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#27: Insertion Euclidean Geometry and Extension Introduction and Mathematics on 12.11.2010

Axiom: Everything possible exists until someone finds out its impossibility.

Definition: Everything, possible, existence, something, different, whole, not and space are substantial terms, which cannot be defined without a circle. A part is something that forms, with something distinct, to wit somehow different, from it, a whole. A point is, according to Euclid, something that has no parts. Two points are identical, if they differ by nothing else than their designation (otherwise non-identical). Then they coincide (otherwise they do not).

Definition: Two non-identical points are called, with two options, starting and end point and, considered as a whole, elementary line segment or pair (of points) . A chain of elementary line segments consists of an initial elementary line segment and possibly further elementary line segments, whose starting point each coincides with the end point of an elementary line segment different from them, where the end points are each pairwise non-identical, as well as starting and end point are uniquely determined.

Definition: A starting point of the chain of elementary line segments is then called, in each case, connected with its end point. A chain of elementary line segments, for that three points of it, chosen freely, satisfy the triangle inequality for every metric (selected in each case) always with equality, is called line segment. Depending on its length, it is called (in-) finite. The maximal line segment, with as many points as the number of the real numbers, defined in set theory, is called straight line. More specifically, a line segment is only a chain of elementary line segments that was cut out from a straight line at its starting and end point. One can define (in-) finite line segments with (in-) finitely many points and obtains this way four different possibilities to define line segments.

Definition: Line segments are called congruent, if their starting and end points have in each case the same Euclidean distance. A point y of a line segment is located between two points x and z different from it and among themselves, if for the Euclidean norm ||x - z|| = ||x - y|| + ||y - z||. If we have additionally ||x - y|| = ||y - z|| = ||x - z||, it builds the middle between x and z. If the three points are not located on a line segment, so each is located between the two others, depending on the perspective. If two not end-point-identical line segments have an identical starting point, and if their end points coincide with the starting resp. end point of another line segment, so all three edge a triangle and are called sides of the triangle, whose three starting resp. end points are called vertices of the triangle.

Definition: If we duplicate a triangle by letting coincide (combining) the two corresponding sides so that the original (starting point) end point coincides with the duplicated (end point) starting point, we obtain a parallelogram, if it is yet ensured that the original triangle vertex that is not on the combined side, has maximum distance from the associated duplicated vertex, with regard to the Euclidean norm. Here, two corresponding sides and their partial line segments are called parallel.

Definition: A point of a line segment is called its inner point, if it is not identical with its starting and end point. Two line segments intersect at a point, if they have only an inner point in common. The geometry is called free of point identity, if we permit in their overall result of the point-set no identical points, otherwise open for point identity.

Remark: Angles are defined via scalar product, Euclidean norm and cosine function as usual, or otherwise, hyperplanes, circles, spheres, etc. by the sufficiently known equations. This can also happen otherwise (for example as by Hilbert). It is generally useful to specify maximum distances of the points. In the non-Euclidean geometry, the definitions partially vary.

Result: For shorter defined straight lines, arbitrarily many counter-examples, with the above, can easily be given for the parallel axiom, Pasch's axiom, the axiom of line completeness and several other axioms and their equivalents. This was to be shown. The parallel axiom is redundant in Euclidean geometry, since a parallel straight line through a further point is uniquely determined by its maximality. Any other straight line through this point has a point that has another distance as that to the original straight line, and can therefore not be parallel to it. The Archimedean axiom is to extend to a trans-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 one considers that a straight line must completely pass the interior of a triangle because of its maximality, including its boundary. There must be only enough inner points per side. Maximality means here that longer straight lines than the real numbers define are not to be considered.

Since individual n-ness belongs to every natural number n that cannot be derived from its predecessors or successors, there is no complete system of axioms in mathematics, because with each new number something irreducible new emerges. If one confines, however, to selected aspects, a finite system of axioms for a finite number of entities can be specified. Each level of infinity refuses completeness all the more.

Theories are based on presuppositions. In mathematics, they are often expressed by axioms that may be true or false, what can possibly be proven by other considerations. Thus, all theories are incomplete and, as the case may be, beyond that contradictory. Instead of explicit axioms, (implicit) definitions are more suitable in that the existence of the specified is tacitly presupposed, until refutation.

© 12.11.2010 by Boris Haase


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