Let's extend
http://www.internationalskeptics.com/forums/showpost.php?p=10841149&postcount=708 as follows:
In my opinion the understanding of Entropy (as observed and defined in, so called, closed systems) may have a profound impact on our survival as complex phenomena, and I think that intuition and rigorous treatment have to be combined in order to understand better Entropy, where infinity is a main factor of this research.
Let's carefully examine the notion of Length.
It is based on two building-blocks, which are:
Length=0
Length>0
There are no intermediate states between length=0 and length>0, such that Length=0 and length>0 are used as building-blocks (non-composed objects) in order to determine the values of each other, for example:
In this example there are 3 length=0 building-blocks and 2 length>0 building-blocks that determine the values of each other.
It means that length=0 and length>0 can't be related to values ≠ 0 without associated with each other (in the example above the 3 length=0 building-blocks are related to values 2, 5 and 7, where the 2 length>0 building-blocks are related to values 3 and 2).
Actually constants like Planck length (
https://en.wikipedia.org/wiki/Planck_length) can't be defined without length=0 and length>0 association, so it is not true that Length=0 is not involved in the determination of Planck length value (which is > 0).
Actually the essence of QM is not some measured value > 0, but it is the lack of intermediate states between length=0 and length>0, exactly because both of them are building-blocks (they are non-composed).
According to this observation, no amount of collection of length=0 building-blocks along a given length>0 building-block, is that building-block or in other words, the continuum is actually not less than a length>0 building-block.
Moreover, being a length>0 building-block the continuum, is exactly the reason of why between any arbitrary closer given pair of length=0 objects along a given length>0 building-block, there is at least one intermediate length=0 object.
The logic of this observation is as follows:
Given length=0 building-block along a given length>0 building-block, it is at the domain of the length>0 building-block, where the given length>0 building-block is at AND not at the domain of the given length=0 building-block along it.
Being at AND not at a given domain is logical contradiction, if the observation is done from the length=0 building-block with respect to the length>0 building-block.
But being at AND not at a given domain is not logical contradiction, if the observation is done from the length>0 building-block with respect to the length=0 building-block.
No matter what amount of length=0 building-blocks are defined along a given length>0 building-block, each length=0 building-block is at the domain of the given length>0 building-block, where the given length>0 building-block is at AND not at the domain of each given length=0 building-block along it.
One claims that two length=0 building-blocks if used as endpoints of a given length>0 building-block, then it is at the domain of these two length=0 building-blocks.
This claim is true not because of these two length=0 building-blocks, but because any given building-block is its own domain, or in other words, length>0 building-block is its own domain, whether the two length=0 building-blocks are defined as its endpoints, or not.
By using the notion of length=0 and length>0 building-blocks, one can observe Zeno's Achilles\Tortoise Race, as follows:
If only length=0 building-block is used than there is no room for race.
If only length>0 building-block is used than there is no race because there are no racers along it.
So the race is possible only if the two building-blocks are associated with each other, as follows:
Case A: Achilles wins against the Tortoise, and the Race stops.
Case B: Achilles does not win against the Tortoise, and the Race doesn’t stop.
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 ∞
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
(The Race continues after Loop K 1, since Tortoise position > Achilles position)
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
(The Race stops after Loop K 2, since Tortoise position < Achilles position)
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 ∞
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
Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.1
(The Race continues after 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
Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.01
(The Race continues after 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
Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.001
(The Race continues after Loop K 3)
...
Loop K ∞
The Race doesn’t stop 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, no matter how many loops (finite or infinite) are used (Achilles position < Tortoise position is an invariant state exactly because no collection of length=0 building-blocks (where each given position is some distinct length=0 building-block) is the continuum, which is not less than a given length>0 building-block).
Since no collection is the continuum, then we define an entropy-free realm, which can’t be defined by Standard Analysis (its fruitfulness can't deal with entropy-free realm).
The same limitation holds among the notion of mapping between ur-elements like length=0 and length>0 building-blocks (where building-blocks are non-composed objects), such mapping actually ignores the difference between being length=0 building-block and being length>0 building-block, and all it cares is to define the 1-to-1 correspondence between them.
