Is 1 = 0.999... ? Really?

“They” have a proof that if (0\times{n}=0), “n” can be any number you want since all numbers multiplied by zero equal zero. Therefore dividing both sides by 0 to get (\frac0{0}) it’s not (\infty) that this equals, but “undefined”.
Except “they” is you too. Maths is objective so it’s your proof just as much as it is anyone else’s.

And as I just said to Magnus, this doesn’t mean temporarily undefined. It means it’ll always be undefinable.

Indeed infinity does not exist. InfA doesn’t either, because it contains within it the “…” that implies “infinitely more times”, which as we’ve just learned is an undefined number of times, which could only lead to an undefined result if it could ever reach any result at all. It has a divergent limit that cannot be definitely specified one way or another - and calling it “infA” doesn’t solve this.

I think that Magnus is exactly right about that and that your claim is exactly wrong.

After we debunk orders of infinity, I’m coming back after silhouette.

That’s nice.

I mean, I just disproved it (again) in the post right at the end of the previous page:

But as long as you just simply “think” that’s exactly wrong, does that make you exactly right?

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.

Magnus,

Your graph is nice, but it’s actually not different from simply stating 0.333… * 3 = 0.999…

I want to stay on orders of infinity with you before really laying into silhouette

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).

Neither of those claims make any sense to me. So you haven’t proven anything to me. Why would your first claim be true?

That one doesn’t make any sense either.

I’ll argue Silhouette for silhouette!

The reason Silhouette sees no difference between .000…1 and zero is because the 1 in 0.000…1 is never arrived at. It’s ALWAYS zero!

Except that ∞, by definition is NOT a number (“n”).

So the “proof” using n/o is invalid.

So you are saying that you never get to the end??

And that would mean that there is always a “9”, never a “0” in 0.999…, wouldn’t it.

Absolutely. But that’s not my focus right now, I’m debating Magnus on orders of infinity, that’s my focus.

I’ll get to Silhouette later.

So if it never has a zero at the end, because it doesn’t have an end, then it cannot be equal to 1.000…, which has zero throughout.

Yes. Correct. Actually, it can be best seen as a problem with operators … some think that problem makes an equality, and some (me) don’t.

I wasn’t using any operators. I was talking about the static situation of an already infinite string.

You don’t understand what you’re saying!

If 1/3 = 0.333…

And 0.333… *3 = 0.999… not 1/3!

That’s an operator problem!

That might be so but I wasn’t multiplying anything. People who tried that as a proof have that issue, not me.

Do you agree that 1 whole number (1) divided by 3 equal 0.333… ?

Do you agree that 0.333… times 3 equals 0.999… ?

If all that is true, then operators don’t work. At least for base-1.

No I don’t. “0.333…” is not a quantized number, a “quantity”. But 1/3 is a quantity.

I agree that math operators do not work on non-quantity items (anything ending with “…”).

So, obsrvr,

So, This is an interesting theory of numbers!

9/3 = 3
10/3 = 3

I’m not seeing where you are getting that.
Why would 10/3 = 3?