Frege's Concept - Script (Begriffsschrift)

[Organic]

p and q are real numbers.

If p < q then [p, q] = {x : p <= x <= q} or
(p, q] = {x : p < x <= q} or
[p, q) = {x : p <= x < q} or
(p, q) = {x : p < x < q} .

A single-simultaneous-connection is any single real number included in p, q
( = D = Discreteness = a localized element = {.} ).

Double-simultaneous-connection is a connection between any two real numbers included in p, q ( = C = Continuum = a non-localized element = {.___.} ).

Therefore, x is . XOR .___.

In Conventional Math 0^0 is not well defined, because each member is D.

Let us say that power 0 is the simplest level of existence of some set's content.

Because there are no Ds in C, its base value = 0, but because it exists (unlike the emptiness), its cardinality = 0^0 = 1.

There are now 3 kinds of cardinality:

|{}| = 0 = the cardinality of the Empty set.

|{._.}| = 0^0 = 1 = the cardinality of C.

|{.}| = 1^0 = 1 = the cardinality of D.

Any point is a D element. Any line is a C element.

It means that there is a XOR connective between LINES and POINTS.

XOR connective between LINES and 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.

We can break C infinitely many times, but always we shall find an invariant structural state of {._.}, which is a connector between any two Ds.
Let power 0 be the simplest level of existence of some set's content.

{._. .} = Dinf = Infinitely many Ds and/or Cs (oo^0 = C XOR D = 1).
{___} = Cinf = Infinitely long C (0^0 = 1).

0^0 = oo^0 = 1 and we can see that we can't distinguish between C and D by their quantitative property.

But by their Structural property Dinf is not Cinf.

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

Any element that is under a definition like "finite or infinitely many ..." cannot be anything but a member of D or Dinf sets, which have the structure of the Discreteness concept.

So, any line's segment is not a container but a connector between any two points {.___.}, and you can find this state in any scale that you choose.

C XOR D, and 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 the XOR between any line to any point), but any two events can be connected by a Continuum.

Take 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 a non-localized X-axis and infinitely many Y(=0)-axises points on (not in) the X-axis.

Through this point of view, the X-axis is a connector (not a container) between infinitely many Y(=0)-axises events.

In general, there are two levels of XOR:
A) ({} XOR {.}) OR ({} XOR {_})
B) {.} XOR {_}

There are 4 important conclusions from the above:

For example, let n = 3 = 1+1+1

A) 0^0 = Continuous 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 connective) between its continuous side (Continuous n) to its discrete side (n Connectors).

So from my point of view, Mathematics is more than variety of systems, where each system has its consistent universe.

As I see it, through this attitude one of the most important things of the evolution is cut out of today's Math.

Any evolution is based at least on two principles, variety and mutation.

The meaning of a mutation is to redefine existing things or familiar terms.

As I see it, the Modern Math has to look at this kind of approach as if it is a mutation in the Continuum concept and not as another axiomatic system where we can get:

|Q| < |R| = Pointed/segmented Continuum < A Pointless/segmentless Continuum .

Pointed/segmented Continuum is simply a contradiction, so as this concept does not exist, we can get a simple solution to the CH problem where:

|Q| < |R| < Continuum(={__}) .

 

[Organic]

The algorithm (which is based on Cartesian product) that was found and programmed by Stratman (by the way he call to these elements Sumbers,which is a combination of a set and a number) can easily find the associations products between {._.} forms and {.} forms:

For any n > 0, the following algorithm calculates the number of transitional states:

ASSOCIATION_LEVEL(int n):Vector

1 V:= new vector
2 if n = 1 then
3 V:=V + the single root of level 1
4 if n = 2 then
5 V:=V + the two roots of level 2
6 else
7 if n >2 then do
8 For each partition vector P in n do:
9 ROOTS(P,V)
10 return V

ROOTS(vector P,vector V)

1 roots :=[]
2 for each element in P do:
3 if element is a root then
4 roots:= roots + element
5 else
6 roots:=ASSOCIATION_LEVEL(integer value of element)
7 return horizontal alignment of the Cartesian product of P and draw new arc around them

You can check it here:

http://cyborg2000.xpert.com/ctheory/

Please don't go beyond 6 or 7.


The Cartesian product adding some left-right forms that can be ignored.

After Ablert Einstein there is no meaning to talk about space without time and vise versa, so any product is space/time protuct.

Through my point of view there is no meaning to talk about quantity without structure and vise versa, so any product is a structure/quantity protuct.

General conclusion:

The internal structure of any given quantity (finite or infinite) cannot be ignored.