Zeno's Achilles\Tortoise Race
First, one has to define what are the minimal conditions that enable a race, in the first place.
Both in Mathematics and Physics there are primitive notions which are themselves undefined, and they are used as the building blocks of deduction or induction in order to avoid endless regression (An endless regression prevents a strict solution of some problem).
In that case we may ask: Is a solution is always a strict solution, or there are also solutions that are not strict (which means that endless regression (or endless progression) is also a possible solution)?
In order to demonstrate these notions, let's carefully observe, for example, point and line as primitive notions under Hilbert's axiom system of Euclidean Geometry:
"1. For every two points
A and
B there exists a line
a that goes through
A and goes through
B ."
If we understand primitive notions in terms of atoms, then atoms are actually non-composed things (they are not defined by other building blocks since they are themselves building blocks).
In that case a line is not a collection of points (it is a non-composed thing exactly as a point is a non-composed thing) which means that the term "goes through" does not define a point as a building block of a line, and no collection of points is a line.
Now, by using these observations, let's carefully observe the minimal conditions that enable a race, in the first place.
If only a line is considered, we can't define points along the line that define at least two competitors.
If only a point is considered, then nor competitors neither a race path are definable.
So, the minimal conditions that enable a race is a line and at least two points along it, such that a point and a line are non-composed (they are not building blocks of each other and yet they can associate through their common property of being non-composed).
The existence of a line as a non-composed thing (as a building-block) is stronger then any collection of distinct points along it.
As a result, in order to actually reach a given point from another given point, it is done by a single non-composed line segment, which simultaneously connects the given pair of distinct points with each other.
Such connections, are always done by finitely many steps. Therefore, a finite collection of points is defined as finitely weaker than the existence of a line (which is a non-composed thing).
Moreover, by this observation, the existence of a line as a non-composed thing, defines it as actual infinity, which is reachable by a collection of finitely many steps (a finite collection is finitely weaker than actual infinity).
In that case a collection of infinitely many points along a given non-composed line is defined as infinitely weaker than actual infinity, and therefore its cardinality can't be defined by strict value (as done in case of finite cardinality), which means that infinite collections are no more than potential infinity that can't reach a given point by infinitely many steps.
Let's use the observation of clapping hands.
By this observation, one of your hands actually reaching your other hand only by finitely many steps, since finitely many steps are finitely weaker than actual infinity (which is the non-composed gap between your hands).
Since potential infinity is infinitely weaker than the non-composed gap between your hands, no infinitely many steps of your hand actually reaching your other hand (since potential infinity is inaccessible to actual infinity, by definition).
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Zeno's Achilles\Tortoise Race
It is argued that Zeno's Achilles\Tortoise Race is not a paradox in real life because we can summarize non-finite values (where each value > 0) that are added to some initial value. By doing that we are able to get an accurate value, which is different from the initial value. For example: 1 is the initial value and 1+1/2+1/4+1/8+…= 2, where 2 is an accurate value that is different from the initial value 1. Actually the whole idea of Limits is somehow motivated by the desire to solve Zeno's Paradox. Let us investigate two different cases of Achilles\Tortoise Race:
Case A: Achilles wins against the Tortoise OR Achilles and the Tortoise are on the same position, and the Race stops.
Case B: Achilles does not win against the Tortoise, and the Race continues (actually forever).
Case A:
Distance = Speed * Time
The next position of Achilles and the Tortoise along the Race = previous position + Distance.
Case A exists only if neither Speed nor Time are changed during the Race. Let us show it by using an algorithm (no particular programming language is used here):
Position X1 = 0
Position X2 = 10
Achilles Speed = Aspeed = 10
Tortoise Speed = Tspeed = 1
Time = 1
Do Loop K from 1 to forever
Achilles position = position X1 + distance ( = Aspeed * Time)
Tortoise position = position X2 + distance (= Tspeed * Time)
Position X1 = Achilles position
Position X2 = Tortoise position
If X1 ≥ X2 then STOP
Next Loop K
Loop K=1:
Achilles position = 0 + (10 * 1) = 10
Tortoise position = 10 + (1 * 1) = 11
Position X1 = Achilles position = 10
Position X2 = Tortoise position = 11
If X1 ≥ X2 then STOP
(The Race continues after Loop K=1, since Tortoise position > Achilles position)
Next Loop K=1
Loop K=2:
Achilles position = 10 + (10 * 1) = 20
Tortoise position = 11 + (1 * 1) = 12
Position X1 = Achilles position = 20
Position X2 = Tortoise position = 12 If
X1 ≥ X2 then STOP
(The Race stops after Loop K=2, since Tortoise position < Achilles position)
Next Loop K=2
Case A is finitely weaker than actual infinitely.
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Case B:
Distance = Speed * Time
In
case B , Time is changed during the Race. Let us show it by using an algorithm (no particular programming language is used here):
Position X1 = 0
Position X2 = 10
Achilles Speed = Aspeed = 10
Tortoise Speed = Tspeed = 1
Time = 1
Do Loop K from 1 to forever
Achilles position = position X1 + distance ( = Aspeed * Time)
Tortoise position = position X2 + distance (= Tspeed * Time)
Position X1 = Achilles position
Position X2 = Tortoise position
If X1 ≥ X2 then STOP
Time = Time / Aspeed (Achilles Speed = Aspeed = 10)
Next Loop K
Loop K=1:
Achilles position = 0 + (10 * 1) = 10
Tortoise position = 10 + (1 * 1) = 11
Position X1 = Achilles position = 10
Position X2 = Tortoise position = 11
If X1 ≥ X2 then STOP
Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.1
(The Race continues after Loop K=1)
Next Loop K=1
Loop K=2:
Achilles position = 10 + (10 * 0.1) = 11
Tortoise position = 11 + (1 * 0.1) = 11.1
Position X1 = Achilles position = 11
Position X2 = Tortoise position = 11.1
If X1 ≥ X2 then STOP
Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.01
(The Race continues after Loop K=2)
Next Loop K=2
Loop K=3:
Achilles position = 11 + (10 * 0.01) = 11.1
Tortoise position = 11.1 + (1 * 0.01) = 11.11
Position X1 = Achilles position = 11.1
Position X2 = Tortoise position = 11.11
If X1 ≥ X2 then STOP
Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.001
(The Race continues after Loop K=3)
Next Loop K=3
...
Loop K n+1
The Race continues forever because any given position of Achilles and the Tortoise that comes next is the result of previous positions + Distances values, where each Distance value > 0 (Achilles position < Tortoise position is an invariant state).
Case B is infinitely weaker than actual infinitely.
For more details, please observe
http://www.internationalskeptics.com/forums/showpost.php?p=12653709&postcount=3302 .
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Traditional mathematicians take such loops as a sequence of partial sums.
For example, they are claim that 0.999... is not a term of the following sequence of partial sums: (0.9, 0.99, 0.999, ...)
Let's check this claim.
The result of
Code:
1
↓ 2
0.9 = 0.9 ↓ 3
0.99 = 0.9 ( + 0.09) ↓
0.999 = 0.9+0.09 ( + 0.009)
...
Is actually the same as
Code:
1 2 3 ...
↓ ↓ ↓
0.9 + 0.09 + 0.009 + ...
or in other words, 0.999...
is embedded in (0.9, 0.99, 0.999, ...) even if it is not one of its exposed terms.
What we care is about the result of the series of infinitely many 0.9+0.09+0.009+... added
finite numbers (where every one of them > 0) which are
embedded in the infinitely many terms of the sequence (0.9, 0.99, 0.999, ...).
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Also we are aware that any infinite collection is infinitely weaker than actual infinity (which is non-composed, as observed in this post).
