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

[CRGreathouse]

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?

Isotopes change neutrons. Charge is about (protons and) electrons. (Maybe you were thinking 'ion' instead of 'isotope'...?)

The total charge and mass inside a black hole all act 'together' for the purpose of interacting with the outside world. We don't really know how they act inside the black hole, but that has nothing to do with the mathematical concept of infinity.

btw what does GR and IIRC mean?

General relativity, Einstein's laws of how the universe functions on a large scale.
IIRC = If I recall correctly

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

If you quote my post you'll see what I typed, then you can type the same thing.

 

[matt grime]

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...

Once more you are confusing different symbols. In arithmetic, 0/0 and 1/0 do not make sense. 3, and 2.5234 do. However it is possible to take limits in f(x)/g(x) at a point where f(x)=g(x)=0 and get a meaningful answer. By laziness people refer to this as 0/0, but it has a strict meaning that you are forgetting.

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.

That is because you are thinking of non-projective geometry and applying the results to projective geometry. They are different things, so you need to have different intuition.

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?

Because it leads to a contradiciton. Not all objects form a set in the mathematical sense of the word, and the class of cardinals is 'too big' to be 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)

The paradox is one about sets of sets, not your misunderstanding of cardinals.

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

Look, infinity should not be considered, by you, as an 'object' in calculus. Stop saying it is. When we write limf(x) = infinity we are using the symbol to describe a specific property of f(x). In calculus if we say two things both 'diverge to infinity' then the usage of infinity is the same but it does not make sense to equate the 'infinities since they are not numbers that can be equated. Once more it is merely a convenient short hand to write lim f(x)=infinity.

Now, there are extended systems in (complex) calculus which add one or two symbols that we can manipulate as though they were part of the real or complex numbers, and then there is exactly one symbol, infinity, or two, plus and minus infinity.

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

Put it in context. Someone's playground idea that infinity plus one is infinity is not the same as the proper usages in mathematics of infinite cardinals or divergent sequences/series/functions or geometry. (It doesn't even make sense to multipliy points in a plane, for instance).

Everytime you see the symbol for infinity ask yourself is this actually saying that something is 'infinity' like sin(0) is 0, or is it saying something about a property of the object, such as 'it is not finite', or 'increases without bound'.

 

[Skhandelwal]

All my questions about infinites has been answered, I finally understand it, but one flaw remains. In black hole, if a 2 gram copper is compressed to infinite density w/ no volume and if a 2 million gold is compressed to infinite density w/ no volume. How can they be the same?

 

[matt grime]

Try asking that in a physics forum, since it has no mathematical content.

 

[CRGreathouse]

would Squre root of infinity also be infinity?

[MEDIA=youtube]7Ft4Ogih2vs[/MEDIA]&mode=related&search=[/URL]
:-p

 

[Skhandelwal]

ok, how about this, since the size of every point is undefined, how can it make up a line? Here is what I have come up, the term, undefined is used b/c its size is not zero, but rather infinitely small. Well, since there are infinite of them, I think it makes up for it. But by that way, zeno's paradox shouldn't make sense.(stating that if you have to go from 1 place to another, you take it's half distance, then you take the half distance of that, and then you keep taking half till infinity but never get there.) I am sorry but this is one place where infinity does confuses me.

 

[Hurkyl]

ok, how about this, since the size of every point is undefined

What notion of "size" are you using? A point has a cardinality of 1, zero length, zero area, and zero volume.

I can't think of any reasonable notion of size that one might ever want to use in the same sentence as the word "point" for which the point's size would be undefined.

how can it make up a line?

Why do you think this has any relationship to what precedes it?

 

[Skhandelwal]

It doesn't, all the rest of my questions has been answered, just this remains. Perhaps an analogy would help to understand a point. in a black hole, when something is sucked, it turns into a point, that is to say, no volume, infinite density. But no matter how many add up, that remain in same region, that is to say, no matter how much black hole sucks, its size doesn't increase. Then how come infiniteous points make up a line if their volume is zero?

The only way it can make sense to me is that if cartesian coordinate system is flawed, thus it needs revision. lol But I don't think I guy like me is going to challange a guy like him.

 

[CRGreathouse]

ok, how about this, since the size of every point is undefined, how can it make up a line? Here is what I have come up, the term, undefined is used b/c its size is not zero, but rather infinitely small. Well, since there are infinite of them, I think it makes up for it. But by that way, zeno's paradox shouldn't make sense.(stating that if you have to go from 1 place to another, you take it's half distance, then you take the half distance of that, and then you keep taking half till infinity but never get there.) I am sorry but this is one place where infinity does confuses me.

A point has a (hyper)volume* of 0. It's not undefined.

I'm not entirely sure how to understand your question. I think the answer is that you are succumbing to the fallacy of composition.

The problem with the Zeno paradox you mention (there are several) is the underlying assumption that the sum of an infinite number of lengths is infinite.

* Actually, I suppose it has a 0-volume of 1, but I suppose that's not what you meant.

Perhaps an analogy would help to understand a point. in a black hole, when something is sucked, it turns into a point, that is to say, no volume, infinite density. But no matter how many add up, that remain in same region, that is to say, no matter how much black hole sucks, its size doesn't increase. Then how come infiniteous points make up a line if their volume is zero?

1. There's no reason to assume that a black hole is actually compressed to a literal point. It acts like a point mass in the same way the Earth does if you aren't inside it. Most physicists I've read are rather agnostic on this point; in fact, considering that serious consideration has been given to the theory that our observable universe is a black hole, there are many who feel physical processes would go on 'as usual' inside black holes. Of course this has nothing to do with math.

2. Given that our observable universe has finite mass, any black hole in same must have finite mass. Thus if it were to have literal infinite density, it must have zero volume, even if our entire universe, as such, fell into it. This, again, has nothing to do with geometry.

3. A line has 0 volume, but I'll assume you mean length (1-volume). Then your question becomes "Why does a line have length if its constituent parts have no length?". Once again this is the fallacy of composition. I'll offer a nonmathematical analogy back: Are any of the atoms that make up your body living beings? If not, then how are you a living being?

 

[Skhandelwal]

Simple, b/c they are make up the molecules, which make up a lot of stuff which make up dna which makes cells and that makes us. See, now we get into biology, you see, when I was referring to black hole, I was just referring there to try to understand point, if I were to ask the same question in physics thread, they'd say go to math. Well, I got to ask somewhere. What I don't get is that "Why does a line have length if its constituent parts have no length?" The analogy you made, I answered it, I don't think that applies to lines though.

 

[matt grime]

if I were to ask the same question in physics thread, they'd say go to math.

No, they would not.

Maths is a language in which we model things that physicsts (and others) care about. It offers no causal explanations. You are not asking about the model you are asking about the physics. If you ask what happens when we go beyond the mathematical bounds of some given model (which you are doing), then that is a question for physicists to asnwer.

What I don't get is that "Why does a line have length if its constituent parts have no length?"

That is not our fault. A line is an uncountable collection of points. There is nothing that says that length is an additive property that commutes with arbitary unions. Indeed it does not. It does not even make sense to think about numerically adding up lengths like this and use your intuition that is developed purely from adding up a finite collection of numbers. There is measure theory if you want to study it, but it is not something you will understand very easily (no one does at first).

 

[Skhandelwal]

So you were saying I should have asked in physics thread why do points make up a line? Now that you have answered my question(to study measure theory), I'll look into that, thx. Btw, what level of study is that?

 

[Data]

Now that you have answered my question(to study measure theory), I'll look into that, thx. Btw, what level of study is that?

The first course my university offers in measure theory is listed as a fourth-year honours math course (I actually wanted to take it, but it doesn't quite fit into my schedule!).

 

[CRGreathouse]

Simple, b/c they are make up the molecules, which make up a lot of stuff which make up dna which makes cells and that makes us. See, now we get into biology, you see, when I was referring to black hole, I was just referring there to try to understand point, if I were to ask the same question in physics thread, they'd say go to math. Well, I got to ask somewhere. What I don't get is that "Why does a line have length if its constituent parts have no length?" The analogy you made, I answered it, I don't think that applies to lines though.

You answered that we're living beings despite our atoms not being so because we're (recursively) built up from atoms. I don't see how that answers my question, but I'll offer the same back to you: lines have length even though points don't because they are built up from points. Neither makes sense to me as answers to their respective questions, but apparently they make some sense to you. If you still have questions about this, please address them in my question to you first. That way I'll hopefully understand what you are asking.

 

[Skhandelwal]

What I am asking is really simple, why are you trying to make it complicated? atoms make up a lot of stuff(molecues, etc.) and they make us. The reason atoms arent living beings is b/c every living being needs to have a dna and dna is made up of atoms.

Everyother person I asked this question to understood at once, you are the only person who is having difficulty.

 

[CRGreathouse]

The reason atoms arent living beings is b/c every living being needs to have a dna and dna is made up of atoms.

Fine. DNA isn't a living being, random cytoplasm & the other stuff that makes us up aren't/isn't a living being, but you are made up of both and are a living being.

I think 'everyone else' agrees that you have fallen to the http://www.nizkor.org/features/fallacies/composition.html.

 

[Skhandelwal]

Look, i don't want to start a war here, so ok, I have fallen so deep in the fallacy of composition that I will never get out. happY?

 

[CRGreathouse]

Look, i don't want to start a war here, so ok, I have fallen so deep in the fallacy of composition that I will never get out. happY?

I just want to understand you and have you understand me. Points need not have length to make up an object with length.

So you were saying I should have asked in physics thread why do points make up a line?

I think that the black hole question should be asked in a physics thread. Points making up a line could be classical geometry, set theory, analytic geometry, measure theory, epistomology, or logic (but not physics).

 

[matt grime]

So you were saying I should have asked in physics thread why do points make up a line?

I was referring to your question about black holes, atoms and such.

If you think I was referring to lines and points, then the answer is: you're talking abuot maths. The answer is 'because it is'. The length of the line segement [a,b] is *defined to be b-a, and the length of the point is therefore 0 (point being the line segment [c,c]). It is a consequence of the definition. If you don't like it, well, I don't know what to say.

 

[Skhandelwal]

I guess what you saying has a point but I still don't like the idea of a point not having any lenght. I guess for that, i would really have to study measure theory to understand that.

I was studying some advanced math last night and then I encountered the term Transfinite numbers. I asked about that earlier in the forum and some folks helped me out but their explanation was so upper level that even my teacher couldn't understand it.(not that it is a bad thing, I wanted a simpler defination, or else, I would have understood it on my own) However, when my teacher looked it up on his on, he told me, that transfinite numbers in which when you add infinity, order matters.
Like 1+infinity=infinity but infinity+1>infinity. Now this doesn't make sense to me.

 

[Hurkyl]

I guess what you saying has a point but I still don't like the idea of a point not having any lenght. I guess for that, i would really have to study measure theory to understand that.

The brief overview is this:
(1) First, you try and figure out how to define the notion of length.
(2) You then prove that points have zero length.

(Aside: it's wrong to say that "points have no length". Points do have a length, and that length is zero. The phrase "X has no length" means that the concept of length isn't even applicable to X)




Anyways the first thing you need to learn is to stop calling things "infinity".


The second thing you need to learn is that, in mathematics, numbers aren't "god-given" -- whenever we want to do any sort of arithmetic, we must first define our numbers.


(Incidentally, "transfinite" is just a synonym for "infinite")


The thing your teacher found is called the "ordinal numbers", which are a subclass of things called "order types". The ordinal numbers describe orderings. For example, the ordering

* < * < * < * < *

is the ordinal number "5". (Yes, we use the same symbols for the natural numbers and for the finite ordinals) (each * denotes an arbitrary object)

Another ordering is

* < * < * < * < ... |

where I've used the pipe (|) to denote that the sequence keeps going infinitely. An example of something with this ordering is the natural numbers. This is the ordinal number $\omega$.

Another ordering is

| ... < * < * < * < *

This one is an order type, but it's not an ordinal number. An example of this ordering is the negative integers.

Another ordering is:

A < B < C < ... | *

Again, each of the symbols denote an arbitrary object. In this ordering, the object "*" comes after every other object. (So, for example, C < *) Note that * has no predecessor. This is the ordinal number $\omega + 1$. An example of a set of numbers with this ordering is
{1} U {1/2, 2/3, 3/4, 4/5, 5/6, ...}


I hope I've adequately described what an order type is. It turns out that there are reasonable ways to define addition and multiplication on order types. (And even exponentiation, I think) I will only describe addition, since it's very easy.

Addition is performed simply by concatenating things. For example, the order type 3

* < * < *

plus the order type 5:

@ < @ < @ < @ < @

is the order type 3+5

* < * < * < @ < @ < @ < @ < @

which is equal to the order type 8. We can add the other way, to get the order type 5 + 3:

@ < @ < @ < @ < @ < * < * < *

When we're looking at finite orders, this all behaves just like the natural numbers. But for infinite things, consider the order type 1:

*

and the order type $\omega$:

* < * < * < * < * < ... |

The two possible ways of adding them gives:

1 + $\omega$:

* < * < * < ... |

$\omega + 1$:

* < * < * < ... | *

You can (hopefully) see that $1 + \omega = \omega \neq \omega + 1$.

 

[matt grime]

I guess what you saying has a point but I still don't like the idea of a point not having any lenght. I guess for that, i would really have to study measure theory to understand that.

But it isn't hard: if we take a line segment (a subset of the real line), from a to b (this is [a,b]), then its length is obviously (what else could it be?) b-a, the end point minus the start point. If the beginning and end are the same point (i.e. a point), then the length is zero.

However, when my teacher looked it up on his on, he told me, that transfinite numbers in which when you add infinity, order matters.
Like 1+infinity=infinity but infinity+1>infinity. Now this doesn't make sense to me.

Why doesn't it? It can only not make sense for one of two reasons

1) you're misapplying some previous knowledge in a situation that it says nothing about

2) you didn't find out what the definition of the objects in question is.

If you don't know what you're talking about (and that is written in the literal not the derogatory sense), then you cannot possible *know anything about it*. It is wrong to say it doesn't make sense to you. It is more correct to say that you dont' know enough about the objects in question to see why this should be.

To draw a crappy analogy. Suppose we have coloured building blocks (Lego^TM), and the colours correspond to some property, and sticking the together is addition. Putting a green block on top of a red on creates a different object that putting a red one on top of a green one.

The main problem is that you think that since we use the symbol +, it must behave with precisely the same properties as before for all new objects on which it is defined. That simply does not have to be true. It is interesting to find out what properties extensions of a definition share with the original, that is what a lot of research is about.

If someone gives you some information like this (that is reasonably reliable), and you find sometihng puzzling, don't say 'that doesn't make sense to me', say 'hmm, that seems strange, I wonder why that is'.

 

[Skhandelwal]

I got it about the transfinite numbers but for line having no lenght(I don't get how saying no length and zero is any diff., even though you specified the reason, a point is undefined, w/ zero volume, meaning, he doesn't have a dimension for lenght.), What you said makes sense to me, you know, doing arithmetic calculations, but it doesn't make sense to me imaginatively. That is to say, you have proven your point by calculative evidence. But when I think about, my mind just have trouble believeing that infinity amount of points w/ no length can make up a line/line segment.

 

[matt grime]

In what sense is a point 'undefined'? What do *you* mean when you say something 'doesn't have a dimension for length'?

 

[Skhandelwal]

Well Hurkyl stated, "(Aside: it's wrong to say that "points have no length". Points do have a length, and that length is zero. The phrase "X has no length" means that the concept of length isn't even applicable to X)" So I was replying to him that if something has 0 lenght, I would assume that the concept of length isn't even applicable to it. If my assumption is wrong, then give me a counterexample.

 

[radou]

Banal analogy: A car rests in the parking lot. It has zero velocity. The concept of velocity isn't even applicable to the car.

 

[matt grime]

A point has a length, that length is zero. Zero is a prefectly valid value for a measurement to take. However, that doesn't answer either of my two queries where you used terms in a manner that I found puzzling.

 

[Skhandelwal]

if I would have said the car has no velocity instead of saying 0 velocity, I agree, that's not valid, but when I talk about lenght, I believe I was being pretty valid.

 

[lafrench34]

Does infinity have direction? Dimensions? As we all know it does, then doesn't it mean that infinity is actually limited? Just like values b/w 1 and 2 x values are infinity but have a total sum.

here we go again with this

 

[matt grime]

I believe I was being pretty valid.

Good for you. You possibly appear to be wrong, but I've lost track of what you were saying.

 

[Skhandelwal]

Ok, I am going to summarize everything up for ya. This guy said that I shouldn't say it has no lenght, I should rather say it has 0 length b/c no length means it is not even applicable for it. Well, I said that if you were talking about velocity, it would make sense but I really don't see how if something has 0 length can be applicable to lenght.

 

[MeJennifer]

A point has a length, that length is zero.

Well Matt Grime I have to respectfully disagree with you on that.

A point has no length just as much as a line has no area or a triangle has no volume.

Length, area and volume are simply not defined for a point.
You cannot simply add some additional dimensions and then say that its length, area and volume adds up to zero. By doing this you add properties to a point that it does not have.

 

[shmoe]

A point has no length just as much as a line has no area or a triangle has no volume.

Area is perfectly well defined for a line (it's zero), as is volume for a triangle (also zero).

I have to wonder what definition of area (and length, and volume) you are using that is different from the one mathematicians use that makes you think it's not defined for a line?

 

[MeJennifer]

Area is perfectly well defined for a line (it's zero), as is volume for a triangle (also zero).

I have to wonder what definition of area (and length, and volume) you are using that is different from the one mathematicians use that makes you think it's not defined for a line?

A point is a zero dimensional mathematical object, the property of length does not exist in zero dimensions, it requires at least one dimension. Similarly with area and volume, those need resp. 2 and 3 dimensions minimally.

Think about it how can for instance a triangle have a volume? It is a two dimensional object, two dimensional objects do not have a volume, not even a volume of 0. It is simply a property that does not exist for a triangle. Only objects that are 3 dimensional (or higher) can have a volume.

 

[shmoe]

So what *are* your definitions for length, area, and volume?