Infinity

[SoccerfreakAB2]

Quote by: ZNFYRH I thought it was precise?

Please be less vague in your short answers. What is "it"? Math?

Math is exactly precise. I think you're misunderstanding the focus of the use of infinity still. Infinity, in and of itself, is not a precise answer, nor is x2 or e^x or (x-2)(x+6). Infinity is not a precise number, so using it in an equation to determine patterns and characteristics of an equation will not give you a precise answer.

[ZNFYRH]

"It" was meant to be math.

And I agree with you.

[nm420]

Quote by: Lullaby Chainer What are your views on infinity? ø Is there such a thing in the real world? ø Is our universe infinite?

Infinity, as I tend to envision it, is process without end. As such, I do think that the universe is infinite, though I recognize this flies in the face of currently accepted scientific ideology, or seems to anyhow. Of course, infinity being a concept, it is not a "point" we can ever reach, rather a ceaseless journey (be it through space or time or whatnot).

Is there such a thing in the math world? ø Can infinity be used in math? ø Can infinity be multiplied, divided, added, or subtracted in a math equation? ø If so, how would that be done?

It most certainly is a mathematical concept, and is used in mathematics all of the time. Cantor was one of the first mathematicians to attempt to formalize the notion of infinity, though Archimedes was well on the way to developing a science of the infinite (infinitely small anyhow) before he was cut down in his prime.

As infinity is a concept, it makes no sense to speak of performing operations on it. When it is said that the limit of some function or sequence is infinite, this is mathspeak for saying it is unbounded; there is no "real" number that is not eventually exceeded by the function in question. It often happens that mathematicians use the extended real number system, which is all real numbers with two extra symbols called positive or negative infinity. It is true that these symbols are used to represent the notion of infinity, and some simplistic operations can be performed on them, but it is understood that these operations are merely shortcuts for the limits they are to represent. Thus, 1/oo=0 in the extended reals, as any sequence (x) of reals which goes off to infinity will make 1/(x) go to 0. The reason oo/oo, or 0*oo, or 1^(oo), are all undefined is that different sequences will make these quantities converge to different numbers, and no ambiguity is tolerated in mathematics.

[iclaudius]

Some stuff I want to address:

Quote by: ZNFYRH Infinity is a concept.

In the same way that numeric value is a concept. It can be attributed to things as condition, a description of quantity, etc.

Quote by: ZNFYRH How many times does infinity go into infinity? Once. ∞ / ∞ = 1

Quote by: Yasa wrong, INF/INF is undefined.

First, "infinity" is not one number -- it's a set of numbers that cannot be quantified by finite numeric value (unlike numbers like "3" or " 7.2"). If a number is infinite, that just means it isn't finite (which means that its value is always unquantifiable, either on a huge scale or an infinitesimal one).

Second, dividing any infinite number by another infinite number does not usually yield a specific number (like 1). For example: if it's unquantifiably huge, how would you know how many times it goes into another unquantifiably huge value? The question is just not specific enough -- we have no value to weigh how many times one goes into another. It's not undefined, it just it's just too broad.

For purposes of clarification: Infinity is a term generally used in math to explain the behavior of an equation. For example: If I asked, "How far up do numbers go?" your response would probably be something like "infinite." That designation means "it goes on forever, and for the purposes of communication, we don't need to bother specifying which infinite number; we need only know that the number it goes up to is definitely not finite, only meaning that it cannot be classified by values like '3'."

Quote by: ZNFYHR How large is a segment if you divide something into an infinite number of segments? Not zero. 1 / ∞ != 0

Correct. It's an infinitely small number.

Quote by: ZNFYRH How many times does nothing go into nothing? Forever. 0 / 0 = ∞

Actually, anything divided 0 times is the singular encompassing value of everything. Not infinity, but a representation of all numeric value.

Quote by: Lullaby Thus the equation is absurd and infinity cannot be treated as a number.

Infinity must be treated as an unquantifiable number, not as a concept or a finite number. Just because you conclude that it doesn't behave like a finite number does not mean that it should be abolished from finite math entirely.

Quote by: Lullaby Chainer Is it even POSSIBLE for space and the real world to be infinite? I remember hearing some pretty good arguments against it in the other thread. (or maybe I'm just remembering my own arguments )

Of course it is. Why wouldn't it be?

Quote by: 5010 According to Infinity (Wikipedia) x/∞ = 0 (where x is a real number) but 0·∞ is indeterminate (except in some contexts). It also states that [x/∞ = 0] is not equivalent to [0·∞=x]. So is there an error in this wiki, or is there disagreement, or is it coming from a different context? And could one say x/∞ = dx? The infinite slice is infinitessimal?

Wiki states, essentially, that x/INF = 0 and x = 0 * INF are not equivalent because that would mean that all numbers are equal. Assuming both equations are correct (they are not), the assertion is true, but wiki fails to acknowledge that they aren't even remotely the same statement. Let's start with the basic beginning, and try to get the second equation from the first (and let us, for sake of debate, assume that the statement is true in the first place):

We start with x/INF = 0. Habit might tempt us to multiply both sides by INF. Let's see what happens when we do that.

x/INF * INF = 0 * INF

x * 1/INF * INF = 0 * INF

x * INF/INF = 0 * INF

What have we established? Nothing. Not only are we not guaranteed that we can even multiply by INF, we must remember that INF/INF is non-conclusive (or, by wiki's standards, undefined). Even if x/INF = 0 were true, the second statement, x = 0 * INF is entirely unlinked. They are not the same, and infinite numbers do not reflexively cancel themselves out, because we have no guarantee that they are the same. We can't divide by x because dividing anything by zero (or even anything that could potentially be zero) yields an incomprehensible mathematic operation. Therefore the first statement exists and the second one is just wrong.

But the first one is wrong, anyway. 1/INF = an infinitely small number, not 0. And for the record, 0 * INF is still 0. And number that is unquantifiably huge may not have quantifiable value, but its pain property is that the value is too large to be quantified. Therefore, it still has value, and that value times 0 is 0.


Any other questions?

[ZNFYRH]

iclaudius,

I see what you're saying. Infinity is more of a limit and representation and not so much a value, right?

On the division of INF by INF, I always look at it as saying, "How many times does everything go into everything?" Then again, I'm not talking about the varying "sets" of infinity.

Quote by: iclaudius Actually, anything divided 0 times is the singular encompassing value of everything. Not infinity, but a representation of all numeric value.

I think this relates to my perception of infinity. I think of infinity as everything. So to speak it out... "How many times does nothing go into nothing?" or "How many empty sets are in an empty set?"

On your evaluation of the equation, I also understand. People assume, as I do, that INF/INF is 1. You're saying that it isn't, and therefore the equation is invalid.

Quote by: iclaudius But the first one is wrong, anyway. 1/INF = an infinitely small number, not 0. And for the record, 0 * INF is still 0. And number that is unquantifiably huge may not have quantifiable value, but its pain property is that the value is too large to be quantified. Therefore, it still has value, and that value times 0 is 0.

I absolutely agree with this. I went to the same system of schooling as another member here who said the same thing.

I never understood how people could think something could become nothing, especially when those same people would say "matter cannot be destroyed". Yet by saying x/INF = 0 they were basically implying that it was possible.

[iclaudius]

Quote by: ZNFYRH iclaudius, I see what you're saying. Infinity is more of a limit and representation and not so much a value, right?

Uh... no. An infinte number, most simply, is a number that is not finite. Infinity, a commonly misunderstood word, is just the quality of being infinite. When you say, "numbers go on to infinity," you're saying is that finite value can be counted, but that the actual value of everything is infinite, meaning you can't ever be finished counting, because you can always count 1 higher than what you've counted. The value of everything is not countable by finite value. That's all you're saying. In common usage, you might hear, "infinity, that means it goes on forever," but that's only half true. Yes, if you were counting finite numbers to infinity, you would be counting forever, but that doesn't mean that infinity is a number that keeps getting larger... it's just a quality. Infinity is not one "largest number." It's a type of number.

On the division of INF by INF, I always look at it as saying, "How many times does everything go into everything?" Then again, I'm not talking about the varying "sets" of infinity.

But that's not what infinity is. Ask yourself this: how many times does finity go into finity? We don't know! Be more specific.

I think this relates to my perception of infinity. I think of infinity as everything. So to speak it out... "How many times does nothing go into nothing?" or "How many empty sets are in an empty set?" On your evaluation of the equation, I also understand. People assume, as I do, that INF/INF is 1. You're saying that it isn't, and therefore the equation is invalid.

A singular representation of all numeric value and the quality of being infinite (infinity) are not the same thing.

I absolutely agree with this. I went to the same system of schooling as another member here who said the same thing. I never understood how people could think something could become nothing, especially when those same people would say "matter cannot be destroyed". Yet by saying x/INF = 0 they were basically implying that it was possible.

Well, actually, matter can be destroyed. It is converted to energy in nuclear reactions.

That aside, that particular law of physics is not strictly applicable here...

[ZNFYRH]

Quote by: iclaudius But that's not what infinity is. Ask yourself this: how many times does finity go into finity? We don't know! Be more specific.

"finity" isn't a word.

What you're asking is, "How many times does something finite go into something finite?" which can be answered.

Quote by: iclaudius Well, actually, matter can be destroyed. It is converted to energy in nuclear reactions.

Energy and matter cannot be created or destroyed, only converted.

Regardless, you missed the point... people who insist that something can't be destroyed (turned to nothing) somehow think there is a way to divide X (something) until it is nothing (0).

[Yasa]

Infinity divided by infinity is undefined, just like 0/0 is undefined.

[iclaudius]

Quote by: ZNFYRH "finity" isn't a word. What you're asking is, "How many times does something finite go into something finite?" which can be answered.

Rules of lexicography are arbitrary. Whether or not "finity" is formally recognized by x number of lexicographers is irrelevent, so long as the term I use is logically consistent. And that argument does not change the fact that it makes no sense to ask, "what is INF/INF?" Just to set everything straight, I wasn't asking how many times "something finite" goes into something else finite... I was asking how many times the quality of being finite goes into the quality of being finite. That's what you are asking when you ask what INF/INF is. If that question still makes sense to you, answer it.

Energy and matter cannot be created or destroyed, only converted. Regardless, you missed the point... people who insist that something can't be destroyed (turned to nothing) somehow think there is a way to divide X (something) until it is nothing (0).

Here's a secret: matter is highly excited energy. When you "convert" matter to energy, it ceases to be matter. Remember, if matter exists at all, it is just a configuration of energy. It's a term that describes how energy behaves. Thus, when something of that configuration ceases to exist, you have destroyed the configuration, and thusly destroyed matter.

But anyway, I didn't miss the point. Matter's ability/inability to be destroyed is not entirely applicable to this topic, because matter cannot be infinitely divided. And "matter cannot be destroyed" is not the same as "something cannot be destroyed."

[iclaudius]

Quote by: Yasa Infinity divided by infinity is undefined, just like 0/0 is undefined.

Did you even read the thread? If you did, how about providing valid argumentation instead of stupid conjecture?

[Yasa]

Quote by: iclaudius Did you even read the thread? If you did, how about providing valid argumentation instead of stupid conjecture?

Yes, I've been in a part of the thread and have read it. I was simply arguing that INF/INF is undifined, because people (I can quote ZNFYRH if you need) throughout the thread seem to disagree. Sorry if you thought that it was a stupid conjecture.

However, I'll guess I'll elaborate using Lullaby's logic. If INF/INF=1 then 1*INF=INF which I agree with. However, he claimed that INF*x=INF (which I also agree with), therefore INF/INF would equal x. x can be anything (except zero) and thus the true answer is undefined. You can't operate on infinity the same way you would with finite numbers due to it's paradoxical nature.

Quote by: iclaudius It's not undefined, it just it's just too broad.

That's exactly why it's undefined.


As for everything else you've said about infinity, I agree.

[iclaudius]

Quote by: Yasa Yes, I've been in a part of the thread and have read it. I was simply arguing that INF/INF is undifined, because people (I can quote ZNFYRH if you need) throughout the thread seem to disagree. Sorry if you thought th at it was a stupid conjecture. [Middle part excised and placed below.] That's exactly why it's undefined. As for everything else you've said about infinity, I agree.

Yasa, you are a smart person ... why didn't you just flesh out your argument like that in the first place?

However, I'll guess I'll elaborate using Lullaby's logic. If INF/INF=1 then 1*INF=INF which I agree with. However, he claimed that INF*x=INF (which I also agree with), therefore INF/INF would equal x. x can be anything (except zero) and thus the true answer is undefined. You can't operate on infinity the same way you would with finite numbers due to it's paradoxical nature.

But again, INF is not one number. In fact, its unquantifiable, so we're not sure that infinity goes into infinity just once. It could go in infinitely many times, or 3. It really depends on the context. And if INF/INF equals x, then you must be saying (based on your logic) that all numbers are equal to 1. Or if you agree with me, that INF is not one number, then you are saying that all numbers are potentially equal, and/or infinite. But let's assume that this was the correct answer... it wouldn't be undefined, because you would have defined it. It equals x. And in that case, it would be indeterminate.

The real answer is that 0/0, INF/INF, and a lot of other problems like that are indeterminate. In mathematics, "undefined" means that we literally have no answer for it, under ANY context. And indeterminate mathematical operation is one where in you must specify which context the statement exists in to obtain an answer. INF/INF is one such context.

I hope all this is clear and didn't confuse you.

[Yasa]

Quote by: iclaudius But again, INF is not one number. In fact, its unquantifiable, so we're not sure that infinity goes into infinity just once. It could go in infinitely many times, or 3. It really depends on the context. And if INF/INF equals x, then you must be saying (based on your logic) that all numbers are equal to 1. Or if you agree with me, that INF is not one number, then you are saying that all numbers are potentially equal, and/or infinite. But let's assume that this was the correct answer... it wouldn't be undefined, because you would have defined it. It equals x. And in that case, it would be indeterminate. The real answer is that 0/0, INF/INF, and a lot of other problems like that are indeterminate. In mathematics, "undefined" means that we literally have no answer for it, under ANY context. And indeterminate mathematical operation is one where in you must specify which context the statement exists in to obtain an answer. INF/INF is one such context. I hope all this is clear and didn't confuse you.

I was not disagreeing with you, I was proving it's indeterminate (undefined) using Lullaby's logic as he claimed that INF/INF is 1 but also that INF times any number is INF, proving that INF/INF could be any number, thus undefined (or indeterminate if you will). They (INF/INF and 0/0) are undefined because they yield incomprehensible answers. We could set both equal to x but x would be unknown as it could be anything.

[iclaudius]

In that case, allow me to point out that "undefined" and "indeterminate" are not the same thing. If you wrote that INF/INF was undefined on a math test, you would be marked wrong by any decent math teacher.

[Yasa]

Quote by: iclaudius In that case, allow me to point out that "undefined" and "indeterminate" are not the same thing. If you wrote that INF/INF was undefined on a math test, you would be marked wrong by any decent math teacher.

Please elaborate, I don't understand the difference. Dictionary.com search results for indeterminate:

5. Mathematics. a. (of a quantity) undefined, as 0/0.
b. (of an equation) able to be satisfied by more than one value for each unknown.

[gw120]

What are your views on infinity?
It hurt's my head if I think about it for too long.
Is there such a thing in the real world?
Yes.
Is our universe infinite?
Yes.
Is there such a thing in the math world?
Yes.
Can infinity be used in math?
To an extent.
Can infinity be multiplied, divided, added, or subtracted in a math equation?
No, infinity is not an actual number.
If so, how would that be done?
Can't be done.

[Lullaby Chainer]

Quote by: gw120 Is our universe infinite? Yes.

That would be impossible.

No point in the universe is infinitely far away from another point.

For the universe to be infinite, this would actually have to be true. There MUST be at least one point that is infinitely far away from another point. Let's assume there is such a point. We'll call this point A. Point B is touching point A and is in between point A and the point that is infinitely far away from point A, point C. Point B must also be infinitely far way from point C.. else.. we have just defined infinity as a number. Obviously that's nonsense. So, point B is also infinitely far away from point C. Now, there is the MidPoint. This point is between point A and point C. It also must be infinitely far away from point C, else, infinity would be defined by the distance between point C and A multiplied by 2.. which is nonsense. Thus, ANY point in a system in an infinite universe would be infinitely far away from every single point. Nonsense. Please note that when we speak of an infinite universe, it is entirely different from infinity in closed spaces (for instance.. there are an infinite number of numbers between 1 and 2). The difference is that in the math realm, numbers can be generated as needed to produced any number of numbers between 1 and 2. An infinite universe demands an external infinity that is already generated.. which is also impossible.

[Yasa]

Quote by: Lullaby Chainer That would be impossible. No point in the universe is infinitely far away from another point. For the universe to be infinite, this would actually have to be true. There MUST be at least one point that is infinitely far away from another point. Let's assume there is such a point. We'll call this point A. Point B is touching point A and is in between point A and the point that is infinitely far away from point A, point C. Point B must also be infinitely far way from point C.. else.. we have just defined infinity as a number. Obviously that's nonsense. So, point B is also infinitely far away from point C. Now, there is the MidPoint. This point is between point A and point C. It also must be infinitely far away from point C, else, infinity would be defined by the distance between point C and A multiplied by 2.. which is nonsense. Thus, ANY point in a system in an infinite universe would be infinitely far away from every single point. Nonsense. Please note that when we speak of an infinite universe, it is entirely different from infinity in closed spaces (for instance.. there are an infinite number of numbers between 1 and 2). The difference is that in the math realm, numbers can be generated as needed to produced any number of numbers between 1 and 2. An infinite universe demands an external infinity that is already generated.. which is also impossible.

Actually, they wouldn't be points if they were infinitely far away. The moment you label something as a point, it cannot be infinitely far away from another point. Example: numbers are infinite, however any two numbers are not infinitely separated. See what I mean?

[bishop]

Quote by: Lullaby Chainer That would be impossible. No point in the universe is infinitely far away from another point. For the universe to be infinite, this would actually have to be true. There MUST be at least one point that is infinitely far away from another point. Let's assume there is such a point. We'll call this point A. Point B is touching point A and is in between point A and the point that is infinitely far away from point A, point C. Point B must also be infinitely far way from point C.. else.. we have just defined infinity as a number. Obviously that's nonsense. So, point B is also infinitely far away from point C. Now, there is the MidPoint. This point is between point A and point C. It also must be infinitely far away from point C, else, infinity would be defined by the distance between point C and A multiplied by 2.. which is nonsense. Thus, ANY point in a system in an infinite universe would be infinitely far away from every single point. Nonsense. Please note that when we speak of an infinite universe, it is entirely different from infinity in closed spaces (for instance.. there are an infinite number of numbers between 1 and 2). The difference is that in the math realm, numbers can be generated as needed to produced any number of numbers between 1 and 2. An infinite universe demands an external infinity that is already generated.. which is also impossible.

actually, the rate of the universe's expansion has been under debate for quite some time. any given point that we can see is limited by the speed of light, which is a finite/fixed rate, and thus results in a finite point in space. and, the distance between two known points is most certainly finite.

the universe is expanding infinitely - meaning that there is no known limit to the extent that it can expand, hence why it's characterized as the infinite universe. there are guesses that the rate of expansion during the universe's early days was greater than the currently unknown rate, but - it's just a guess at this point.


looking at a simple exponential curve, it is true that the curve will never actually hit the limit.

[saltinespike]

The only infinite thing I have totally thought through and have agreed with is an infinite series. Let's say you shoot an arrow at a bullseye. It makes it half way there then an quarter of the way there, then an eigth, and a sixteenth. Will the arrow ever hit? Yes. Let's suppose you shot the arrow from 3 feet away and hit right on bullseye. It's an infinite series. You could go to 1/209674326098 and it still would not have hit yet, but it has to hit, what's going to stop it?

Infinite series = 1.