# Foundations of Geometry, D. Hilbert (trans. L. Unger), Open Court Publ.

Axioms
Axioms of Incidence
Postulate I.1.
For every two points A, B there exists a line a that contains each of the points A, B.

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!
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I think you understand the idea of a smooth (pointless) segment, but you still have problems to understand that when we deal with the non-finite, the limit concept (which is a natural property of any finite mathematical object) is replaced by the proportion concept.

The simplest way to understand the proportion concept (which is valid in both non-finite andr finite mathematical objects) is to use 0_1 as the basic unit of any other mathematical object.

0_x/0_1 = 0_x and 0_1/0_x is the multiplicative inverse of 0_x.

Sine any segment is indivisible by definition, we have to understand '/' not as a division operation but as a ratio between indivisible segments.

For example Circumference/Diameter is the ratio between a close segment and a non-closed segment, and since this ratio is very important to us we give it a unique name, which is: Pi.

The base method is a measurement tool that takes 0_1 and uses some invariant proportion according to it that can be found on infinitely many different scale levels that have self similarity over scales, where this self similarity is determined by the base value n where n = 2 → ∞ .

In other words base_2 fractal is not identical to base_3 fractal ... etc. ad infinitum.

When we understand fractals and the invariant proportion that is naturally related to them, we can understand something, which is simply amazing and beautiful:

a) We know that 0_1 is a finite mathematical element, when it is compared to itself (or in other words, we use 0_1 as the ratio of itself, or the scale level of itself, for example 0_1/0_1).

b) Let us say that we use base_2 fractal, and in this case we use smaller (fractions) or greater (integers) segments, to construct each scale level of the Base_2 fractal.

c) Please pay attention that any base_n fractal is made of segments, where each segment is a finite mathematical object, which is length > 0 where 0 is not a finite segment, but a point.

d) The basic difference between a point and a segment is this:

A segment has a direction, for example: we can move forward of backward along closed of opened segment. Also we can talk about the right edge or the left edge of a segment, top and bottom, ... atc.

We cannot do that in the case of a point, because a point has no direction at all (not 0 direction, but no direction at all).

e) Since any base_n fractal is based on segments, and only on segments, this basic property cannot be changed, even if we have a non-finite base_n fractal.

f) At the moment that we understand this fundamental fact about the base method, we immediately understand, that any non-finite path along this fractal cannot change the fact that any base_n fractal is made only by segments, where each segment > 0.

g) 0 is used only as an initial place, which helps us to the base_n fractal as a measurement tool, but when we go beyond the initial place we are in the kingdom of segments.

h) when we understand all of what is written above, we immediately understand that 0.111... [base_2] < 1 or in a more accurate way, since we are in the kingdom of segments the accurate notations for this notion is:

[0.111..., _1) < 0_1 (where the notation '_1)' of [0.111..., _1) is not natural number 1, but an infinitesimal and non-reachable segment, which permanently prevents from 0.111... to reach the exact value of 0_1).

Each [0.###, _#) has [0.000..., _1) which is an infinitesimal segment that is the complement of
[0.###, _#) to 0_1.

We have to understand some very interesting facts about this complement segment.

a) If the complement segment is not used (by an addition operation), then [0.###, _#) is a non-finite unique sequence along some base_n fractal.

b) If the complement segment is used (by an addition operation), then immediately we reach 0_1, but than [0.###...,_#) is changed to [0.###...,_#] or in other words, we get a finite sequence of segments, where the last segment is = or > than the previous first segment.

But a deeper understanding of the above is this:

Let us use base_10 in order to understand it, but it holds for any other base.

By understanding [0.999..., _9) and [0.000..., _1) we get another beautiful insight, which is:

If we do not use an addition between [0.999..., _9) and [0.000..., _1),
then [0.999..., _9) is a non-finite fractail where [0.000..., _1) is an infinitesimal segment.

When we use an addition between [0.999..., _9) and [0.000..., _1), then and only then [0.999..., _9) and [0.000..., _1) become finite objects ( notated as [0.999..., _9] and [0.000..., _1] ) that their exact length is segment 0_____1.

By this insight we understand that our own action can change a mathematical object from a non-finite state to a finite state and vise versa.

In other words, the mathematician himself is not an objective observer, but an active participator of any explored mathematical universe, which his influence on this universe must not be ignored anymore.

"All people have names."
"Each person has a name."

(sinθ)^2+(cosθ)^2=1

s=(x+y)/2
r=(x-y)/2