If I must walk halfway to the door before exiting the door, and then halfway again, how do I ever reach it?

Question

If I start 10 feet away, I have to go half way first, 5 ft away. Then, i have to go half way again: 2.5 feet. Eventually it'll get to the point where I'm only .000000000001 inches from the door. But there's always a half way: .0000000000005 inches in this case. How do I ever cross the threshold?

Answer 1

Zeno thought something like this.

This can be rephrased:

How can we pass an infinite number of points in a finite time?

My answer:
The infinite number of points exist anywhere, hypothetically, so you are basically combatting infinity with infinity (your shoes have infinite number of points too)... and you win!

Answer 2

You reach the door because contrary to what Euclid said about points having no extent (exact words are: "A point is that which has no part"), points do have an extent and by taking each step, you are traversing those points finitely.

The fact that Limit 1/2 + 1/4 +.... = 1 has nothing to do with you exiting the door. Zeno's paradox is not really a paradox because Zeno assumes (erroneously like most so called mathematicians do today) that points have no extent. In reality, there is no such thing as half-way because once there is a half-way, then you have already divided the distance into two parts and consequently you have created two points each of the same extent. But points do not have an extent?! Make sense?

Answer 3

This paradox is solved by summing up an infinite series. Suppose that it takes you 1 second to walk 5ft, .5 seconds for 2.5ft, etc. Then the total time it will take you to walk 10 feet will be:

∑_{i = 0}^{∞} (1/2)^i

This is a geometric series whose sum is 2. Therefore it will take you two seconds. Of course you get the same answer by noting that if you can walk 5 feet in 1 second, you can walk 10 feet in 2. The proof that the above geometric series converges to 2 can be done, for instance, using calculus.

Answer 4

The answer is simple - your travel time is linear. You do not stop at each halfway point, which is exactly why Zeno's paradox fails. When you think of it, you unconsciously assume that each step takes an equal time, no matter how short it is.

So the first half of the trip takes one unit of time, and the second unit of time covers all the other halves in one stroke.

Answer 5

This is an old paradox: Zeno's Paradox. But the point is that you do reach it. At some point your step is greater than the denominator you're dividing the distance by: at that point you've reached or passed the doorway.

Answer 6

this is the principle of limits and something tending to zero and not equal to zero!!!

Answer 7

try one giant leap for mankind

then the door will suck you into its vortex before you know it