Infinity Problem

Ok, I used to be an ATS veteran, but now I am turned noobie. I left the board for a sports board. Anyways, I apologize if someone already came up with this question!

People have been talking about big things, and one of my theories is going to rock our worlds! Bear with me!

Let's say I am going to the kitchen to get a sandwich. I am 50 feet from the fridge. How can I get to the kitchen if the space can be infinitly divided? This either questions the truth of fractions or the truth of infinity, or perhaps something else.

Another scenario. How does time move. How can a second be called a second when there are infinite decimal points. Where is Brian Greene when you need him!


-Zuz-

Thanks for answering my questions ATSers!

That's an old one:

The dichotomy paradox
Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
This description requires one to complete an infinite number of steps, which for Zeno is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.
Zeno's paradoxes

But here's one for you. If a point has no extent how can we say that a line segment (made of an infinity of points) has a finite, but not zero lenght? And how come a straight line also made of an infinity of points has an infite length?

edit
Oh, and another one!
How come if the square root of 2 = 1.4142.... is an irrational number a segment of that lenght is preciseley determined since is a diagonal in a square with side segments of length 1.

[edit on 25-8-2006 by Apass]

Originally posted by MysticalUnicorn
Ok, I used to be an ATS veteran, but now I am turned noobie. I left the board for a sports board. Anyways, I apologize if someone already came up with this question!

People have been talking about big things, and one of my theories is going to rock our worlds! Bear with me!

Let's say I am going to the kitchen to get a sandwich. I am 50 feet from the fridge. How can I get to the kitchen if the space can be infinitly divided? This either questions the truth of fractions or the truth of infinity, or perhaps something else.

Another scenario. How does time move. How can a second be called a second when there are infinite decimal points. Where is Brian Greene when you need him!

-Zuz-

Deal MysticalUnicorn,

when you divide the remaining distance by 2, then you don't ever reach the far part of the division.

The question you posted is actually a trick question.

I was going to type a response but in fear of not being able to commence it or to finish it I didn't start - but I did and have, or have I....?!

It’s the decision, more than the action perhaps.

Apologies

[edit on 30-8-2006 by one_small_step]

Simple. Decide to walk twice the distance, and give up halfway.

Regarding the distance, our understanding is only hampered by a concept of measurement and fractions. You say we must walk 1/8th, but to walk that, we need to walk 1/16th... Yet we do. Instead of focusing on a number that will be ever smaller, focus on a unit of measurement that matters - if a fridge is three metres away, take the first step that covers a certain amount of those three metres, whether it's 30cm or 50cm.

You don't wanna walk a few picometres each step, hey? If we did, then I can see that paradox having validity - but the point is, there is no barrier saying we "must" travel half the distance. We however, "do" travel the distance. You only travel half the distance if you want to travel half the distance. Saying we must travel half the distance first, then half the distance of that... is only changing your destination - if you say you must first walk half way to the fridge, then your destination isn't the fridge - it's halfway to the fridge... if you say you must travel halfway to halfway, then you're only decreasing the amount of distance to your own destination, and your original point of reference is lost.

Just like someone who's main goal is to produce a logo for a company, if he focused on the end result, he'll finish it. If he only focused on getting the lineart, that's as far as he'd get. It's a matter of focus and perspective - if you focus on what you want, you get it. If you keep making yourself go further and further (or closer and closer as the paradox says) then you'll perpetually be doing that and never achieve the main goal.

Yeah. one_small_step summed it up nicely.


And regarding irrational numbers, that's just how they are. It's a real number, it just can't be expressed accurately by any means other than √2. I'm not sure what question you posed, but it's that because of pythagoras' theorem, a^2+b^2=c^2, therefore,

1^2+1^2=c^2
2=c^2
√2=c

And there you have the diagonal.

Here's a weird one, adding to the diagonal thing, or im jsuit illustrating, not sure.

you start off with a staircase

__
__|
__|
_____|
_____|
________|
________|

the lenght of which is 1 unit, and the height of which is 1 unit. altogether the perimiter is two units

you divide the staircase as so:

_
_|
__|
___|
____|
_____|
______|

it's still a perimiter of two units.
Now you do this an infinite number of times, and you get this:

\
_\
__\
___\
____\
_____\
______\

is the perimiter still 2, or is it as common sense and pythagorean thorem would dictate square root of 2?


[edit on 8.30.06 by ProveIt]

I don't think perimeter's the right word, but I know what you mean.

Since it's not a triangle, and you're not dividing all available lengths by two, you can't keep dividing the length and height of each "step" and still claim it's the same object. I think it's more a logic problem, understanding the question, instead of the question itself.


But onto what you said - it depends what you define a diagonal as. Technically, if all atoms are "rounded" as such, is a straight line really straight? Same with a diagonal. You just need a point of reference to determine if a line is "straight" or "diagonal". In the case of the triangle, if you added more and more steps, no, you wouldn't get a diagonal no matter how far you go, in theory. However, in all practical purposes, it is a diagonal, just like how for all practical purposes, pi is roughly 3.14.


Most of these problems are just about understanding the questions posed, and getting your heads around 'em, because they do have logical answers.

A trickier question that'll stump both logic and understanding, without resorting to nonsensical answers, is "can you imagine anything beyond the barriers of the universe?"

Ok, I'm far from a mathematician, but here's a question I've wondered about for awhile now.

What number correctly defines 1/2 of infinity?

or maybe better phrased like this,.....

infinity divided by two = ????

very interesting thread by the way.

Later,.... Ausable_Bill

From what I've read, in mathematics, infinity isn't a number - it's a limitless set, basically. Like [..., -3, -2, -1, 0, 1, 2, 3, ...], the integer set, it's bounds are infinite. So half of infinity would be where

x = [..., ... < 0 < ..., ...] (infinite set of numbers)

therefore
x/2

You'd have to use algebra. Or take the ∞ sign in place of x, and put ∞/2.

That's my reasoning, anyway.

Originally posted by Xar Ke Zeth

And regarding irrational numbers, that's just how they are. [....] I'm not sure what question you posed....

It wasn't actualy a question, it was only an observation by wich I tried tu underline that even if it's difficult for somebody to apprehend irrational numbers that doesn't mean that these numbers don't exist or are not well defined. About this particular number √2, I saw somewhere a demonstration that it's not a rational number made I think by Pythagoras himself in wich he proved that √2 (or an approximation of it) is an odd and an even number in the same time or something like that..I don't remember exactly.

Regardig "can you understand anything beyond the universe"

Answer is no. The universe is placed in three tangible dimensions of space and one dimension of time, with six "hidden" dimensions. In these dimensions, the universe is infinite. However, this infinity is a mere bubble in an infinite-dimensional space/time complex... thing. Don't know what to call it. Basically, we cannot understand what is "outside" the universe becasuse we live "inside" the universe.

It's like a cartoon character trying to escape out of a flat piece of paper. No matter which way he goes, he can't because he does not posess the ability to move in a direction that is not parralel to the dimensions of his flat 2D universe. Same thing with us, we cannot move in a dimension that is unnatural to us, thus we cannot break free of this invisible barrier. Our universe is likely intersected with infinite dimensions, but we just can't reach them.

Question that'll doodle your noggin is, could you be right now an inch away in an extra-universal dimension from a blazing star in another universe and not even notice it