Dichotomy paradox – Wikipedia

The Dichotomy paradox is a paradox formulated by Zénon d’Elée during Antiquity:

“When equal masses move at the same speed, some in one direction, others in the opposite direction, along equal masses and which are motionless, the time that the first put to cross the immobile masses is equal to the double of the same time. »»

Different version: the paradox of the stone launched towards a tree, is a variant of the previous one. Zénon is held eight meters from a tree, holding a stone. He launches his stone in the direction of the tree. Before the pebble can reach the tree, it must cross the first half of the eight meters. It takes a while, not zero, to this stone to move over this distance. Then, she still has four meters left to travel, of which she first accomplishes half, two meters, which takes her some time. Then the stone advances by one more meter, progresses after half a meter and still a quarter, and so on to infinity And each time with a non -zero time. Zeno concludes that the stone will not be able to hit the tree, since it would be necessary for an infinite series of steps to be effectively crossed, which is impossible. The paradox is resolved by maintaining that the movement is continuous; The fact that it is infinitely divisible does not make it impossible.

In addition, in modern analysis, the paradox is resolved using the fact that the sum of an infinity of strictly positive numbers can be finished, as is the case here where the numbers are defined as the terms of a suite tending Towards 0.

Paradox resolution by modern analysis [ modifier | Modifier and code ]

Note

${displaystyle t_{0}}$

the time put by stone to browse half the distance from the tree,

${displaystyle t_{1}}$

The time taken to cover half the remaining distance, etc. The total duration of the journey is the sum of the durations of the path pieces, that is to say

${displaystyle t_{0}+t_{1}+…+t_{n}+…}$

Zénon thought that this sum was necessarily infinite, hence the paradox. Now we know that this total duration is not infinite since the pebble reaches the tree. It therefore seems that the general term series

${displaystyle t_{n}}$

converge. This is what we propose to demonstrate by assuming that the speed of the stone is constant.

The speed being constant, we have

${displaystyle t_{1}=t_{0}/2}$

,

${displaystyle t_{2}=t_{1}/2}$

and more generally

${displaystyle forall nin mathbb {N} }$

${displaystyle t_{n+1}={frac {t_{n}}{2}}}$

So the rest

${displaystyle (t_{n})_{n}in mathbb {N} }$

is geometric where we pull

${displaystyle forall nin mathbb {N} }$

${displaystyle t_{n}={frac {t_{0}}{2^{n}}}}$

Partial sums therefore have a simple expression

${displaystyle sum _{n=0}^{N}t_{n}=sum _{n=0}^{N}{frac {t_{0}}{2^{n}}}=t_{0}{frac {1-(1/2)^{N+1}}{1-(1/2)}}}$

We deduce that

${displaystyle sum t_{n}}$

converges and that

${displaystyle sum _{n=0}^{infty }t_{n}=t_{0}{frac {1}{1-(1/2)}}=2t_{0}}$

The stone therefore makes an infinity of journeys in a finished and equal time to

${displaystyle 2t_{0}}$

, Note that

${displaystyle 2t_{0}}$

is double the time it took stone to travel the first half of the journey.