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Euclidean Geometry

Euclidean Geometry

Definition: Two different points x and z of a Euclidean space (in the following shortly: space) as a subspace of ℝn with n ∈ κℕ* (see set theory) are called pair (of points). A line segment is a pair (x, z) that is completed by all inner points y, different from the starting point x and the end point z, and thus lie between x and z, because they satisfy ||x - y|| + ||y - z|| = ||x - z|| with the Euclidean norm || ||.

Definition: Two line segments intersect if they have exactly one inner point in common. This applies even if the latter emerges only after completing the line segments by all further inner points of the line segment from the ℝn. A (one-dimensional) point-set in the space, where each point has gapless at least one and at most two neighbouring points, is called line and a two-dimensionally maximal subspace of the space is called plane.

Definition: Line segments are called straight lines if both their starting and their endpoints are part from the boundary of the space, but for the moment not their inner points. Two line segments are called parallel, if each point of one line segment has the same shortest distance to the other one, and their included line segments, too. All line segments parallel to the straight lines defined above of the space are also called straight lines.

Definition: Two lines in a plane are called parallel if each point of the one line has always the same distance to the other one and vice versa, or one line emerged by translation from the other one. Two planes are called parallel here if they are each contained in one plane of ℝn that can be changed into each other by translation.

Result: For shorter defined straight lines, arbitrarily many counter-examples can easily be given, with the above, for Pasch's axiom, the axiom of line completeness and several other axioms and their equivalents. The parallel axiom is redundant in euclidean geometry, since a parallel straight line through a further point is uniquely determined by its shortest distance to the original straight line.

If two straight lines are only parallel iff they are in a plane and do not intersect, the parallel axiom is false, provided the reciprocal of the distance of the further point to the original straight line is at least infinite or smaller than |κℕ|, because then infinitely many different straight lines can be put through the further point without intersecting the original straight line.

The Archimedean axiom is to extend to a infinite natural number of ablations of a line segment, which cannot be done beyond the starting and end point of a straight line, or to replace by the Archimedean proposition (in the finite case). Pasch's axiom is also dispensable if we consider that a straight line must completely pass the interior of a triangle because of its maximal length, including its boundary.

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