Is 1 = 0.999... ? Really?

[Silhouette]

In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits. It is part of the standard algorithm to add numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column and the "1" is carried to the left. When used in subtraction the operation is called a borrow.

That's a Wikipedia article (not a "Math for Kids" website) describing a very specific way to find the sum of two quantities. It's definitely not the only way. You can add "6" and "7" together by placing six apples and seven oranges in front of yourself and then counting all of them. That's not the same as the above.

Addition is a much more general concept, and saying that I'm talking about addition in general is not the same as saying that I'm talking about some specific algorithm for adding numbers. I did not the do the former, I did the latter.

Thanks for the masterclass on carrying and borrowing - you could have stuck to your kid's website though - it's that basic. No wonder you're more qualified than professional mathematicians to determine these issues.
Are these processes how you're distinguishing the standard algorithm from addition "in general"?

You realise that carrying and borrowing are only superficial accomodations for numerical systems that use positional notation? Without positional notation, let's say we're using a conceptual numerical system that had an infinite number of digits, you wouldn't need to carry or borrow, because they don't actually do anything to the quantity itself that is being denoted. The standard algorithm is addition with superficial decoration - your argument that they're different is like saying a house is different to a house with graffiti on it.

This is probably indicative of a major difference between mathematicians and non-mathematicians: that mathematicians understand quantity at a deeper level since they traverse across multiple different representations depending on the job at hand and the best way to go about it. They don't get hung up on the extraneous and incidental, they don't confuse appearance for essence. This will be why non-mathematicians get so bewildered by \(1=0.\dot9\) because their intuitions lack adaptability to different representations of the same thing.

By contrast, you even quoted one of my explanations on the last page.

Yes, and it's wrong, and I explained why, even though it's unnecessary. (One doesn't have to prove more than it's necessary.)

It's not wrong and as mathematicians will tell you - explaining why is always necessary.
Simply saying it's wrong backed by a vague objection and saying it's unnecessary to prove it more is a far cry from mathematical proofs that can require obnoxious amounts of backing just to be complete.

Your main objection is that \(\frac{9.\dot9}{10}\neq0.\dot9\) right?
As above, you're getting "carried" away by the superficial - thinking that \(\frac{9.\dot9}{10}\) literally shifts all the digits 1 space to the right (or the decimal point 1 space to the left) instead of that just being appearance, meaning there's an extra "unmatched 9" at the "end" of the endless sequence of 9s compared to the "end" of the endless sequence of 9s in \(0.\dot9\)

The whole reason I explicitly used this notion for \(\frac{s}{10}=\frac{9/{10^0}+9/{10^1}+9/{10^2}+...}{10}=\frac{9/1+9/{10}+9/{100}+...}{10}=9/{10^1}+9/{10^2}+9/{10^3}+...=\sum_{x=1}^\infty \frac9{10^x}\) was to emphasise the one-to-one correspondence for "each decimal place" in both \(\frac{s}{10}\) and \(s\) but by representing each as a fraction instead of a decimal place - to try and help you from getting confused about superficial appearance of decimals.
This matches the \(9/{10^1}\) in \(s\) with the \(\frac{9/{10^0}}{10}\) in \(\frac{s}{10}\) (i.e. each 9 in the tenths column together for both s and s/10)
It does NOT match the \(9/{10^1}\) in \(s\) with the \(\frac{9/{10^1}}{10}\) in \(\frac{s}{10}\) (i.e. not the 0.9 in s with the 0.9/10=0.09 in s/10)

No doubt this still went over your head and you still got confused by the appearance of \(9/{10^1}\) looking the same in either, even though one was divided by 10 in \(\frac{s}{10}\) (obviously) and the other in \(s\) wasn't (obviously).

Quite obviously I know what I'm talking about, so maybe you should just stop insisting I don't, what do you think?

This is one of the most pointless things you can say in a forum discussion.

This was exactly why it was a rephrasing of something you said to me, that "You have no clue what you're talking about, so maybe you should just stop it and instead stick to the topic, what do you think?"

The whole point of me saying it was to demonstrate to you that what you just said was "one of the most pointless things you can say in a forum discussion."

God, you miss literally everything...

You wonder why I'm getting so frustrated?
And you think I'm proud of it?!
I couldn't be further from proud of the fact that even the most clear and obvious explanations mean nothing to you.

I love how you've just brought up Hilbert's Hotel too, because it's just another one of those thought experiments that demonstrates the indeterminate sizelessness of infinity (the hotel with infinite rooms in it) and therefore disproves all your nonsense about sizes of infinity. There's still only 1 infinity (the hotel) whatever happens to it, and regardless of any finite operations/adjustments that you make to its rooms, it's counterintuitively still full whilst also able to have spaces. Operating on infinity does nothing definite to the fact that it's still infinite.

This is what indeterminate/undefinable/infinite/endless amounts to.

You can emphasise that, no matter how many 9s in \(0.\dot9\) fill the gap between it and \(1\), there's always an infinite number of smaller infinitesimal gaps that could be filled, and I can emphasise that infinite recurring 9s will always fill these gaps and both/neither of us would be fully right, because infinity is undefinable, which is why things like \(\frac1\infty\) are invalid because it's the same as saying \(\frac1{undefinable}\).
And since there's nothing definite about how many 9s there are in \(0.\dot9\) (including any "spare 9s" compared to \(9.\dot9\)), this is the quality of endlessness that changes all your finite understandings of progressions like \(0.9+0.1=1\) and \(0.09+0.01=1\) and so on such that \(1=0.\dot9\) - there's no gap because the indeterminacy of endlessness/infinity means there's no way to determine any gap.

Yet another set of explanations that really really should put this topic to rest.
No doubt it won't. Disappoint me further.

[Aegean]

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

You are correct.
I was hasty in my expressions.
There is no movement towards the absolute, but away form it. Linear time is a movement towards increasing chaos, not increasing roder - that lies in the indeterminate past, as a hypothetical point in space/time - Big Bang.
It is a nihilistic inversion corrupting human language.
The more accurate image would be a movement towards absolute nil - away from the theoretical one.
The arrow of time points away from a singularity - the near absolute point we use to measure time - and represented by Yin/Yang.

Even in the metaphor of the line, the individuals in line are deteriorating - fragmenting, as they wait.

One is reference to order, since the mind cannot conceive chaos.
All language, including mathematics, measures order, so it tends to be positive, when, in fact, order is declining over space/time, not increasing.

[Ecmandu]

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

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

[Silhouette]

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

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 know I sound like an incomrehensible crazy person...

I’m working again on the 0.999... /= 1 argument, and if as Magnus states, silhouette is the “ecmandu whisperer”...

It’s a lot to ask silhouette, but can you help me disprove your proof that 0.999... does equal 1??!!

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.

[Ecmandu]

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)

[Magnus Anderson]

This will be why non-mathematicians get so bewildered by \(1=0.\dot9\) because their intuitions lack adaptability to different representations of the same thing.

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.

Simply saying it's wrong backed by a vague objection

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.

and saying it's unnecessary to prove it more is a far cry from mathematical proofs that can require obnoxious amounts of backing just to be complete.

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

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

Your main objection is that \(\frac{9.\dot9}{10}\neq0.\dot9\) right?
As above, you're getting "carried" away by the superficial - thinking that \(\frac{9.\dot9}{10}\) literally shifts all the digits 1 space to the right (or the decimal point 1 space to the left) instead of that just being appearance, meaning there's an extra "unmatched 9" at the "end" of the endless sequence of 9s compared to the "end" of the endless sequence of 9s in \(0.\dot9\)

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.

The whole reason I explicitly used this notion for \(\frac{s}{10}=\frac{9/{10^0}+9/{10^1}+9/{10^2}+...}{10}=\frac{9/1+9/{10}+9/{100}+...}{10}=9/{10^1}+9/{10^2}+9/{10^3}+...=\sum_{x=1}^\infty \frac9{10^x}\) was to emphasise the one-to-one correspondence for "each decimal place" in both \(\frac{s}{10}\) and \(s\) but by representing each as a fraction instead of a decimal place - to try and help you from getting confused about superficial appearance of decimals.
This matches the \(9/{10^1}\) in \(s\) with the \(\frac{9/{10^0}}{10}\) in \(\frac{s}{10}\) (i.e. each 9 in the tenths column together for both s and s/10)
It does NOT match the \(9/{10^1}\) in \(s\) with the \(\frac{9/{10^1}}{10}\) in \(\frac{s}{10}\) (i.e. not the 0.9 in s with the 0.9/10=0.09 in s/10)

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:

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

You wonder why I'm getting so frustrated?

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

And you think I'm proud of it?!

Considering that you talk about it publicly, yes.

I love how you've just brought up Hilbert's Hotel too, because it's just another one of those thought experiments that demonstrates the indeterminate sizelessness of infinity (the hotel with infinite rooms in it) and therefore disproves all your nonsense about sizes of infinity. There's still only 1 infinity (the hotel) whatever happens to it, and regardless of any finite operations/adjustments that you make to its rooms, it's counterintuitively still full whilst also able to have spaces. Operating on infinity does nothing definite to the fact that it's still infinite.

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.

This is what indeterminate/undefinable/infinite/endless amounts to.

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

infinity is undefinable, which is why things like \(\frac1\infty\) are invalid because it's the same as saying \(\frac1{undefinable}\)

That's not true.

And since there's nothing definite about how many 9s there are in \(0.\dot9\) (including any "spare 9s" compared to \(9.\dot9\)), this is the quality of endlessness that changes all your finite understandings of progressions like \(0.9+0.1=1\) and \(0.09+0.01=1\) and so on such that \(1=0.\dot9\) - there's no gap because the indeterminacy of endlessness/infinity means there's no way to determine any gap.

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

Yet another set of explanations that really really should put this topic to rest.
No doubt it won't. Disappoint me further.

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.

[obsrvr524]

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).

[gib]

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

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

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.

[gib]

Building the representation of \(0.\dot9\) needs endlessness

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.

[Ecmandu]

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).

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)

[Ecmandu]

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

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

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.

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!

[obsrvr524]

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

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

[Magnus Anderson]

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

If I say, "suppose you had a queue of 50 people," do you say, "wait... okay, the 50th person has just been added. Now we can talk about it." What if I say, "suppose you had a queue of 100 people"? Do we have to wait twice as long before continuing with the conversation?

0.999... is simply notation. It just stands for the idea of an infinite number of 9s. You're supposed to imagine you already have an infinite number of 9s.

(And really, it doesn't even represent that; it represents a quantity; the debate in this thread is: what is that quantity? Is it 1 or the next number before 1?)

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.

[obsrvr524]

the dot in \(0.\dot9\)--which in turns just ends up being a representation of the quantity.

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.

[Magnus Anderson]

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?

[Ecmandu]

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

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

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

[obsrvr524]

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.

[gib]

\(0.\dot9\) approaches but never attains \(1\).

"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

[Magnus Anderson]

"Approaches" sounds to me like "builds up to".

They are the same.

Ok, I see what you mean. When you move the first point to fill the first gap, you get a gap after that point two points long. Then to fill that gap, you need to move the next two points, which leaves a gap four points long. It seems that as you move the points down, you get an ever grow gap moving in the opposite direction. That indeed brings into question what the line ends up looking like at the end. Do you really get an identical line, point for point, or do you get a line with an infinitely long gap at the other end (somehow still an infinite number of points away)?

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.

[Ecmandu]

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.

[obsrvr524]

Magnus’ claim: infinities can be quantified

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.

[Ecmandu]

Magnus’ claim: infinities can be quantified

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.

[obsrvr524]

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

[Ecmandu]

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

[gib]

Ecmandu,

An infinite expansion doesn’t imply an irrational number.

I didn't say it does.

(a repeating pattern, an irrational number)

^ That was meant to be read as "a repeating pattern or 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.

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

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

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,

"Approaches" sounds to me like "builds up to".

They are the same.

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.

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.

Did you read this:

Come to think of it, I'm not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it's at the end of the line. But then what's at the end of the other line? More line? For all intents and purposes, if we're talking about "the end of the line", I'd say it's fair to say that's where the lines end. So the gap has effectively been push out of the line and the two are once again identical.

And where do the gaps disappear to in this scenario:

This problem arises when you imagine each point taking its turn to fill the gaps. If each point takes its turn, you'd need an eternity to complete the thought experiment and answer the question above. But what about each point moving at the same time? This is how we are to imagine Hilbert's Hotel. Each guest moves to the next room simultaneously, not one after the other. Of course, in the case of the gaps in the line, each point would have to move a different amount. The first point moves one position, the second point moves two positions, the third point moves three positions, etc..

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.

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

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.)

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.