Is 1 = 0.999... ? Really?

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.

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…

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…}

(\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.)

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.

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

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).

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

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?

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 !!!

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

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

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.

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.)

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.

Yes it is a quantity.

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

Not blue, or large, or charismatic.

Or negative, or irrational.

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

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

No it isn’t.

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.

That seems to be the fact of it.

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.

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.

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;
<_____________________________>
[list][list]…[/list:u][/list:u]

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?

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

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.

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?

and barely even the cruder points of arythmetic as well. i told ya i used to huff spray paint after school everyday in the eighth grade, right? i probably destroyed more brain cells in that year alone than i would throughout the rest of my life. but the shit you see while high on inhalents is something you’ll never regret. this one time me and matt were in the back yard and we saw this cloud turn into the starship enterprise and engulf a helicopter. pulled it right in with a tractor beam.

but check this out. when the old man put me into the looney bin (‘holly hill’), i had to take an entry I.Q. test. thirteen years old and determined NOT to cooperate. i didn’t even try. 123. now that’s nothing spectacular, but you gotta remember i had a brain saturated with spray paint. what got me was the math. if i’da known anything about that math, i’m sure i woulda broke a buck thirty at least. i mean math’s a big chunk of that test, bro.

then i took another one a few years back at the processing camp before they put me in medium security. scored 118 on that one. fucking math again. i swear to god if it weren’t for math i’d be a near genius. so, i’ve spent my entire life compensating for this failure and shame by producing philosophies that prove numbers are dumb. hey, you gotta do what you gotta do, right? think i’m gonna sit here and let you nerdy fucks make me look like a dummy? yeah i got your numbers, buddy. i got your numbers hangin.