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.
