Another point of view on Zeno's paradox

If we use the idea of sets and look at their contents from
a structural point of view, we can find this:

{} = Emptiness = 0 = Content does not exist.

Let power 0 be the simplest level of existence of some set's content.

{__} = The Continuum = An infinitely long indivisible element = 0^0 = Content exists (from a structural point of view, the Continuum has no elements in it, so its base value = 0, But because it exists (unlike the emptiness) its full notation = 0^0).

{...} = The Discreteness = Infinitely many elements = [oo]^0 = Content exists.


Any transformation from {} to {__} or {...} is based on phase transition, because we have 0(= content does not exist) to 1(= content exists) transition.

So, from a structural point of view, we have a quantum-like leap.


Now, let us explore the two basic structural types that exist.

0^0 = [oo]^0 = 1 and we can see that we can't distinguish between
Continuum and Discreteness by their Quantity property.

But by their Structural property {__} ~= {...} .

I think that Zeno's paradox is the result of using {...} to measure
{__} by the Quantity concept, and by doing this we force {__} to be
expressed in terms of {...}, and we get a system which is closed on itself under the Discreteness concept.

In the Common Math the Continuum is a container of infinitely many points with no gaps between them, but if you think about the meaning of "points with no gaps..." you find a simple contradiction when you connect the word "points" to "no gaps".

Through the structural point of view, the Continuum is not a container but a connector between any two points {.___.} and you can find this state in any scale that you choose.

Through this approach you don't have any contradiction between the Discreteness and the Continuum concepts, because any point is not in the Continuum, but an event that breaks the Continuum.

The Continuum does not exist in this event, but any two events can be connected by a Continuum, for example the end of a line is an event that breaks the line and it turns to a nothingness, so from one side we have the Continuum, from the other side we have the nothingness, and between them we have a break point, that can be connected to another break point that may exist on the other side of the continuous line.

Another way to look at these concepts is:

Let a Continuum be an infinitely long X-axis.

Let a point be any Y(=0)-axis on the X-axis.

So what we get is an indivisible X-axis and infinitely many Y(=0)-axises points on (not in) the X-axis.

Through this point of view we get an indivisible X-axis, as a connector (not a container) between any two Y(=0)-axises events.



Now, through the above I think we can solve Zeno's paradox like this:


Let a Continuum A be a connector between any two runner A positions.

Let a Continuum B be a connector between any two runner B positions.

Let the start time be equal for both runners.

Let the rest time in any position, be equal for both runners.

Let a connector A > connector B .

Because there are no discrete elements between any two positions, we have no paradox.



In general, through the structural point of view, we have two levels
of XOR retio:

Level A is: ({} XOR {.}) or ({} XOR {_})

Level B is: {.} XOR {_}



More of the Structural point of view on Math languge, you can find here:

http://www.geocities.com/complementarytheory/CATpage.html




Doron

Fundamentally, you are correct- the answer to Zeno's paradox is that he is incorrectly mixing "discrete" and "continuous"- although he had neither of those concepts. That's pretty much covered in any calculus text.

Your explanation would be clearer (perhaps too clear? It might be obvious that it's already been said many times?) if you would not use words in non-standard ways or without defining them. You talk about "base value" without defining it, "power 0", assert that a continuum is an "infinitely long, indivisible element" (a continuum, in its standard definition, does not have to be either infinitely long nor indivisible- the interval from 0 to 2 is a continuum, is neither infinitely long nor indivisible: it can be divided into many different continua).

I have tried a number of times to open the URL you give without success. In any case, if this post is an example, I suspect that your "structural view of math language" consists of undefined or poorly defined terms and vague concepts.

Hi HallsofIvy,


You can break an infinitely long continuum infinitely many times, but always
you will find an invariant structural state of {.___.} which is a connector
between any two points.

The transition between point A to point B is a quantum leap because a connector
(by definition) is an indivisible element.

Through this model, a movement is the sum of the rest times of all explored break points (quantum leap positions) along the continuous path.

the interval from 0 to 2 is a continuum, is neither infinitely long nor indivisible: it can be divided into many different continua).

By my structural point of view I distinguish between The Continuum (which is
an indivisible and infinitely long) and a connector, which is an indivisible finite
element between any two break points.

Through my system, any point is a break point because of the XOR ratio between
the discrete and the continuous.

So, between break point 0 and break point 2 there may exists a connector
of length 2, and in this case we have a quantum leap of length 2.

If you continue the breaking process forever, you always find {.___.} which is an
indivisible element of quantum leap between any two break points (as you wrote too).

As I wrote, the Common Math explores the continuum by using its
opposite concept, which is the discreteness concept (the mangitude
of an interpolation between infinitely many points).

As I showd above, because of the XOR ratio between {...} and {___},
an infinite interpolation magnitude {.-->.<--.} never
reaches the {___} state and {.-->.<--.} can reach {___} only
by phase transition.

The use of "power 0" to distinguish between 0=(set's content does not exist)
to 1=(set's content exists) by the quantity concept, is simple and clear.

And immediately after that I show that we can't distinguish between
{...} and {___} by the quantity concept, and it can be done by the
structure concept. So I do not see any thing which is vague here.

Also the "base value 0" simply says that there are exactly 0 points in The Continuum.

Doron,

Your approach to description is, to say the least, very non standard. It can confuse people if left here, so I'm sending it to the "Theory Development" forum (which is under Theoretical Physics).

Hi ahrkron,

Thank you.

If you have any remarks or insights on my Ideas, I'll be glade to get them.

Yours,

Doron

I think that Zeno's paradox is the result of using {...} to measure
{__} by the Quantity concept, and by doing this we force {__} to be
expressed in terms of {...}, and we get a system which is closed on itself under the Discreteness concept.

That about covers it. Nothing more needs to be said. You deserve an award for this.

Yes! I shall bestow on you the [pi] award.

Hi Arc_central,


I'll be glad to get your remarks to another thread (please start from its top):

https://www.physicsforums.com/showthread.php?s=&postid=57361#post57361


Thank you,

Doron

An easy way to say all this?

What Zeno says is true, up to a point. Where the paradox comes in, and the leap you have to take in your mind is that you are somehow committed to acting out the scenerio he puts forward. Of course you're not committed to it .. you can, and do, simply chose to move otherwise.

Here's Nacho's Paradox ..

If I said yall had to pay me $1.00 each time you posted here, would you feel or be committed to acting that out? ;)

Hi Nacho,

What Zeno says is: "Any movment is an illusion".

But because he uses infinitely many points (0 length elements) there is no paradox, but some sysem that it closed under 0 length by definition.

So, through its system you can't move from your starting point, there is no race that cen be examined, so there is no paradox.