Zeno's paradox.?

Question

If I were to tell you that you would be making a trip, on this trip you will be crossing a certain number of checkpoints. Then I was to tell you that this number of checkpoints is infinity, it would be safe to assume that the total trip would be of a infinite distance, and it would take you a infinite amount of time. But, the trip is only 100 feet long. What I did was drew a starting line and finishing line exactly 100 feet apart. Then I put the first check point halfway, at 50 feet, then the next checkpoint between the halfway mark and the end at 25 feet. Then the next at 13 1/2 feet, then the next at 6 3/4 feet... etc, each exactly half way between the last checkpoint and the finishing line. Since a number can be divided in half a infinite amount of times, then there could be a infinite amount of checkpoints along a 100 foot stretch. I know this logic is wrong, but I've yet to find a web site explaining why or how it's wrong.

Another example is a house with the bottom floor being one story tall, the second floor is half a story, the third is 1/4 a story..and so on for infinity. So the total house is made up of an infinite amount of floors but it never reaches the height of two stories. So if you put a roof on this house, over which floor is it built?

Answer 1

Sounds like you are about to, or just have, started to learn about limits.

If you want to make it even weirder, consider that the time it takes to go half a remaining distance also decrease by 1/2 (for constant motion) so eventually time would have to stop for you to get to the 100 ft. mark (that was how this paradox was presented to me first).

You should be able to derive an answer of this by setting up a limit to the condition.

Answer 2

There's no paradox. You can put as many checkpoints as you want provided they are small enough. The start and finish lines are still 100 feet apart no matter what. Ultimately, the size of your checkpoint will limit the number you can fit. If your checkpoints are truly just points, then you can have an infinite number. So what?

Same with the house. The roof will be a certain height. How many floors you can cram in is an engineering issue, not a paradox.

As someone pointed out before, it only seems like a paradox until you learn enough math to understand that an infinite series (sum of the numbers in an infinite sequence) can have a finite value.

Answer 3

That kind of problems represented a headache for philosophers until the invention of infinitesimal calculus by Fermat, Newton and Leibniz. Then they could explain that sometimes a sum of infinite terms of infinitesimal order can result in a finite value.

On the other hand you could actually divide the distance in an infinite number of parts. Space is not continuous as some greeks believed. You will be forced to stop at the atom level.

Answer 4

Its a paradox thats the point :) You know the logic is wrong but the mathematics say otherwise. The math involved in the particluar Zenos paradox (there are 8) that you are talking about is this...ok, Zenos mathematical sequence is like this:

{...., 1/16, 1/8, 1/4, 1/2, 1}

mathematically, since its an infinite sequence its impossible to have a starting point to even begin the trip. I know its wierd but thats why its called a paradox :)

Answer 5

When you draw a line, it can be in an infinite number of positions between start and end. However, when the lines got too close together the atoms would eventually touch and you would end up with one solid line.

Basically, although space is infinitely divisible, atoms are not and therefor you cannot keep drawing lines and have them still seperated by space.

Answer 6

what you have is the difference between mathematics and engineering (or any other real world application of math)
In 'real world' problems there is a resolution that is acceptable, depends on scale of the situation. walking across a room , a half at a time, while mathematically infinite, the person gets to the other side rather quickly. so you can wind up with some what non-logical outcomes when modelling real world phenomena with theoretical concepts, like 99.999% pure or 99% confidence interval

Answer 7

I have asked that exact question. I use the analagy of moving your thumb and index finger towards each other, but only closing half the distance each time. Your two fingers will continue towards one another for eternity, but never touch.

Answer 8

Wish I had a dollar for every time I've heard the phrase "A series converges if its sequence of partial sums converges".

Answer 9

It is built over all of them.