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.

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

{...} = Discreteness = Infinitely many elements = infi^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 jump.


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

0^0 = infi^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 discrete 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 segment A be a connector between any two runner A positions.

Let a Continuum segment 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 in Math languge, you can find here:

http://www.geocities.com/complement...ry/CATpage.html




Doron

Your entire post seems to be in opposition to:

"In the Common Math the Continuum is a container of infinitely many discrete points "

which is NOT true. Points in a continuum are not discrete- that's that whole point of being a continuum. (And is itself an answer to Zeno's paradox- it's not quite fair to say that he is confusing "discrete" and "continuous"- he didn't have either of those concepts- but that's basically the problem.

That's also why calculus is not part of "discrete matematics"!

Hi HallsofIvy,




. Points in a continuum are not discrete- that's that whole point of being a continuum

I'll be glade if you show me how points can be a continuum,
or how points can't be discrete.

That follows directly from the definition of continuum. Perhaps it ould help if you would give the definition YOU are using since it does not appear to be standard.

Certainly the set of all real numbers is a continuum: and the real numbers are NOT discrete.

(The rational numbers ARE discrete- the set of all rational numbers is NOT a continuum.)

It makes no sense to speak of individual points (or objects in general) as being "discrete". Discreteness is a property of a set: a set is discrete if and only if it is either finite or countably infinite.

A continuum is a set having a topology in which it is connected.
A continuum CANNOT be discrete.

There is no way to associate between a discrete set {…} and a Continuous set {___} by means of the Quantity concept, without forcing the Continuum concept to be expressed in terms of the Discreteness concept, and what the Common Math does is:

{. <-- . --> .} = Extrapolation over scales = elements with finite magnitude = N, Z.

{. --> . <-- .} = Interpolation over scales = elements with finite or infinite magnitude, where those with an infinite magnitude are built on repetitions over scales = Q.

{. --> . <-- .} = Interpolation over scales = elements with infinite magnitude without repetitions over scales = R.

But the infinite { . --> . <-- . } magnitude never reaches the {___} state, and this is an axiomatic fact that no mathematical manipulation (which is based on the quantity concept) can change.

For example, please show me how we can use the bijection method between {...} and {__} ?

We find that |R| > |Q| by using the bijection method, and for this, the strucrute of each compared elemant in both sides
MUST BE {. <-- . --> .} or {. --> . <-- .}, so we are closed under {...} and can't conclude that |R| = {__} = Continuum.

All we can conclude is that the magnitude of the infinitely many elements of |R| is bigger than the magnitude of the infinitely many elements of |Q|.


Here are again my non standard basic definitions:

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

{} = The 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 the structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, But because it exists (unlike the emptiness) its full notation = 0^0).

(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 break points.)

{...} = The Discreteness = Infinitely many elements = infi^0 = Content exists.

Any transformation from {} to {__} or {...} is based on phase transition, because we have 0(=does not exist) to 1(=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 = infi^0 = 1 and we can see that we can't distinguish between
Continuum and Discreteness by their Quantity property.

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

From the above we can learn that the Structure concept have
more information than the Quantity concept in Math language.


Discreteness is a property of a set: a set is discrete if and only if it is either finite or countably infinite.
Any element (with finite or infinite magnitude) that is under a definition like "infinitely many ..." can not be but a member of {...}, which is the structure of the Discreteness concept.

Because we can't know the exact value of any R member, we can't use any R member as a common boundary between any two sets.

Therfore we can't conclude that |R| = C .

More than that, we can take any R member which is being used as a set's boundary, and has an infinite magnitude, and we find this:

A member with an infinite magnitude never reaches 0 by definition.

It means that there is always an unclosed gap
for some line's segment {0_____R member}.

Therefore we can conclude that |R| < C .

I find your thread very insightful and potentially useful to a hightened level of understanding of these seemingly incompatable mathematical genre. Good thread.

Hi errandir,

Tnank you very much.

Another thread of mine, which is close to this subject, you can find here:

http://www.physicsforums.com/showthread.php?s=&postid=51682#post51682



I'll be glad to get your remarks and insights on a new theory
of numbers of mine, that those threads are based on .

The address is:

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


Yours,

Doron

There was a lot of termonology that I didn't understand on that thread. I think that the first thing I should ask is, "What is bijection?"

Then, I would also like to know what is meant by "topology," "cardinal," and the difference between rational and natural numbers. These concepts/terms seem vaguely familiar, but it's been a while since I've heard them.

How did you arrive at the conclusion:

0⁰ = infinity⁰ = 1

?

When you say that Csim and Dsim are opposites, can you define "opposites?" They seem more complementary than opposite.

"...Csim XOR Dsim are exclusively = 1."

What does this mean?

Also, I don't see how any of this relates to symmetry. Maybe I don't know what symmetry really is.

A bijection is a 1 to 1 correspondence between members of two sets, in a way that no member is left out.


A cardinal is the number of elements that can be found in some set.

For some set with an infinitely many elements the cardinal is measured by its magnitude.

Through Topology Theory we can measure the transformations
between elements and their symmetry degrees, for example a perfect ball looks the same from any point of view, so it is the most symmetrical element (by symmetrical we mean that some
explored property remains unchanged through exploration).

Any change in the shape of a perfect ball put it in a less symmetrical degree, that can be measured by using Topology Theory.

Through my structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, but because it exists (unlike the emptiness) its full notation = 0^0 = 1 = set's content exists.

Infi^0 = 1 = set's content exists and we can conclude that
0^0 = infi^0 = 1 = content exists (the opposite is 0 = no content).

Before association Csim and Dsim are opposite to each other (they prevent each other).

When associate, they are in a complementary relations
(CAT is Complementary Associations Theory) .

XOR = eXclusive OR . It means that some elements with XOR
ratio among them, can not be in their "pure" self state in the same place/time.

So Csim XOR Dsim are exclusively (or separately) = 1.



Thank you.

Originally posted by Doron Shadmi
For some set with an infinitely many elements the cardinal is measured by its magnitude.

What is the magnitude of a set?




Originally posted by Doron Shadmi
Through Topology Theory we can measure the transformations
between elements and their symmetry degrees, for example a perfect ball looks the same from any point of view, so it is the most symmetrical element
It does not look the same if I observe it from two distinct distances: the size appears to change. Can you reword the definition without using the phrase, "looks the same?"




Originally posted by Doron Shadmi
(by symmetrical we mean that some
explored property remains unchanged through exploration).

Any change in the shape of a perfect ball put it in a less symmetrical degree, that can be measured by using Topology Theory.
I don't understand the term "explored." If I look at any shape, the act of looking does not change that shape (QM not withstanding). Can you explain "explore" and give an example of a "property?"




Originally posted by Doron Shadmi
Through my structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, but because it exists (unlike the emptiness) its full notation = 0^0 = 1 = set's content exists.

Infi^0 = 1 = set's content exists and we can conclude that
0^0 = infi^0 = 1 = content exists (the opposite is 0 = no content).
This seems to be more a restatement of the conclusion that the logical reasoning behind it. All that I'm seeing is, "since A then B." I'm not seeing the logical connection between A and B. Can you elaborate a little on 0⁰ = 1? According to my mathematical background, this is one of those undefined quantities (it depends on the kind of zeros used), yet you seem to be defining it absolutely.




Originally posted by Doron Shadmi
Before association Csim and Dsim are opposite to each other (they prevent each other).

When associate, they are in a complementary relations
(CAT is Complementary Associations Theory) .

XOR = eXclusive OR . It means that some elements with XOR
ratio among them, can not be in their "pure" self state in the same place/time.
I don't understand this terminology at all. What do the terms "associate," "ratio," and "self state" mean?

There are more members in the set of all real numbers (called R)
than the set of all natural numbers (called N).

Since we are talking about a difference between two sizes of infinity, we use the word "magnitude" instead of "quantity".

About the perfect ball, put your point of view in the middle of the
ball, and think of yourself as the center of this ball (I mean you have a zero dimension).

By using the word "explore" I mean mind exploration, for example
think about some pattern of a fractal, so no metter what zoom in or out you do in your mind, you will see the same pattern.

It means that this pattern is symmetrical over scales.

Any interval from X to ~X can be opend or closed only by a quantum-like leap, and no number which is not X, can close it.

Only a continuous line can close this interval ,so if, by analogy, we think of this line as a wave, then there are 0 particles(points) in this wave.

XOR ratio
------------
00 -> 0
01 -> 1
10 -> 1
11 -> 0

XOR ratio
------------
0(LINE) 0(POINT) -> 0-(No information)
0(LINE) 1(POINT) -> 1-(Particle-like information)
1(LINE) 0(POINT) -> 1-(Wave-like information)
1(LINE) 1(POINT) -> 0-(No clear information)

Originally posted by Doron Shadmi
There are more members in the set of all real numbers (called R)
than the set of all natural numbers (called N).

Since we are talking about a difference between two sizes of infinity, we use the word "magnitude" instead of "quantity".

I understand intuitively that any subset is "smaller" than or "equal in size" to any set of which it is a subset. But, in order for the idea magnitude to be useful, the criteria for judging the magnitude should be established. I don't see any such establishment.




Originally posted by Doron Shadmi
About the perfect ball, put your point of view in the middle of the
ball, and think of yourself as the center of this ball (I mean you have a zero dimension).

That makes sense. A ball is just the region that contains all points less than some constant, fixed distance from the point of observation?




Originally posted by Doron Shadmi
By using the word "explore" I mean mind exploration, for example
think about some pattern of a fractal, so no metter what zoom in or out you do in your mind, you will see the same pattern.
I still don't get it. You are defining the word "explore" by using the word "explore." That only increases the specificity. How about this: what is the difference between "explore" and "observe?"




Originally posted by Doron Shadmi
Any interval from X to ~X can be opend or closed only by a quantum-like leap, and no number which is not X, can close it.

Only a continuous line can close this interval ,so if, by analogy, we think of this line as a wave, then there are 0 particles(points) in this wave.
This means nothing to me. I don't see any interval. What does it mean to "close an interval?" If I think of the line (what line?) as a wave, then things become far more complicated in may mind. A wave may propogate through a medium of particles (i.e. a sound wave), or it may propogate through a medium without particles (i.e. a light wave). Also, I wave may be thought of as a disturbance in a continuum (classical field theory), or it can be thought of a the travel of discrete packets of energy (bosons). Is this what you mean: the wave in the classical sense?




Originally posted by Doron Shadmi
XOR ratio
------------
00 -> 0
01 -> 1
10 -> 1
11 -> 0

I am familiar with the basic truth tables of digital logic. I'm not saying this to be sassy, only so that you don't waste any more time trying to explain it to me.




Originally posted by Doron Shadmi
XOR ratio
------------
0(LINE) 0(POINT) -> 0-(No information)
0(LINE) 1(POINT) -> 1-(Particle-like information)
1(LINE) 0(POINT) -> 1-(Wave-like information)
1(LINE) 1(POINT) -> 0-(No clear information)
This is what I don't understand. By "1," I'm assuming that you mean "true," and by "0," I'm assuming that you mean "false." But, what is it that you are saying is true or false? "Clear information exists?" What is unclear about having one line and one point?

Hi errandir,

About "magnitue", please look here:

http://www.factmonster.com/ce6/sci/A0849267.html



You can change "explore" by "observe"



Do we see another thing between X to NOT-X which is not an interval ?

No infinitely many points can close this interval because any point has exectly 0 size, so the interval can be closed only by an element ~=0, and this element can't be but a quantum-like leaps of continuous smooth lines.

In the middle of a qauntum leap there are exectly 0 points.

So, we can notate a line as 0^0.

XOR ratio
------------
0(LINE) 0(POINT) -> 0-(No information) -> no conclusion.
0(LINE) 1(POINT) -> 1-(Clear Particle-like information) -> conclusions on points.
1(LINE) 0(POINT) -> 1-(Clear Wave-like information) -> conclusions on lines.
1(LINE) 1(POINT) -> 0-(No clear information) -> no conclusion.