Is 1 = 0.999... ? Really?

[Ecmandu]

Okay, let's say for the sake of argument that you can have more symbols than your number system can count. As if that makes any sense in itself.

You think there aren't more symbols than there are natural numbers? The set of natural numbers is the set of all symbols?

How about \(\{a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3, \dotso\}\)? That's \(3\) times more than \(\{1, 2, 3, \dotso\}\).

But I understand. You don't think that \(3 \times \infty > \infty\).

There you go again!

What’s an infinite orange?

What’s an infinite orange + an infinite pear??

What are two infinite oranges?

[obsrvr524]

i got your PM, ecmandu, but i don't yet feel comfortable with publishing. there still seems to be much disagreement among our mathematicians and i'd like to wait until i see unanimous agreement.

maybe you guys'll figure out if 1 = 0.999 THIS FUCKING YEAR, and i can proceed with the project.

Do you really believe that any of these guys will change their mind?
I don't.

[Ecmandu]

i got your PM, ecmandu, but i don't yet feel comfortable with publishing. there still seems to be much disagreement among our mathematicians and i'd like to wait until i see unanimous agreement.

maybe you guys'll figure out if 1 = 0.999 THIS FUCKING YEAR, and i can proceed with the project.

Do you really believe that any of these guys will change their mind?
I don't.

Obsrvr! I could say the same about you. Think before you post!

[Pedro I Rengel]

Psh.

Promethean has no understanding of the finer points of arythmetic.

[phyllo]

You think there aren't more symbols than there are natural numbers? The set of natural numbers is the set of all symbols?

At the end of the day, you can't count them with the natural numbers. Which was important in context because you had a 1 for each symbol. Your 1+1+1+... is infinite whether you use symbols or not.

But I understand. You don't think that 3×∞>∞.

You have to make up your mind whether you will consider all infinities the same, in which case 3*infinity=infinity. Or you will consider them to be distinct, in which case you need a way to track the size of each infinity.

That's the problem with the math here:

∞+1=∞ // subtract ∞ from both sides
∞+1−∞ =∞−∞ // substitute ∞−∞ with 0
1=0

...

But where's the mistake?

The first line says that all infinities are equal. "If I add a grain of sand to a heap of sand, then I still have a heap of sand."

The second line says that each infinity is distinct and the distinct quantity can be subtracted on both sides of the equal sign. "I know how many grains of sand are in a heap."

Both lines cannot be true. If you think that line one is true then line two is false. If you think that line two is true then line one is false.

One has to make up one's mind in order to be consistent. One can swing either way but standard math says all infinities are the same.

[obsrvr524]

One can swing either way but standard math says all infinities are the same.

So you believe that two infinite lines depict the same amount of locations as one? An infinite plane has the same amount of locations depicted as an infinite line? There is not a one-to-one correspondence.

[Ecmandu]

One can swing either way but standard math says all infinities are the same.

So you believe that two infinite lines depict the same amount of locations as one?

Yes. It’s called 1:1 correspondence.

[obsrvr524]

One can swing either way but standard math says all infinities are the same.

So you believe that two infinite lines depict the same amount of locations as one?

Yes. It’s called 1:1 correspondence.

No. there is not a one-to-one correspondence between a single infinite line and two infinite lines.

[Ecmandu]

Obsrvr wrote:

No. there is not a one-to-one correspondence between a single infinite line and two infinite lines.

Ecmandu writes:

Then you don’t know much about 1:1 correspondence!

1,3,5,7,9...
2,4,6,8,10...

Have the same correspondence as:

1,2,3,4,5,6,7,8,9...

[obsrvr524]

There are a lot of things that I don't know but one thing I do know is that -

{a1,a2,a3,a4,...}

does NOT have a one-to-one correspondence with -

{a1,a2,a3,a4,...
b1,b2,b3,b4...}

[Magnus Anderson]

The first line says that all infinities are equal. "If I add a grain of sand to a heap of sand, then I still have a heap of sand."

The second line says that each infinity is distinct and the distinct quantity can be subtracted on both sides of the equal sign. "I know how many grains of sand are in a heap."

Both lines cannot be true. If you think that line one is true then line two is false. If you think that line two is true then line one is false.

One has to make up one's mind in order to be consistent.

\(\infty + 1 = \infty\) can be interpreted in two different ways.

In mathematics, to say that two numbers are equal (\(a = b\)) is to say that their difference is equal to zero (\(a - b = 0\)). So if we're saying that \(\infty + 1\) is equal to \(\infty\) in the mathematical sense of the word "equal", then the conclusion necessarily follows.

\(\infty + 1 = \infty\) // subtract \(\infty\) from both sides
\(\infty + 1 - \infty = \infty - \infty\) // replace \(\infty - \infty\) with \(0\)
\(1 = 0\)

But if what that expression means is "If you add one to an infinite quantity, you will still have an infinite quantity" we're no longer dealing with the mathematical concept of equality and it no longer follows that if \(a = b\) that \(a - b = 0\). Take a look at the statement "If you add one to an infinite quantity, you will still have an infinite quantity". The first infinite quantity is not necessarily the same as the second one. It's not saying they are the same value. It's merely saying they are both endless. Thus, if you subtract them, the result won't necessarily be \(0\). So in this case, the conclusion does not follow (but some other do.)

[gib]

Is it because the latter evaluates to \(\infty\) or because you think the former equals to a finite number or because of something else?

Nope, you've pretty much got it. If it evaluates to a finite number, you can use it in arithmetic. If it evaluates to \(\infty\), you can't.

And how did you come to your conclusion?

Because arithmetic deals with numbers, quantities. \(\infty\) is not a quantity.

Why is it okay to say that \(10 \times (0.9 + 0.09 + 0.009 + \cdots) = 9 + 0.9 + 0.09 + \cdots\) but not okay to say that \(10 \times (1 + 1 + 1 + \cdots) = 10 + 10 + 10 + \cdots\)?

Well, you can, but here's where the other rendition I talked about comes in: if you want to do arithmetic on \(\infty\), you have to play by different rules. \(10 \times (1 + 1 + 1 + \cdots) = 10 \times \infty = \infty\). <-- In what case, with normal arithmetic using normal numbers, can you say 10 x n = n?

Note that those weird rules don't apply to the case of \(0.\dot9\): \(10 \times (0.9 + 0.09 + 0.009 + \cdots) = 10 \times 0.\dot9 = 9.\dot9\).

If I understand you correctly, we can do arithmetic with \(0.9 + 0.09 + 0.009 + \cdots\) because it's not equal to \(\infty\).

Why is that so?

Because \(0.9 + 0.09 + 0.009 + \cdots\) is a quantity and you can do arithmetic with quantities. \(\infty\) is not so you cannot.

So don't do arithmetic with \(\infty\) or accept that different rules apply. <-- Take your pick.

But you didn't explain why.

You can only do arithmetic with quantities. If you choose to do arithmetic with things other than quantities, you gotta figure out a different set of rules that makes sense with those things.

For example, suppose you wanted to do arithmetic with 'red'. One good starting point might be to ask: what does adding 1 to red mean? red + 1 = ??? You gotta figure out what that could possibly mean. Are you adding 1 unit of saturation to red? If so, then maybe the rule is: red + 1 = brighter red. Maybe adding 1 means adding 1 more square foot of red (like if you were painting). Then red + 1 = more surface area of red.

^ It would be a different kind of arithmetic, not the traditional one that everyone is used to.

As I said, \(\infty\) is a property of sets, the property of being endless. I keep asking you what \(\infty\) + 1 means. I'm asking what it means to add something to an infinite set such that it's property of being endless changes somehow. I'm asking this in much the same vein as asking what does red + 1 mean. I need to understand what's being conveyed by the statement "red + 1" in order to know what it equals. Same with \(\infty\) + 1. What am I supposed to be imagining. The best I can do is to imagine adding something to a collection of things that's already infinite and asking myself: does this change the property of being endless. And the answer seems to be: no, it's still endless. If you say, it makes it more endless, then I have to ask about that in the same vein: what am I imagining when I imagine something being more endless than something else that's endless?

[Ecmandu]

There are a lot of things that I don't know but one thing I do know is that -

{a1,a2,a3,a4,...}

does NOT have a one-to-one correspondence with -

{a1,a2,a3,a4,...
b1,b2,b3,b4...}

Well, relearn math then.

1:1 correspondence can happen for an infinite amount of sets (not just 2)

1.) 1
2.) 2
3.) 3
4.) 4

Etc...

1.) a1
2.) a2
3.) a3
4.) a4

Etc...

1.) b1
2.) b2
3.) b3
4.) b4

Etc...

ALL in 1:1 correspondence !!!!

[Magnus Anderson]

In what case, with normal arithmetic using normal numbers, can you say 10 x n = n?

\(10 \times \text{finite number} = \text{finite number}\)

But note that this isn't arithmetic. And neither is \(10 \times \infty = \infty\).

Because \(0.9 + 0.09 + 0.009 + \cdots\) is a quantity and you can do arithmetic with quantities. \(\infty\) is not so you cannot.

A couple of questions that need to be addressed in order to resolve the disagreement.

- What makes you think that \(0.9 + 0.09 + 0.009 + \cdots\) is a quantity and \(1 + 1 + 1 + \cdots\) is not?

- What is a quantity?

- Are you merely saying that \(0.9 + 0.09 + 0.009 + \cdots\) evaluates to a finite number whereas \(1 + 1 + 1 + \cdots\) does not? If so, what makes you think that \(0.9 + 0.09 + 0.009 + \cdots\) evaluates to a finite number? Most importantly, what does it mean for an infinite sum to evaluate to a finite number?

But I think I can predict your answers. Basically, what you're saying is:

\(0.9 + 0.09 + 0.009 + \cdots\) evaluates to a finite number and this simply means that its limit is a finite number.

\(1 + 1 + 1 + ...\) does not evaluate to a finite number because it is a divergent series (i.e. it has no limit.)

This is one of our main points of disagremeent: I think that the result of an infinite sum is not the same thing as its limit. The limit of an infinite sum is the value that it approaches but that it does not necessarily attain. The result of an infinite sum, on the other hand, is the value that it attains (i.e. it does not merely approach it, it actually attains it.)

\(5 + 0 + 0 + 0 + \cdots\) is an infinite sum that evaluates to \(5\). It does so because it attains \(5\). It does not merely approach it. Its limit is also \(5\).

\(0.9 + 0.09 + 0.009 + \cdots\) is an infinite sum that does not evaluate to a finite number. This is because there is no finite number it attains. Its limit, however, is \(1\) because it approaches \(1\).

In other words, neither \(0.9 + 0.09 + 0.009 + \cdots\) nor \(1 + 1 + 1 + \cdots\) evaluate to a finite number.

But what really interests me is the following: why does an infinite sum must evaluate to a finite number in order for us to be able to do arithmetic with it?

I don't think you can answer this question. You don't really know. And you don't really know because you did not reach this conclusion logically. You reached it by trying to justify a popular opinion. That's what I think.

[Magnus Anderson]

I'm asking what it means to add something to an infinite set such that it's property of being endless changes somehow.

I don't know what that means. This shouldn't come off as surprise to you for the simple reason that I never said such a thing. I never said that adding an element to an infinite set means changing its property of being endless. What I said is that it increases the number of elements it has (the set remains endless.)

The best I can do is to imagine adding something to a collection of things that's already infinite and asking myself: does this change the property of being endless. And the answer seems to be: no, it's still endless. If you say, it makes it more endless, then I have to ask about that in the same vein: what am I imagining when I imagine something being more endless than something else that's endless?

I don't know what it means to say that a set is "more endless" than it was before. Sets are either endless or they are not. There are no degrees fo endlessness.

What happens to an infinite set when you add a new element to it is the same thing that happens to finite sets: you make them bigger. They have everything they had before + this new thing.

The problem is that infinite sets are endless so you cannot simply list of all their members (something you can do with finite sets) which makes it all too easy to pretend that adding a new element to an infinite set does not change that set.

[Pedro I Rengel]

Yes it is a quantity.

[Pedro I Rengel]

Infinity exists in relation to 1, or 2, or 57.

Not blue, or large, or charismatic.

Or negative, or irrational.

[phoneutria]

Some infinites are bigger than others
Some infinites are bigger than others
Some infinite's mothers are bigger than other infinite's mothers

[obsrvr524]

There are a lot of things that I don't know but one thing I do know is that -

{a1,a2,a3,a4,...}

does NOT have a one-to-one correspondence with -

{a1,a2,a3,a4,...
b1,b2,b3,b4...}

Well, relearn math then.

1:1 correspondence can happen for an infinite amount of sets (not just 2)

1.) 1
2.) 2
3.) 3
4.) 4

Etc...

1.) a1
2.) a2
3.) a3
4.) a4

Etc...

1.) b1
2.) b2
3.) b3
4.) b4

Etc...

ALL in 1:1 correspondence !!!!

Apparently you don't understand one-to-one correspondence.

Yes it is a quantity.

No it isn't.

Infinity exists in relation to 1, or 2, or 57.

Only in that it is not like any of those.

Where do parallel lines cross? That is where infinity exists.
Infinity doesn't exist as a quantity at all. It is not quantified or even quantifiable without more information about it being given. That is why maths do not work on it.

Some infinites are bigger than others
Some infinites are bigger than others
Some infinite's mothers are bigger than other infinite's mothers

That seems to be the fact of it.

[Ecmandu]

Apparently I don’t understand 1:1 correspondence !?

Wow, your ignorance is appalling!!

It’s anything that can be listed by the well ordered set of counting numbers.

[obsrvr524]

Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. More at "Wikipedia"

[Magnus Anderson]

Only in that it is not like any of those.

It's not merely unlike 1, 2 and 57. It's actually greater than all of these numbers. The word "red", on the other hand, does not represent something that can be said to be greater than or less than some number.

Where do parallel lines cross? That is where infinity exists.

Certainly, infinity is not an ordinal number. There is no such thing as infinity-th position. But I don't see a problem with treating it as a cardinal number.

[obsrvr524]

Certainly, infinity is not an ordinal number. There is no such thing as infinity-th position. But I don't see a problem with treating it as a cardinal number.

If it is assigned a fixed infinite expression, such as was done with infA, maths can then be used. If it is just left as "infinity" or "endless", as you can see from this discussion, any cardinal value is ambiguous.

infA = <_____________________________>

infA + 2 = <_____________________________> + ..

Or better;
<_____________________________>

..

And 2 * infA =
<_____________________________>
<_____________________________>


And infA^2 = an infinite plane.

The coordinate value {infA, infA} is clearly not on an infinite cardinal line {infA}


But it must be assigned a fixed, unambiguous infinite expression first.

Wasn't the same kind of thing done with imaginary numbers? How could you do maths with imaginary numbers until they were adequately defined and limited to a specific meaning?

[phyllo]

∞+1=∞ can be interpreted in two different ways.

In mathematics, to say that two numbers are equal (a=b) is to say that their difference is equal to zero (a−b=0). So if we're saying that ∞+1 is equal to ∞ in the mathematical sense of the word "equal", then the conclusion necessarily follows.

∞+1=∞ // subtract ∞ from both sides
∞+1−∞=∞−∞ // replace ∞−∞ with 0
1=0

It doesn't follow based on a simple expansion of the equation
$$\infty+1=\infty$$

But if what that expression means is "If you add one to an infinite quantity, you will still have an infinite quantity" we're no longer dealing with the mathematical concept of equality and it no longer follows that if a=b that a−b=0. Take a look at the statement "If you add one to an infinite quantity, you will still have an infinite quantity". The first infinite quantity is not necessarily the same as the second one. It's not saying they are the same value. It's merely saying they are both endless. Thus, if you subtract them, the result won't necessarily be 0. So in this case, the conclusion does not follow (but some other do.)

Infinity is the concept of a big "value" but not an exact fixed value.

Who knows where "endless" comes into this?

You can have a series (with an infinite "number" of terms) which evaluates to an infinite "value". You can have a series (with an infinite "number" of terms) which evaluates to a finite value.

But infinity does not require there to be any series involved.

[phyllo]

Certainly, infinity is not an ordinal number. There is no such thing as infinity-th position. But I don't see a problem with treating it as a cardinal number.

If you have the series 1+1+1+...

then the "last" 1 is at the infinity-th position and total "value" is infinity.

How can it be otherwise?