#### Doron Shadmi

If we take any R member (with an infinitely many digits) which is being used as a set's boundary, we find this:

A member with an infinitely many digits never reaches 0 by definition.

It means that there is always an unclosed interval

for some line's segment {0_____R member with infinitely many digits}.

More than that, 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.

No infinitely many points can close this interval because any point has exactly 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.

It means that there is a XOR ratio between LINES to POINTS.

XOR ratio between LINES to POINTS

-------------------------------------

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.

There is exactly a one and only one way to cover a ___ by a .

We have to stretch the point until it overpals the line.

But than you have no longer a point and a line, but two lines.

Through topology there can be another solution.

If a point and a stretched_point are the same through topology eyes, than,

by definition, there cannot be any points in the middle of a

stretched_point, and we have a quantum leap element.

More than that, every number can exists(=set's centent ~=0) only by its relation to {}=0, so through this point of view, no number is a point but some distance from zero, and

---------------------------------------------------------------------------

An axiom:

In the middle of a

{} = 0 = Set's content does not exist.

~0 = Set's content exists.

Any transition from 0(= Set's content does not exist)

to ~0(= Set's content exists) can not be but a quantum leap.

Anything that cen exactly overlap(close) a quantum leap = quantum leap.

A quantum leap can be exactly overlapped(closed) only by

a continuous line.

Now we can conclude that some continuous line = some quantum leap.

Therefore, In the middle of a continuous line there are exactly 0 points.

QED.

---------------------------------------------------------------------------

It means that there are exactly 0 points

Now we have 3 structural types of set's contents:

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

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

{__} = The Continuum = An 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 = [oo]^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 = [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 {__} ~= {...} .

From the above we can learn that the Structure concept have

more information than the Quantity concept in Math language.

Any element that is under a definition like "infinitely many ..." can not be but a member of {...}, which is the structure of the Discreteness concept.

Through my point of view, the Continuum is not a container but a

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.

Therefore we can conclude that |R| < C = Continuum .

There are 4 important conclusions from the above:

Let n = 3 = 1+1+1

A) 0^0 = Contiunuous 1

B) 1^0 = 1 Connector

C) n/1^0 = n Connectors (.__.__.__. = 1 1 1)

D) n/0^0 = Continuous n ( ________ = 3)

Through this approach, each natural number is the associations (AND ratio) between its continuous side (Continuous n) to its discrete side (n Connectors).

More detailed information about the structures that you can get from

those associations, you can find here:

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

Please tell me what do you think ?

Yours,

Doron

A member with an infinitely many digits never reaches 0 by definition.

It means that there is always an unclosed interval

for some line's segment {0_____R member with infinitely many digits}.

More than that, 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.

No infinitely many points can close this interval because any point has exactly 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 any qauntum leap there are exactly 0 points.**It means that there is a XOR ratio between LINES to POINTS.

XOR ratio between LINES to POINTS

-------------------------------------

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.

There is exactly a one and only one way to cover a ___ by a .

We have to stretch the point until it overpals the line.

But than you have no longer a point and a line, but two lines.

Through topology there can be another solution.

If a point and a stretched_point are the same through topology eyes, than,

by definition, there cannot be any points in the middle of a

stretched_point, and we have a quantum leap element.

More than that, every number can exists(=set's centent ~=0) only by its relation to {}=0, so through this point of view, no number is a point but some distance from zero, and

**any distance is a quantum leap between 0(=set's content does not exist) to ~0(=set's content exists).**---------------------------------------------------------------------------

An axiom:

**In the middle of any qauntum leap there are exectly 0 points.****Theorem:**In the middle of a

**continuous**line there are exactly 0 points.**Proof:**{} = 0 = Set's content does not exist.

~0 = Set's content exists.

Any transition from 0(= Set's content does not exist)

to ~0(= Set's content exists) can not be but a quantum leap.

Anything that cen exactly overlap(close) a quantum leap = quantum leap.

A quantum leap can be exactly overlapped(closed) only by

a continuous line.

Now we can conclude that some continuous line = some quantum leap.

Therefore, In the middle of a continuous line there are exactly 0 points.

QED.

---------------------------------------------------------------------------

It means that there are exactly 0 points

**in**the continuum, and we can define a new set's content, which is {__} = C = Continuum.Now we have 3 structural types of set's contents:

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

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

{__} = The Continuum = An 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 = [oo]^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 = [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 {__} ~= {...} .

From the above we can learn that the Structure concept have

more information than the Quantity concept in Math language.

Any element that is under a definition like "infinitely many ..." can not be but a member of {...}, which is the structure of the Discreteness concept.

Through my 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**(because of a XOR ratio beween any line to any point), 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.

Therefore we can conclude that |R| < C = Continuum .

There are 4 important conclusions from the above:

Let n = 3 = 1+1+1

A) 0^0 = Contiunuous 1

B) 1^0 = 1 Connector

C) n/1^0 = n Connectors (.__.__.__. = 1 1 1)

D) n/0^0 = Continuous n ( ________ = 3)

Through this approach, each natural number is the associations (AND ratio) between its continuous side (Continuous n) to its discrete side (n Connectors).

More detailed information about the structures that you can get from

those associations, you can find here:

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

Please tell me what do you think ?

Yours,

Doron

Last edited by a moderator: