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Let's start form two versions of Dichotomy paradox

Version 1: Anyone can't walk though any distance without walk though half of it and so on.

solution 0 (?) ... which is false I think, that's not what the statement talking about

For any distance d we have

d/4 + d/8 + ... + d/(2^n)+... = d/2

That we can walk though half of d

Therefore we can walk though any distance d

solution 1

Let f : T -> V be a function with domain time(T) and codomain velocity(V)

Suppose we started from 0

For any distance d that we want to walk though have:

Exists 0,c_0 in T if we take integral from 0 to c_0 on f(x) it equal to d

Alse exists 0,c_1 in T s.t. take integral from 0 to c_1 on f(x) have d/2^1

...(Keep doing this to n)

Alse exists 0,c_n in T s.t. take integral from 0 to c_n on f(x) have d/2^n

...(Keep doing this to infinite that as n->inf, c_n=d/2^n=0)

That will have, exists 0 in T s.t. take integral from 0 to 0 on f(x) have 0

In another word ... Zeno states that in order to move any distance d, what we have to do first is just walk though 0 infinite many times, but infinite many 0 distance add together will still be 0.

But is this just the wrong way to do limit.

Or is there something else I missed

Zeno's arrow paradox: ...(the mistake is similar to Version 1 of Dichotomy paradox)


Version 2: Even anyone can walk though half of any distance, he still have to walk though the half of the rest and so on.

solution 1 (Calculus--James Stewart pg.6 The sum of a series)

For any distance d we have

d/2 + d/4 + ... + d/(2^n)+... = d

Suppose we can indeed walk though all of those sumed distances in a finite time

That implies that we can walk though any distance d


solution 2

Let f : T -> D be a function with domain time(T) and codomain distance(D)

For any people who walk with constant velocity v

If it's the case that range(f)=[0,d), for some d in R

It's either time being defined differently or space being defined differently

For example we define Zeno's time as the following:

Let c,t be some real number then

Zeno's time(T) = t iff Normal time = (L-(1/c)^t)/v where c > 1

Have f(t) = (L-(1/c)^t)

That with finite Zeno's time no one can reach d.

Similarly we can also construct a Zeno's space s.t. range(f)=[0,d)

Basicly, this solution want to say: paradoxes caused by using different definitions.

Achilles and the tortoise: ... (This is similar to second version of Dichotomy paradox)


In both versions, I think solution 1s are fine, are they correct?

And please tell me if you know where I can find some formal proof in calculus which shows the mistakes of Zeno's paradoxes

Any help or suggestion would be appreciated.

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