitely small gap is greater than zero – by definition. No partial product of (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots) is equal to (0). Note that this is different from (0 \times 0 \times 0 \times \cdots) where every partial product is equal to (0). Both are infinite, never-ending, products but one evaluates to a finite number that is (0) (is equal to it) whereas the other doesn’t evaluate to any finite number and is provably greater than (0).
If the answer is yes, subtract (1 + 1 + 1 + \cdots) from both sides.
What do we get?
(1 = 0)
But if the answer is no, it appears to me that it follows that one of the two sides of the expression is greater than or less than the other – and that means that infinities come in different sizes.
Assuming that I’m wrong, can you help me understand what I’m doing wrong?