Fractal Point Space
Because the repetitive nature of the argument, as well as the argument being observed from any angle leading to any point within the order of the argument no numbers will be applied in the step process.
The minimum number of logical symbols will be applied as well and inherent within the progress of one step leading to another.
x. One point progresses to another point, with both points effectively being the same point. This effectively is the point.
∙1(∙1 → ∙1)
x. The progress of one point to itself results in two points as 1 point, with this progression of the 1 point to two halving the point through itself. The point exists as its own fraction/fractal.
∙1(∙2) ↔ (∙1/∙2)
x. The 1 point exists as 3 points.
∙3
All whole numbers are fractals of a point. The problem occurs that the point must shrink or expand in size if it is to exist in a fractal state. So even if we have x points as 1 point, the point never really changes. "x" points will always be the 1 point.
∙1 → ∙1 = ∙1(∙2)
∙1 → ∙2 (∙1/∙2) = ∙1(∙3)
∙1 → ∙3 (∙1/∙3) = ∙1(∙4)
****This progression from 1 to 2 to 3, etc. can be observed as 1 point going to 2 points, from the 2 points 1 point goes to 3 points (considering if both points of 2 project it goes to 4 points. The same applies for 3 to 4, etc.) In these respects the number line from 1,2,3,4, etc. observes the point continuing as a fractal/fraction of itself but maintains a whole nature unless each of these points in there multiple states are viewed as extensions of a whole.
Dually the point as a fractal/fraction observes it's replication observing infinite gradation of the point.
∙1 → ∙2 necessitates "→" as an infinite series of fractions:
∙1.1 → ∙1.2 → ∙1.3 ...
∙1.11 → ∙1.12 → ∙1.13 ...
∙1.111 → ∙1.112 → ∙1.113 ...
Where each fraction of the point, it in itself, is a set of points as 1 point:
∙1.1 = ∙1(∙11) → ∙1.2 = ∙1(∙12) → ∙1.3 = ∙1(∙13) ...
∙1.11 = ∙1(∙111) → ∙1.12 = ∙1(∙112) → ∙1.13 = ∙1(∙113) ...
∙1.111 = ∙1(∙1111) → ∙1.112 = ∙1(∙1112) → ∙1.113 = ∙1(∙1113) ...
Hence each projection of 1 point to another exists as an infinite number of points. So the one point projecting to another point is defined by one infinite series of points, however the beginning and end point are the same as the infinite series of points.
Each point, however, is an infinite series of points considering the fractal nature of one line relative to another, far enough down a continuum, results in all points being an infinite series in themselves.
Hence Point A, the line between Point A and Point B=A, and Point B=A are 3 points where the point itself is a fractal and all lines are observations of fractions.
So the statement:
∙1 → ∙1 = ∙1(∙2)
∙1 → ∙2 (∙1/∙2) = ∙1(∙3)
∙1 → ∙3 (∙1/∙3) = ∙1(∙4)
changes too:
∙1 → ∙1 = ∙1(∙2) = ∙3
∙1 → ∙2 (∙1/∙2) = ∙1(∙3) = ∙4
∙1 → ∙3 (∙1/∙3) = ∙1(∙4) = ∙5
But a paradox occurs in the respect each of these set of points as a continuum its in itself a point relativistically;
∙1 → ∙1 = ∙1(∙2) = ∙1(∙3)
∙1 → ∙2 (∙1/∙2) = ∙1(∙3) = ∙1(∙4)
∙1 → ∙3 (∙1/∙3) = ∙1(∙4) = ∙1(∙5)
hence the "point" is a paradox of definition and no-definition and exists as a continual progression through itself as itself. A line effectively can be observed as a fractal point, where the point as a quantity exists as a fractal state through the line. This is considering the point as defined, observes the line as undefined.
****Too be edited because of Time