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traveling half distance

Kahvan
Posts: 1,339
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3/24/2010 10:17:28 AM
Posted: 6 years ago
It is theoretically possible for one object to travel for an infinite amount of time in one direction. This is possible because the object can always travel half the distance to the object making it travel forever in that one direction without it reaching its destination.
Floid
Posts: 751
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3/24/2010 10:32:57 AM
Posted: 6 years ago
At 3/24/2010 10:17:28 AM, Kahvan wrote:
It is theoretically possible for one object to travel for an infinite amount of time in one direction. This is possible because the object can always travel half the distance to the object making it travel forever in that one direction without it reaching its destination.

It sounds like you are mixing one of Zeno's paradoxes with a truism.

It is true that one can travel for an infinite amount of time in one direction because we know of nothing that limits one abilities to travel any further in a particular direction. So if you traveled and infinite time in a given direction you would be traversing an infinite distance in that direction.

Now, one of Zeno's paradoxes states that it is impossible to go from point A to point B because between point A and point B there is a midpoint C, then between point A and point C there is a midpoint D, etc. So between any two points there is an infinite number of midpoints. Logically, in order to travel to point B from point A one must pass through every midpoint, but since there is an infinite number of midpoints this is impossible. Of course, if you really think about it there really isn't that much of a problem, it merely highlights the fact that we can't discretely model a continuous world with 100% accuracy.
mongeese
Posts: 5,387
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3/25/2010 7:42:11 PM
Posted: 6 years ago
At 3/24/2010 10:32:57 AM, Floid wrote:
At 3/24/2010 10:17:28 AM, Kahvan wrote:
It is theoretically possible for one object to travel for an infinite amount of time in one direction. This is possible because the object can always travel half the distance to the object making it travel forever in that one direction without it reaching its destination.

It sounds like you are mixing one of Zeno's paradoxes with a truism.

It is true that one can travel for an infinite amount of time in one direction because we know of nothing that limits one abilities to travel any further in a particular direction. So if you traveled and infinite time in a given direction you would be traversing an infinite distance in that direction.

Now, one of Zeno's paradoxes states that it is impossible to go from point A to point B because between point A and point B there is a midpoint C, then between point A and point C there is a midpoint D, etc. So between any two points there is an infinite number of midpoints. Logically, in order to travel to point B from point A one must pass through every midpoint, but since there is an infinite number of midpoints this is impossible. Of course, if you really think about it there really isn't that much of a problem, it merely highlights the fact that we can't discretely model a continuous world with 100% accuracy.

The problem there is that points have no length of time, so it takes no length of time to pass through a point. The length of time between two points is the only thing that concerns anybody.
homework
Posts: 31
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4/6/2010 3:37:13 PM
Posted: 6 years ago
At 3/25/2010 7:42:11 PM, mongeese wrote:
At 3/24/2010 10:32:57 AM, Floid wrote:
At 3/24/2010 10:17:28 AM, Kahvan wrote:
It is theoretically possible for one object to travel for an infinite amount of time in one direction. This is possible because the object can always travel half the distance to the object making it travel forever in that one direction without it reaching its destination.

It sounds like you are mixing one of Zeno's paradoxes with a truism.

It is true that one can travel for an infinite amount of time in one direction because we know of nothing that limits one abilities to travel any further in a particular direction. So if you traveled and infinite time in a given direction you would be traversing an infinite distance in that direction.

Now, one of Zeno's paradoxes states that it is impossible to go from point A to point B because between point A and point B there is a midpoint C, then between point A and point C there is a midpoint D, etc. So between any two points there is an infinite number of midpoints. Logically, in order to travel to point B from point A one must pass through every midpoint, but since there is an infinite number of midpoints this is impossible. Of course, if you really think about it there really isn't that much of a problem, it merely highlights the fact that we can't discretely model a continuous world with 100% accuracy.

The problem there is that points have no length of time, so it takes no length of time to pass through a point. The length of time between two points is the only thing that concerns anybody.

The length is halved every time you have to get a new midpoint, so time (no matter how small) is still used to do this travelling. Yes, we can't discretely model a continuous world 100%, but this still doesn't explain why the logic does not conform to the reality. Even though the time to travel to each subsequent midpoint is smaller, there are still an infinite number of journies, which means it should theoretically still take an infinite amount of time.
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Puck
Posts: 6,457
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4/7/2010 1:56:59 AM
Posted: 6 years ago
At 4/6/2010 3:37:13 PM, homework wrote:

The length is halved every time you have to get a new midpoint, so time (no matter how small) is still used to do this travelling. Yes, we can't discretely model a continuous world 100%, but this still doesn't explain why the logic does not conform to the reality. Even though the time to travel to each subsequent midpoint is smaller, there are still an infinite number of journies, which means it should theoretically still take an infinite amount of time.

If you are travelling, you are travelling at the speed of something. So no, it's not infinite.
Ore_Ele
Posts: 25,980
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4/7/2010 5:52:50 PM
Posted: 6 years ago
At 4/6/2010 3:37:13 PM, homework wrote:
At 3/25/2010 7:42:11 PM, mongeese wrote:
At 3/24/2010 10:32:57 AM, Floid wrote:
At 3/24/2010 10:17:28 AM, Kahvan wrote:
It is theoretically possible for one object to travel for an infinite amount of time in one direction. This is possible because the object can always travel half the distance to the object making it travel forever in that one direction without it reaching its destination.

It sounds like you are mixing one of Zeno's paradoxes with a truism.

It is true that one can travel for an infinite amount of time in one direction because we know of nothing that limits one abilities to travel any further in a particular direction. So if you traveled and infinite time in a given direction you would be traversing an infinite distance in that direction.

Now, one of Zeno's paradoxes states that it is impossible to go from point A to point B because between point A and point B there is a midpoint C, then between point A and point C there is a midpoint D, etc. So between any two points there is an infinite number of midpoints. Logically, in order to travel to point B from point A one must pass through every midpoint, but since there is an infinite number of midpoints this is impossible. Of course, if you really think about it there really isn't that much of a problem, it merely highlights the fact that we can't discretely model a continuous world with 100% accuracy.

The problem there is that points have no length of time, so it takes no length of time to pass through a point. The length of time between two points is the only thing that concerns anybody.

The length is halved every time you have to get a new midpoint, so time (no matter how small) is still used to do this travelling. Yes, we can't discretely model a continuous world 100%, but this still doesn't explain why the logic does not conform to the reality. Even though the time to travel to each subsequent midpoint is smaller, there are still an infinite number of journies, which means it should theoretically still take an infinite amount of time.

Limits is what disproves this, the first thing you learn in first year calculus.

Here's a crash course on this particular example.

Assuming you are walking at a constant speed (it can still be proven with acceleration, but that only makes it more complex). You are walking from point A to point B (we'll call it AB) and it takes X time. This "paradox" (going 1/2 then 1/2 of whats left, then 1/2 of what's left) doesn't truely take infinate time, it takes X time, just broken down into infinate points of time.

When you are 1/2 AB, it takes 1/2 X, then when you do 1/2 of whats left, that is 1/4 AB more (1/2 or 1/2 = 1/(2^2) = 1/4) which takes 1/4 X, and so on to 1/8 AB and 1/8 X and so on some more (1/16, 1/32, 1/64, 1/128... 1/1048576...).

Well, that comes out to 1/2 X, then 1/4 X, then 1/8 X, and so on. And if you add that all together (to get the total time, rather then just each increment of time) you get X(1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...). Now, you can do the math yourself for the first few to see a pattern. 1/2 -> 3/4 (1/2 + 1/4) -> 7/8 (1/2 + 1/4 + 1/8) -> 31/32 -> 63/64 ->...-> 1023/1024 ->...-> 1048575/1048576 ->... You'll see that it get's closer and closer to 1, but never quite reaches it.

What does that mean? That the time it takes to walk AB, as you approach your destination (point B) gets closer and closer to time X (which is defined, not infinate).
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omelet
Posts: 416
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4/7/2010 10:38:56 PM
Posted: 6 years ago
At 4/6/2010 3:37:13 PM, homework wrote:
Even though the time to travel to each subsequent midpoint is smaller, there are still an infinite number of journies, which means it should theoretically still take an infinite amount of time.

No. Just take the sum of the infinite series 1/(2^n) from 1 to infinity.

That's 1/2 + 1/4 + 1/8 + 1/16, etc., with infinite terms. And guess what the sum is? It's one. It's not infinity. There may be an infinite number of terms, but the fact that they get smaller and smaller makes it so that, in this case and many others, the sum is finite rather than infinite.