Is Infinity Truly Limitless or Does It Have Boundaries?

[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?

 

[MeJennifer]

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

Sorry but to understand that a point has no length, area or volume, or that a line has no area or volume, or that a triangle has no volume is mathemathics 101.
I suggest you start with Euclid, he is good!

 

[shmoe]

Sorry but to understand that a point has no length, area or volume, or that a line has no area or volume, or that a triangle has no volume is mathemathics 101.
I suggest you start with Euclid, he is good!

Should I assume that you in fact don't have a definition for length, area, or volume and yet you feel you can declare which objects they are or isn't defined for? Does that not seem strange?

 

[MeJennifer]

Should I assume that you in fact don't have a definition for length, area, or volume and yet you feel you can declare which objects they are or isn't defined for? Does that not seem strange?

The only thing strange here is that some people wish to assert that objects of dimensionality n have properties that apply only to dimensions higher than n.

 

[shmoe]

The only thing strange here is that some people wish to assert that objects of dimensionality n have properties that apply only to dimensions higher than n.

Huh? So 'area' is a 2 dimensional property that should only apply to things of dimension 2 or higher (I guess you are actually saying strictly greater than 2)? That's what you are saying? I'd first ask what's the area of a cube. I'd then ask again what your definition of "area" is that you are willing to conclude what objects have this property?

It's not that difficult a concept. If you want to discuss "area" or whatever thing and you don't even have a definition for this thing, then you are just talking nonsense about nothing. These concepts don't have intrinsic definitions or properties, they have whatever properties that follow from the definitions that we give them, before that they are meaningless.

 

[shmoe]

Huh? So 'area' is a 2 dimensional property that should only apply to things of dimension 2 or higher (I guess you are actually saying strictly greater than 2)? That's what you are saying? I'd first ask what's the area of a cube. .

Ahh, nevermind, ignore this bit. I see what you're saying. The rest still applies.

 

[Hurkyl]

The only thing strange here is that some people wish to assert that objects of dimensionality n have properties that apply only to dimensions higher than n.

What's strange is that some people think we're talking about a property that applies only to "dimensions higher than n".

There really is no room for debate here -- if you have a definition of "length", then we simply appeal to the definition to see whether or not a point has length. And if you don't have a definition of length, then (mathematically speaking) you cannot say whether or not it applies to a point.

When you check the definitions of "length" usually used in mathematics, you find they apply to points. There is no way around this fact.

We can debate metamathematically about the merits of different ways we could define the word "length", and whether or not is desirable to choose a definition that allows us to measure the length of a point, but none of this has any bearing on the fact that "length", as used in mathematics, applies to points.

(e.g. sometimes you might want to measure the length of something when you don't know its dimensionality. And if you find that its length is finite and nonzero, that's a proof that it is one-dimensional)

 

[CRGreathouse]

(e.g. sometimes you might want to measure the length of something when you don't know its dimensionality. And if you find that its length is finite and nonzero, that's a proof that it is one-dimensional)

Quite. An example would be a recursvely defined object that may be a fractal. If it is in fact a fractal it is in some sense 'between' dimensions, having (for example) finite area but an infinite perimeter. The perimeter is more than 1 dimensional.

 

[MeJennifer]

There really is no room for debate here --

That is fine, you folks keep thinking about the length of points and the volume of triangles as if there is nothing wrong with that and leave me in error thinking that a 1 dimensional geometric object does not have a length and that a 2 dimensional geometric object has no volume.
Silly me, I must be far gone.

An example would be a recursvely defined object that may be a fractal. If it is in fact a fractal it is in some sense 'between' dimensions, having (for example) finite area but an infinite perimeter. The perimeter is more than 1 dimensional.

Utter nonsense, a fractal has no area since it is a set of disconnected points.

 

[CRGreathouse]

Utter nonsense, a fractal has no area since it is a set of disconnected points.

...!

Look up Hausdorff dimension and tell me if you still don't believe in infinite length and finite area, or fractonal dimensions for that matter.

 

[matt grime]

Think of a point as emebedded in the real line (or higher dimensions), and its n-dimensional measure (which is it's n-volume) is zero. Same with lines in 3-d and higher. Sorry, MeJennifer, but you are mistaken.

You are also mistaken to say that fractrals have bound no area, and that they are a set of disconnected points.

 

[MeJennifer]

You are also mistaken to say that fractrals have bound no area, and that they are a set of disconnected points.

Where did I mention "bound no area"?
Or do you imply that the area of something and the bound area are identical things?
If they are different things, then perhaps you could explain why you attempt to suggest that I am mistaken about something I did not write?

 

[matt grime]

Sorry, I changed my choice of words half way through. Yes, now I see what you're saying, and I would hate to put words in your mouth.

However, fractals are not, necessarily, disconnected sets. The cantor set is totally disconnected (and is the unique blah blay with this property). However, given a fractal F it is not possible, in general to write to find two open non-intersecting sets A and B with AnF and BnF non-empty. (Of course, the real line is a fractal*, and that certainly is not disconnected.)

* for me fractal means something with self similarity. Perhaps some definitions would expressly exclude the real line as a possible fractal, but it is a matter of convention.

 

[Hurkyl]

That is fine, you folks keep thinking about the length of points and the volume of triangles as if there is nothing wrong with that and leave me in error

It's not my fault that you don't want to listen.

 

[shmoe]

That is fine, you folks keep thinking about the length of points and the volume of triangles as if there is nothing wrong with that and leave me in error thinking that a 1 dimensional geometric object does not have a length and that a 2 dimensional geometric object has no volume.
Silly me, I must be far gone.

Very far gone indeed. Don't worry though, I'm sure burying your head in the sand will make it all go away.

Utter nonsense, a fractal has no area since it is a set of disconnected points.

In case the real line wasn't a satisfying example for a fractal that isn't a bunch of disconnected points,

http://mathworld.wolfram.com/KochSnowflake.html

 

[MeJennifer]

Yes, some fractals are not a disconnected set of points like the Koch snowflake.
The Koch snowflake is a 2 dimensional object, independent of how many iterations you go through.
A Hausdorf dimension is just a different way to define a dimension.