Infinity: Limitless or Limited? Exploring the Concept of Infinity in Mathematics

[Hurkyl]

In topology, we can, for example,add a "+ infinity" and "- infinity" to the real line (the "Stone-Cech compactification"), making it topologically equivalent to a closed interval.

That's the extended reals. The Stone-Cech compactification adds a whole bunch of new points; it's not a very pretty space.

 

[Skhandelwal]

is speed of light infinity in terms of velocity?

 

[StatusX]

is speed of light infinity in terms of velocity?

The speed of light is about 3x10⁸ m/s, which is less than infinity. But you can think of Newtonian mechanics as the limit of relativistic mechanics as c->infinity. Since c is very large by everyday standards, this isn't a bad approximation.

 

[Skhandelwal]

I am sorry, I should have been more specific, is speed of light infinity by standards of velocity for weight objects?

 

[StatusX]

"by the standards" of objects with non-zero rest mass, the speed of light is simply a speed that can't be reached. As it is approached, the mass of the object increases so that it takes more work to accelerate it more, and no amount of work can get it up to c. But the speed is just the number I gave above.

 

[Data]

I am sorry, I should have been more specific, is speed of light infinity by standards of velocity for weight objects?

The speed of light isn't infinite. For example, it takes light about eight minutes to get from the sun to the earth.

It is true that you can't accelerate massive objects to or past the speed of light, because as speed approaches the speed of light, the required kinetic energy increases without bound (ie. to infinity).

 

[CRGreathouse]

The kinetic energy and mass as v --> c both approach infinity, so in a sense yes. In the literal sense, c is quite finite.

 

[matt grime]

is speed of light infinity in terms of velocity?

No, it is a number about 3x10^8ms^-1. That is not even a big number, never mind 'infinity' (please, let us know what you are using infinity to mean, since for everyone else it is something to do with 'not finite').

 

[Skhandelwal]

Btw, one of you said few post ago that we pretty much understand infinity, well, if that is so then how do explain cantor's paradox?

 

[arildno]

Btw, one of you said few post ago that we pretty much understand infinity, well, if that is so then how do explain cantor's paradox?

Eeh, what do you mean now?

 

[Skhandelwal]

Whenever someone introduces a new term like this, it is wise to look it up on google before asking, but then again, I am the one to be blaim b/c I shouldn't expect everyone to look it up. Here is the link: http://en.wikipedia.org/wiki/Cantor's_paradox

 

[arildno]

Well, the answer is given in the link you provided:

Like many mathematical "paradoxes" it is not actually contradictory but merely indicative of a mistaken intuition, in this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of naïve set theory and therefore demonstrates that this theory is insufficient for the needs of mathematics. The fact that NBG set theory resolves the paradox is therefore a point in its favor as a suitable replacement.

 

[matt grime]

Yes, and what is the problem there? I don't see it is either paradoxical or in anyway shows that the definition "something is infinite if it is not finite" is at all not well understood.

 

[CRGreathouse]

Btw, one of you said few post ago that we pretty much understand infinity, well, if that is so then how do explain cantor's paradox?

What's to explain? It just says that there is no largest cardinal infinity. Is there something you don't understand about it?

 

[Skhandelwal]

It is a paradox, that means that it is not understood. Nevermind, that wasn't my point. Since you guys understand infinity, I was hoping if you could clear my questions about it.
1. What does it mean when one reaches infinity faster than the other?(ex. y=x^3 graph compared to y=x^2.
2. For objects that have mass, is reaching the speed of light infinity? If it is then that means infinity have their limits. Just like infinite sum. But wouldn't that mean that infinities aren't really infinity, they are just infinity in their own way? Like whenever you take steps, and decided to take half the step you took before and so on, you will be never get where you want to go. meaning not that you can't go, but not in the way you want to. Am I right?
3. Is infinity times 0=0 for sure or it depends?(are there any exceptions?)
4. Why doesn't one infinity=another?
5. As the parabola y=x^2 increases, if you take the slope of the secant line from its origin to some point. And then if you pick a farther point, slope increases, and then it keeps increasing, well, if you get to infinity, will the slope be undefined? If it would then how would y=x^2 be a function? Because it would fail the vertical line test.

Oh heck, I should post the problem I am having understanding cantor's paradox. As far as I understand, it states that there is one set which consists of all the infinites(the amount of them is infinite by itself) and this set is greater than all of those infinities. Well, how can it be greater than all of them if that set itself is in it?(since all the infinities are in it)

 

[CRGreathouse]

1. If lim f(x) = infinity and lim g(x) = infinity, then f(x) reaches infinity 'faster' when lim f(x)/g(x) > 1.

2. No, the speed of light is not infinite.
3. Infinity times zero is undefined.
4. Different infinities can be of different sizes, just like finite quantities can be of different sizes. This is a philosophical question, not a mathematical one.
5. The slope increases without bound. The limit of the slope is infinite. f(x)=x^2 is a function in the reals and the exteded reals; it passes the vertical line test in the extended reals because f(infinity) = infinity and f(infinity) is not equal to any finite value.


You misunderstand Cantor's paradox. There is no 'set of all sets'; no set contains all sets. There is a proper class of all sets, but as a proper class it need not contain itself (since it isn't a set). There are other set theoretical foundations which solve this differently; type theory is an example. I'm not well informed about type theory, though; you'd haveto ask someone else.

 

[Skhandelwal]

1. I know that but since infinity can't be reached, does it really matter if you are faster or slower?
2. How do you figure that?
3. So what would be a mathematical answer?
4. Does that mean that even though y=x^2 always reaches infinity, it never gets there, b/c if it would, it wouldn't be a func. anymore?

Can someone explain cantor's paradox to me? I thought I had it figured out.

 

[CRGreathouse]

1. I know that but since infinity can't be reached, does it really matter if you are faster or slower?

Yes, because knowing that the functions limit as I described tells you things about their behavior in the finite realm. For any N, there exists m such that for any x > m, f(x) > N * g(x) or something like that.

2. How do you figure that?

The speed of light is less than, say, 10^100 m/s, which is still quite finite.

How about this: at infinite speed travel time would be 0 for any finite distance, but this is not the case for light. It still takes, say, 8 minutes to get from the sun to the earth.

3. So what would be a mathematical answer?

"Undefined", just like 0/0.

4. Does that mean that even though y=x^2 always reaches infinity, it never gets there, b/c if it would, it wouldn't be a func. anymore?

You'll have to rewrite this in a sensible manner before I'll be able to answer. f(x)=x^2 is never infinite for finite x, but it's infinite for infinite x. f is closed in the reals and closed in the extended reals, though.

Can someone explain cantor's paradox to me? I thought I had it figured out.

What do you not understand?

 

[Skhandelwal]

3. I thought #/0 is undefined and 0/0 is undeterminate.
4. What I mean is that if you keep taking secant line from the origin of the function to a farther and farther point, the slope keeps increasing, when the graph achieves(theoretical) infinity, the secant line will be undefined, but that can't happen b/c if that occurs, y=x^2 wouldn't be func. anymore, therefore, y=x^2 never gets to infinity, it always reaches it. So seacant line keeps getting bigger till infinity but never gets undefined. Right?

Well, you basically agreed w/ me on the theory except the fact that the set doesn't contain itself, but if a set consist any number equal to its ordinal number(infinity), each infinity, and if a set consist of infinities of those, well, how is it bigger than those? Since one infinity can't be bigger than other? Sorry I know it is really confusing if you don't understand it, I'll just type it again b/c I am really tired right now.

 

[CRGreathouse]

3. I thought #/0 is undefined and 0/0 is undeterminate.

Use whatever terminology you prefer, though I haven't heard that one. In general both forms are undefined (though values limiting toward those can act differently.)

but that can't happen b/c if that occurs, y=x^2 wouldn't be func. anymore

No, that has nothing to do with x^2 being a function. Further, the secant does exist -- there's no reason that it can't be infinite valued if f is.

My problem with your terminology is that you're not explaining in what number system you're working, nor what your functions do. I've assumed so far you're using the extended reals (R plus two distinguished points +infty and -infty with the usual properties).

y=x^2 never gets to infinity, it always reaches it.

That makes no sense.

So seacant line keeps getting bigger till infinity but never gets undefined. Right?

Yes, it's not undefined at any point. It becomes infinite-valued, in this example, when x is infinite.

Since one infinity can't be bigger than other?

Where do you get this idea? In fact a rather strong converse is true: any (possibly infinite) set is strictly smaller than its power set. That is, no bijection exists from any set to the set of that set's subsets.

 

[matt grime]

I1. What does it mean when one reaches infinity faster than the other?(ex. y=x^3 graph compared to y=x^2.

To make this make sense, we will say f(x) diverges to infinity faster than g(x) (as x tends to something) if f(x)/g(x) tends to infinity too.

2. For objects that have mass, is reaching the speed of light infinity?

"reaching the speed of light" is not "infinity" in any sense of the word. One is a gerund the other a noun. It is the same as saying "is baking a cake" the same as "table"

If it is then that means infinity have their limits. Just like infinite sum. But wouldn't that mean that infinities aren't really infinity, they are just infinity in their own way? Like whenever you take steps, and decided to take half the step you took before and so on, you will be never get where you want to go. meaning not that you can't go, but not in the way you want to. Am I right?

Something is infinite if it is not finite. What is causing you to have issues with that?

3. Is infinity times 0=0 for sure or it depends?(are there any exceptions?)

There is no system of arithmetic I know of where you can even ask this question. It is not something that is permitted.

4. Why doesn't one infinity=another?

Now you want to talk about cardinals? By definition different cardinals are different. They are not anything to do with slope on tangent lines, though, or calculus.

5. As the parabola y=x^2 increases, if you take the slope of the secant line from its origin to some point. And then if you pick a farther point, slope increases, and then it keeps increasing, well, if you get to infinity, will the slope be undefined? If it would then how would y=x^2 be a function? Because it would fail the vertical line test.

infinity is not on the real plane, if you want to do this you need to learn projective geometry.

Oh heck, I should post the problem I am having understanding cantor's paradox. As far as I understand, it states that there is one set which consists of all the infinites(the amount of them is infinite by itself) and this set is greater than all of those infinities. Well, how can it be greater than all of them if that set itself is in it?(since all the infinities are in it)

Not a bad summation. But it just means that the class of all cardinals cannot be a set.

 

[Skhandelwal]

These questions might seem random but they aren't
1. I read a big paper on this that 1/0 is undefined b/c even if 0 tries infinitely, it can never catch upto a real number except zero. Therefore, it is undefined, b/c we havn't really defined it. However, 0/0 is undeterminate b/c it can be any answer or any amount of answer. Depending on the answer... I have seen in calculus too that when it is 0/0, then that means that fraction was factorable and some expressions could cancel out, hence when a fraction has 0/0, it does have an answer, just unsure answer, unlike 1/0 which is like no sol. Therefore, 1/0 is undefined, and 0/0 is undeterminate. However, everywhere I go, even in school, everyone says undefined for both of them, I used to think that they are wrong and I don't want to brag about my intelligence over them on a single topic. But when one of you said that you never heard of it, I have started doubting myself, since I read that so long ago, I am not even sure where it is on the internet, except the fact that I read it from mathworld

2. I read in projective geometry,(has just gotten a book after talking to you guys) that parallel lines meet at infinity, that doesn't make sense to me.

No more "random" questions

3. What you said sort of doesn't make sense to me matt grime, not grametically or even mathematically, but logically, the answer to the last question. I mean why can't all infinities be in a set? Since they started using a symbol for each infinity, how is that a problem? I thought the paradox was that the set is bigger than all of those infinities since infinites are suppose to be equal to each other.(a fact I believed was true but proven to be wrong)

4. Why doesn't 1 infinity = another?(calc.)

5. I heard somewhere that infinity square is undeterminate, how do you figure that?

Btw, I didn't know such a cool subject as projective geometry existed, I don't want to miss out on more stuff I don't know, is there a place where I can get a list of all of kinds of science and math subjects and their descriptions? Like a two column list? Online would be fabulous.

 

[StatusX]

1. I read a big paper on this that 1/0 is undefined b/c even if 0 tries infinitely, it can never catch upto a real number except zero.

Even if it tries infinitely? Poor guy, he just can't win.

 

[Skhandelwal]

lol sometimes, some are just better.:tongue2:

 

[Hurkyl]

Therefore, 1/0 is undefined, and 0/0 is undeterminate. However, everywhere I go, even in school, everyone says undefined for both of them, I used to think that they are wrong and I don't want to brag about my intelligence over them on a single topic.

There's a detail you're missing: the indeterminante form 0/0 isn't a string of symbols that represents the result of dividing one real number by another real number. It is a string of symbols that represents a form that an expression can take in the limit.

For computing limits, you have a theorem that if the limiting form makes arithmetic sense, then doing the arithmetic gives you the answer. Some other forms tell you the limit doesn't exist. Some others give you no information at all, and there are even more bizarre forms. But in all cases, the limiting form is not an arithmetic expression; it is a description of the form of the limit.

Although 1/0 and 0/0 make sense as limit forms, they make absolutely no sense as arithmetic expressions. In the arithmetic sense, they are both undefined.

2. I read in projective geometry,(has just gotten a book after talking to you guys) that parallel lines meet at infinity, that doesn't make sense to me.

The Euclidean plane sits inside the projective plane. The projective plane has "more" points though, which all lie in a single (projective) line. These points are called the "points at infinity", and that line is called the "line at infinity".

When they say "two parallel lines meet at infinity", what they really mean is that if you take two Euclidean lines, and extend them into projective lines, the resulting projective lines will intersect.

One of the axioms of projective geometry is "Every two distinct lines intersect in exactly one point".

Exercise: prove that the process of extending a Euclidean line to become a projective line consists of adding a single point to it.

 

[Skhandelwal]

wow I am not ready for proofs right now, I would need to review old theorems and memorize new ones, thanks for the exercise though, I will try it some time. Btw, 3,4, and 5 still to go. Thanks a lot for your help guys so far.

 

[CRGreathouse]

However, everywhere I go, even in school, everyone says undefined for both of them, I used to think that they are wrong and I don't want to brag about my intelligence over them on a single topic.

Yeah, I wouldn't recommend bragging.

2. I read in projective geometry,(has just gotten a book after talking to you guys) that parallel lines meet at infinity, that doesn't make sense to me.

Hurkyl explains this one pretty well. [itex]\mahbb{P}^1[/tex] is just the real number line with an extra point ("at infinity"), and as you increae dimension you add further lines/planes/etc. "at infinity". The behavior you find strange happens at those points.

I mean why can't all infinities be in a set?

Are you talking about the 'set' of all sets?

http://mathworld.wolfram.com/SetClass.html
http://en.wikipedia.org/wiki/Proper_class

The essential answer is that there are too many.

4. Why doesn't 1 infinity = another?(calc.)

You see, this is where we get into the problem with definitions once again. When I first learned calc, infinity was not a number by my text's definitions. A limit equal to infinity had a particular meaning, as did something 'approaching' infinity, etc.

Certainly, a system can be defined where all infinite quantities are equal. It would be $\mathbb{R}\cup\{\infty\}$. This is similar to the calculus infinities, but not nearly the same. For example, in this system, you have

$$\lim_{x\rightarrow0^+}\frac1x=\infty=\lim_{x\rightarrow0^-}\frac1x$$

which is counterintuitive.

 

[CRGreathouse]

5. I heard somewhere that infinity square is undeterminate, how do you figure that?

In the extended reals, $\infty^2=\infty$. In set theory, $\aleph_0^2=\aleph_0$.

Btw, I didn't know such a cool subject as projective geometry existed, I don't want to miss out on more stuff I don't know, is there a place where I can get a list of all of kinds of science and math subjects and their descriptions? Like a two column list? Online would be fabulous.

http://www.math.niu.edu/~rusin/known-math/index/beginners.html

 

[Skhandelwal]

thanks a lot for the link. Btw, would Squre root of infinity also be infinity? Somebody said that all infinities don't equal each other? How did he figure that out?

Hey Crgreatcourse, how is that counterintuitive, it makes perfect sense to me.

 

[CRGreathouse]

thanks a lot for the link. Btw, would Squre root of infinity also be infinity? Somebody said that all infinities don't equal each other? How did he figure that out?

In the extended reals $\infty^2=\infty$, making infinity a square root of itself. Of course

$$x^2=\infty\Rightarrow x\in\{-\infty,\infty\}$$.

Hey Crgreatcourse, how is that counterintuitive, it makes perfect sense to me.

f(x)=x
g(x)=-x

For all real x > 0, f(x) > g(x). Further, for any fixed y, f(x) > y + g(x) for x > y/2. Yet their limits are equal in this system.

 

[Hurkyl]

One thing that might help is to stop using the word "infinity" -- at the moment, the only time I can remember that noun ever being used in mathematics is:

(1) In the extended real line, where we have the two points $+\infty$ and $-\infty$
(2) In the projective line, where we have a single point $\infty$ at infinity.


Other forms of the word are more appropriate in other circumstances. For example, one often uses the adjective "infinite" to describe something. Infinite literally means "not finite".

This adjective is usually used when there's a natural way to measure the "size" of something (such as counting the number of elements in a set, or the area of a shape). If that size is smaller than some natural number, the object is said to be finite. Otherwise, it's infinite. It is not appropriate to use the word "infinity" to talk about infinite things, and IMHO it's a major source of confusion for the layperson.


Another adjective phrase is "at infinity". I've already shown you where this phrase is used. Keep in mind that the phrase only makes sense as a whole: there is no place called "infinity" for anything to be at.

Btw, 3,4, and 5 still to go.

Think about the answers to (1) and (2) -- if you followed them, you should be able to give a reasonable answer at least two of these questions already.

 

[Skhandelwal]

Well, I don't know how to make that horiz. 8 symbol. I guess all my questions about infinity are solved except one. in black hole, when it compresses something, it compresses it to the point where it has no volume and infinite density. Well, what would be the diff. b/w the infinite density of a 2 gram copper and a 2 million grams of gold?

 

[Hurkyl]

Incidentally, in the projective plane, the line at infinity is tangent to your parabola.

Well, what would be the diff. b/w the infinite density of a 2 gram copper and a 2 million grams of gold?

That's not math, that's physics. (and beyond the domain of applicability of our current physical theories!)

Since we're talking black holes, we're talking GR. The difference between the objects is "frozen" into the gravitational field as the objects pass the event horizon. Whatever happens after that has absolutely no effect on the outside universe.

Furthermore, IIRC, the only qualities that survive this process are mass and charge -- to us observers, redshifting will eventually smear out any other differences the objects might have had. (You might want to ask this question in the relativity forum, though, and maybe someone better qualified will answer)

 

[CRGreathouse]

Well, I don't know how to make that horiz. 8 symbol.

Quote my post.

The lemniscate is $\infty$ or

$$\infty$$

In HTML it's ∞, but the boards don't seem to allow that.

 

[Skhandelwal]

what I don't get is how can two different amount can turn into the exact same?(like gold and copper) About the charge, are you saying if there were a specific isotope of copper, the object will mantain that charge? btw what does GR and IIRC mean?

How do I do it w/o quoting your post?