Is 1 = 0.999... ? Really?

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.

I don’t think that you should expect dogs to distinguish red from green.

Oh, the real debating hasn’t even really started yet.

For example…

If you split an infinity every other, it’s 1/2 infinity, not 2 infinity. Nothing has been added or subtracted.

But well really start to debate later…

Let’s cover basic ground first

Ecmandu,

I didn’t say it does.

^ That was meant to be read as “a repeating pattern or an irrational number”.

Perhaps an example might help. How do we “explore” to find the number 1?

Sure, this requires abstraction. (I assume then that step 1 is more concrete? We explore in the world and find objects that can then be quantified?) But how is this a “build up”? I mean, at least in the sense I’m talking about “build up”? Every argument I’ve heard about why 0.999… doesn’t equal 1 goes along the lines of: the 9s never end, they can’t get to 1… which sounds to me like they’re imagining the 9s being “built up”, one after the other, sort of “growing” in an ever larger decimal expansion.

Magnus,

Yeah… you realize notation doesn’t stand for a process, right? It stands for a quantity, a static quantity, something we are to presume is already there. There is only a process in the attempt to visualize the notation. Our minds find that they need to keep adding more 9s to fully visualize the entire series. But the notation they’re trying to build already represents the quantity whether it’s complete or not. The quantity itself doesn’t need to “build up”, it’s just there.

Did you read this:

And where do the gaps disappear to in this scenario:

Woaw, woaw, woaw! What happened to the step of moving the remaining points to fill the gaps?

By definition? At best, you might get away with saying by implication, but again you’re generalizing. You’re taking what seems obvious with finite sets and just assuming it carries over to infinite sets.

See, this is why it’s you who is ignoring the counter-examples I bring up. You’re response is to repeat the same original logic. You keep saying: it’s just logical that if you remove stuff from other stuff you have less stuff. Then you make a leap from finite examples (which I agree is trivially true) to infinite examples. My counter-examples hold in the latter case, in the case of infinite things. I’m not refuting your original logic–of course if you subtract a number of things from a set of things, you get less things–but I’m refuting your right to carry that logic over to infinite things. Your response to that is just to reiterate the original logic and repeat the generalization–as if doing so a number of times will eventually invalidate my counter-examples. You need to address my counter-examples (which granted, you are), otherwise you’re not arguing anything new.