#### Doron Shadmi

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,

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

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

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