Is 1 = 0.999... ? Really?

Because that’s how any number is abstracted in the first place - a build up. (Step 1)

Then we use inferential logic for the value and say, “hey, this never ends” (step 2)

People (rightfully or wrongly) bounce back between the two ways.

If you’re going to articulate a disproof of a “proof” of an infinite sequence, you need to go back to step 1, there’s no other way to do it.

That’s exactly what I did with my argument.

It doesn’t mean that I don’t think step 2 is invalid, step 1 just needs to be revisited/redefined

There’s quantity, and there’s representation.

Representations are constructions that you have to build up before the construction “equals” a quantity (that was conceptually “already there” and didn’t need building).

Building the representation of (0.\dot9) needs endlessness, so writing out the representative construction never ends (or “ends” with indefinite continuation implied) with the quantity that it does equal, which is 1.
Any “gap” is at most hinted at by the representation, not the quantity that it’s building up to, but even then it is only hinted at to the uninitiated who impatiently second guess what the representation would look like if it were to be completed at any given finite point(s) along the way - and they thus conclude that it’ll never get there and there will always be a gap.

You don’t go (\frac{circumference}{diameter}\neq3.141592…\neq\frac4{1}-\frac4{3}+\frac4{5}-\frac4{7}+\frac4{9}-\frac4{11}+…\neq4\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}) because there’s always got to be some “gap” at any conceptual “end” point to the representation implied by that contradiction in terms, that prevents it from ever equalling (\pi). No, they all equal the quantity “pi” because of the fact the construction contains some “incomplete” indetermination to it, not in spite of it.

A fraction with finite terms (C\div{d}) isn’t resolved, the explicit decimal places format isn’t finished, the infinite expansion of explicit fractions doesn’t terminate, the infinite series never gets to that “infinity” - the representation of (\pi) just pretends the irrational and transcendental quantity of pi is finite to make it easier and more compact to accept that all these incomplete constructions really do equal (\pi) in exactly the same way as (0.\dot9) really does equal (1).

Yet somehow, non-mathematicians will generally all accept any of these notations of (\pi) but not the notation of (1) as (0.\dot9). Forgive the pun, but this is entirely irrational.

I’m only able to make sense of some of what you say because the intelligence tests on which I score highly are mostly testing how well someone can extract patterns and sense from increasingly obscure sets of information. When Magnus says to me “It seems like you’re one of those rare people who can understand Ecmandu”, he seems to think that even though my ability to understand even extends as far as it does with you, I don’t understand his extremely simple explanations - the poor guy’s understanding is so consistently backwards.

I don’t have “man up” to address any argument, whether it’s one I support or not - I’ll address it fairly and rationally without partiality either way. I’d be quite happy to analyse any new arguments in this same way, but unfortunately I can’t quite get to the bottom of what you’re building here. Sometimes I feel like I’ve grabbed hold of something, but it goes away when I look into it a bit further.

In the seeming absence of some of your workings, and apparent jumps in your “notes”, I think maybe it boils down to some non-standard terminology that you’re using, like “convergence theory” (I don’t understand the exact process of getting the progressively different results that you’re presenting) or terminology that you’re using in a non-standard way such as the exact distinction “carrying left to right” and “carrying right to left”. Maybe an understanding of them will help me piece together these seemingly bizarre equations of yours.

Silhouette,

I like you, but damn are you arrogant sometimes! By saying this; I mean, man, can you shoot yourself in the foot sometimes!

“Convergence theory” is the idea that you can bound infinities to make them quantities …

For example:

0.999…

Is not a quantity.

1

Is a quantity.

Convergence theory states that an infinite sum (non convergent) equals a whole number (convergent)

What they lack is ability (or at the very least, incentive) to abandon logic when dealing with this subject.

The subject isn’t so complicated that it requires mathematical expertize. You don’t need mathematical expertize to understand that “2 + 2 = 4”. A bunch of mathematicians can’t declare that “2 + 2 = 5” and claim that the dissenters are wrong because they are merely amateurs.

The claim is that the two symbols ((0.\dot9) and (1)) do not represent the same quantity. A number of arguments have been put forward. You’re supposed to point out their flaws. The quoted doesn’t do that.

It’s important to note that it’s not necessary to show why Wikipedia proofs are wrong. There’s a much simpler proof that shows that they must be wrong – whatever their real flaws are.

I’m examining Wikipedia proofs for fun, not out of necessity. I might, in fact, be wrong about why Wikipedia proofs are flawed, but such a scenario wouldn’t render the main proof (which is a very simple proof) invalid.

This isn’t to say that I’m not sure about my claims regarding Wikipedia proofs. I’m pretty confident about them.

Thankfully, not everything is complicated in life, so not everything requires extensive proofs.

Again, the above isn’t pointing out flaws within an argument.

That’s correct.

((9 + \underline{0.\dot9}) \div 10 \neq \underline{0.\dot9})

The two underlined numbers don’t have the same number of 9’s.

In other words, the two infinite sums don’t have the same infinite number of non-zero terms.

Therefore, they aren’t the same number.

That’s my claim. It’s up to other people (which includes you) to point out flaws within my argument if they wish to do so.

That’s precisely the point of our disagreement. The (0.9) in (s) is paired with (0.09) in (s \div 10).

Let’s take (9.\dot9) and divide it by (10) and see what we get:

$$
9.0 \div 10 = 0.9\
0.9 \div 10 = 0.09\
0.09 \div 10 = 0.009\
\cdots
$$

I don’t wonder, I know. You suffer from low tolerance threshold. Not something to be proud of.

Considering that you talk about it publicly, yes.

If you can’t accept that Hilbert’s Hotel is nonsense, there is, I am afraid, no hope for you.

Notice that it’s not merely counter-intuitively full and not full. It does not merely go against intuition. It goes against logic.

Intuition is neither right nor wrong on its own. It’s that thing that is sometimes wrong and sometimes right. You can never know. So if a claim is counter-intuitive, it does not necessarily mean that it’s right or that it’s wrong. But if it goes against logic, then we have a problem, Sir.

A hotel that is both full and not full. Not a logical contradiction at all.

Now you’re conflating different concepts. Infinity/endless is not the same as indeterminate/undefinable.

That’s not true.

If you can’t determine whether there is a difference between two numbers, it does not follow that there is no difference between them.

Sorry to disappoint you.

Here’s a couple of super simple arguments for you to address. And when I mean “address”, I mean “point out their flaws”.

The argument that “Infinite quantity A - 1 < Infinite quantity A”.
viewtopic.php?p=2754790#p2754790

The argument that Hilbert’s Hotel is nonsense:
viewtopic.php?p=2754844#p2754844

Let me see you (or anyone else) point out their flaws.

Who ever said that the lines have to be added end to end? That was the point of having apples, oranges, and pears, each an infinite set.

And Ecmandu, there is a serious difference between “the concept orange” and “the concept of AN orange”.

Equally, there is a significant difference between “AN infinite line” (a finite quantity of a thing) and “the infinity of a line” (a quality of a single thing not being finite).

Explain this to me. What are your step 1 and step 2?

Step 1: build up a sequence of numbers in your mind (a decimal expansion).

Step 2: realize that the build up never ends (a repeating pattern, an irrational number).

This is how we come to realize that certain sequences are infinite, but we don’t have to do this every time. We remember. We say, “suppose you have (\pi),” which we know has an infinite decimal expansion because we’ve proven it already (or someone has).

Besides, we can also propose an infinitely expanding decimal, like 0.999… We say that 0.999… has an infinite number of 9s because we define it that way (and notate it with the …). We don’t need to build it up in our minds to confirm this, we propose that it just is infinite.

Even then, we get around that by inventing a representation of the representation–the dot in (0.\dot9)–which in turns just ends up being a representation of the quantity.

I’m not going to disagree with you here except to note:

What is an infinite orange? (which, if I didn’t say it that way, was a typo)

Gib, you’re showing holes in your knowledge here.

An infinite expansion doesn’t imply an irrational number.

What I mean by step 1 and step 2, is that even to find a finite number like the number 1, we have to explore.

Step number 2 is a completely different exploration and cognitive function, we abstract that it never ends!

Did someone say anything about an infinite orange? And why would he?

It’s merely a convenient way to describe certain properties of infinite sums.

(0.\dot9) approaches but never attains (1). This means two things: 1) not a single one of the partial sums of (0.9 + 0.09 + 0.009 + \cdots) (which is equivalent to (0.\dot9)) is equal to (1), and 2) the greater the number of terms that constitute a partial sum of (0.9 + 0.09 + 0.009 + \cdots), the closer it is to (1).

On the other hand, (1.\dot0) (which is an infinite sum equivalent to (1)) does attain (1). This merely means that it has a partial sum equal to or greater than (1). (In fact, if you’re evaluating the infinite sum from left to right, every single one of its partial sums is equal to (1).)

This alone proves that the two numbers aren’t equal.

I think that is the fundamental problem. I don’t think that it represents a “quantity”. It represents merely a quality. Quanta means finite and discrete.

Two infinite sets added maintains the quality of being infinite. But a quality does not affect quantity.

Here’s a simple visual representation of why (10 \times 0.\dot9 \neq 9 + 0.\dot9).

The image shows (10) instances of (0.\dot9).

The green line indicates the first (0.\dot9) among the ten instances. The number of terms of the equivalent infinite sum that is (0.9 + 0.09 + 0.009 + \cdots) will be represented using (\infty). So every occurence of that symbol from now on represents the number of terms of this sum (and no other infinite quantity.) That would be my standard infinity.

The blue line indicates (10) instances of (0.9) which is (9).

The red line indicates something interesting. It represents (0.\dot9) but is it the same quantity as the one indicated by the green line? Obviously not. You can see that the red rounded rectangle is narrower than the green rounded rectangle. In other words, the number of terms is not (\infty) but (\infty - 1).

Certainly, this proof can’t be accepted if one does not accept that (\infty + 1 > \infty) where every occurence of (\infty) represents the same infinite quantity. So maybe we should settle that question first?

Yeah, it makes no sense does it?

However, Magnus argues “orders of infinity”, which ultimately implies “the infinite orange”

It’s up to him to explain it to us

I’m certain that you are misunderstanding him. Gauging from other threads, you seem to have a habit of that.

For my 2 cents, relying on James’ posts, “orders of infinity” is like infA^1 as first order and infA^2 as second order and so on.

You appear to me to have the problem of not being able to see that one set can be infinite and still be a single, finite item, “A set” - a single item even though it contains or lists an infinity of things. And so when you have two of those sets, by definition, you have more than only one of those single items.

A spacial graph can have 3 infinite axes; x, y, z. That is more space represented than a plane graph of 2 infinite axes; z, y. There are more location points being identified by the 3 dimensional graph than by the 2 dimensional graph. Those would represent a third and second order infinity of locations, infA^3 and infA^2, respectively.

{infA, infA, infA} represents a location that does not exist in {infA, infA}.

I can’t understand why that would be hard to understand.

“Approaches” sounds to me like “builds up to”.

Anyway, I wanted to draw your attention to a couple responses of mine which I think you might have missed in the shuffle:

viewtopic.php?f=4&t=190558&start=1300#p2754877
viewtopic.php?f=4&t=190558&start=1300#p2754895

They are the same.

Depends on what you do. You can take (\circ \bullet \circ \bullet \circ \bullet \cdots) and split it into (\bullet \bullet \bullet \bullet \bullet \bullet \cdots) and (\circ \circ \circ \circ \circ \circ \cdots). I think this is what you’re trying to do. In such a case, you’d have a line that appears, but is not really, identical to that other line. The gaps didn’t disappear. You merely pushed them out.

The line is not identical to that other line (the one we started with, the one that didn’t have the odd points taken out) for the super-simple reason that by definition it’s half the size that other line. The fact that you can make them LOOK identical doesn’t make them so. You can make all kinds of things look convincing. That’s not logic, that’s rhetoric (or sophistry, or magic.)

You had an infinite number of points. Then you removed every odd point. To say that the resulting line is the same line is to say that you removed no point from it, which is a logical contradiction.

This is why you should go back to this argument.

Everything else would be running in circles.

Ok obsrvr,

Before we get more complex, let’s try to keep it simple.

Magnus’ claim: infinities can be quantified

If there cannot be an infinite orange then his claim is either false or has unspecified exceptions. It’s up to him from there to explain himself.

Let’s look at this a little deeper:

An infinite 1.

What is that? Does it make any sense.

Looks like Magnus is going to have to go back to the drawing board with his claim.

I image that he means that you can have one infinite set, perhaps all odd numbers. That is one single set - quantity of 1.

You can also have a different set of all even numbers. That is another single set - quantity of 1.

With both sets, you have 2 infinite sets - quantity of 2.

That’s fine, however, Silhouette and I (probably gib as well) are are arguing that this makes as much sense as 2 infinite oranges.

The thread will progress. I want to see what Magnus has to say as well.