Is 1 = 0.999... ? Really?

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.

but i didn’t even tell you about the special powers it gave me. basically what happened was the composite chemical compounds of the paint formed a symbiotic bond with the membranes of my psuedounipolar neurons… essentially preserving them as well as enhancing the conductive element of the axon fibers. which means, positively charged ions that travel across this membrane have a much slower decay rate. and this is why i can see much further into things… why i can process philosophical information at much greater speeds. you could say that the paint (‘rustoleum’) did to my brain what the adamantium did to wolverine’s skeletal structure. well with the exception of the regions responsible for processing mathematical information. but that was a sacrifice i had to make to gain these powers.

Listing with counting numbers is a demonstrative way to prove 1:1 correspondence for an infinite number of sets.

We’re splitting hairs here.

There is no “last” number. That is where “endless” comes into this. And that is also why maths don’t work with infinity. The only thing you know about the value is that it is higher than all numbers, which actually means that it doesn’t exist except as a direction.

It doesn’t work that way. You have to directly relate each item in one set to a unique item in another set with nothing left over. There are only two sets. Counting them doesn’t mean anything.

The counting implies the correspondence without ambiguity.

Like I said, were splitting hairs here.

Counting makes it clearer than an infinity number of sets are in correspondence, meaning that an infinite number of sets can be bijected.

I’ll leave you in your bubble.

It’s implied. The problem you have with correspondence working on more than 1 list is that it refutes your argument.

I hate to be the one to break it to you:

1:1 Correspondence can work for an infinite number of sets.

You’re the one in the bubble here. I’m trying to pull you out of it.