• Jan 23rd 2008, 03:37 AM
uraknai
On the wikipedia site, the solution to one of the paradoxes is outlined as follows:

For the Dichotomy assume that each of Homer's steps takes a time proportional to the distance covered by that step. Suppose that it takes time t seconds for Homer to complete the last half of the distance to the bus; then it will have taken t/2 sec for him to complete the second-last step, traversing the distance between one quarter and half of the way. The third-last step, covering the distance between one eighth and one quarter of the way to the bus, will take t/4 sec, and so on.

based on this I worked out the time as follows:

Total time, $T = t + t/2 + t/4 + ... + t/n$

so $T = t(1 + 1/2 + 1/4 + ... + 1/2^n)$

Let $S = 1 + 1/2 + 1/4 + ... + 1/2^n$

and so $1/2S = 1/2 + 1/4 + .. + 1/2^m, [where m = n+1]$

Subtracting 1/2S from S gives:

$1/2S = 1 + 1/2^m, [where m = n+1]$

So S = 2 as m tends to infinity.

This gives $T = 2t$

My problem is that at the start, we say that t is the time to travel from the halfway point to the end. This will be exactly the same time to travel from the start to the halfway point since he's travelling at a constant speed(hence the solution 2t). The problem involves you to show that to travel from the start to the halfway point is finite but we already assume that t is finite at the start of the solution.

To me this seems fundamentally flawed and the proof is pointless. Can someone explain where I've gone wrong.

Cheers
• Jan 23rd 2008, 03:47 AM
JaneBennet
Quote:

Originally Posted by uraknai
The problem involves you to show that to travel from the start to the halfway point is finite

No, that is not what the problem involves. The problem, rather, involves showing that the whole journey (not just part of it) can be completed in a finite time. That is what the problem is about.

The paradox comes about because we have made things complicated by dividing Homer’s journey into an infinite number of parts so that it appears Homer will have an infinite number of steps to take and may never complete his journey. Fortunately mathematics shows us that the sum of an infinite number of parts can be finite (as it is in this case) and thus resolves the problem we have created for ourselves. ;)