|Axiom: principle or rule for linking elements of a theory that is not proven within the theory. It is assumed that axioms are true and evident. Adding or eliminating axioms turns a system into another system. Accordingly, more or less statements can be constructed or derived in the new system. See also axiom systems, systems, strength of theories, proofs, provability._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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|I 15 ff
Axioms/Euclid/Waismann: Among the axioms of Euclid, two groups can be distinguished:
A) General size axioms e.g. "Two sizes equal to a third are also equal to each other. The part is smaller than the whole, equal added to equal results in equals".
B) The actual geometrical axioms
1. Each point can be connected to every point by a straight line.
2. Every straight line can be extended beyond each of its endpoints.
3. A circle can be drawn around each point with any radius.
4. All right angles are equal to each other.
5. If two straight lines are intersected by a third such that the angles on the inner side of the two lines to one side of the third give a sum less than two right angles, then the two straight lines intersect, sufficiently extended, on the mentioned side. ("Parallel Axiom")
Parallel axiom/Euclid: in history, the parallel axiom (5th axiom Euclid's) was always controversial, and because of the complexity one tried to derive contradictions, without success. On the contrary, it could be proved that
E.g. similar figures would be impossible if it were not true, or
E.g. there should be a largest triangle in the plane.
E.g. Lambert: without the parallel axiom there would be a length unit distinguished by nature. As absurd as all these results sound, they are not a logical contradiction (this time in favor of the axiom).
Bolanyi and Lobatschefsky then developed consequently conclusions from the omission of the 5th axiom and did not encounter contradictions but a new geometry! Non-Euclidean Geometry.
New problem: how do we know that assumptions will not lead to contradictions in the future? For the first time the problem of non-contradiction in mathematics arose. A direct proof of the consistency is obviously not an option, for this, infinite conclusion chains would have to be considered.
Non-Euclidean Geometry: Felix Klein found in 1870 that the whole system of non-Euclidean geometry can be mapped to the Euclidean, so that any contradiction in the new system would lead to a contradiction in the old.
According to a prescription, a concept of Euclidean geometry is assigned to each concept of non-Euclidean geometry as its image, just as every sentence of one theory corresponds to a sentence of the other,...
...so that both theories have the same logical form.
Within the Euclidean geometry a "model" has been established for the non-Euclidean geometry.
E.g. We imagine in the Euclidean plane a fixed circle k. We now make a lexicon:
By a point we mean a point inside k
By a straight line we understand the piece of a straight line that runs within k.
Additional provisions regulate the possibility that an arbitrary distance on a straight line can be copied infinitely often without leaving the circle. This distance, measured according to the Euclidean scale, is, of course, always smaller. A being that moves from the center of the circle to the periphery becomes smaller and smaller and can never reach the circle's edge (but not zenon).
Proof: This vivid reflection has nothing to do with the power of proof.
For a geometry thus defined, all Euclidean axioms apply except for the fifth.
Fig. I 17
Circle, with rays from a point in the inner of the circle to the outside, somewhere secant. The rays fall into two classes that cut the secant, and those who do not. These two classes are separated by two straight lines (also rays), which we call "parallels", because they intersect the secant (with which they at first glance form a triangle) only non-euclidically at infinity.
All the theorems of Euclidean geometry, with the exception of the fifth axiom, are in the circle consistent.
But this is not an absolute consistency-proof.
If there was a contradiction in the Euclidean geometry, the latter would also have to be applied in the theory of the real numbers._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976