Axiom system, Euclidean geometry
The axiom system in Euklid (and Hilbert) has not been chosen arbitrarily, but is an abstraction from thousands of years of human experience. The associated geometry is therefore the geometry of our visual space.
Until the end of the 19th century, all natural science and technology was based on this Euclidean geometry.
The axiom system in Euklid and also in Hilbert was not chosen arbitrarily, but an abstraction from the world of human experience over thousands of years. The associated geometry is therefore the geometry of our visual space - do my assignment for me . Until the end of the 19th century, all natural science and technology was based on this Euclidean geometry. It was only modern natural science that showed that we had to move on to more general geometries in order to describe the new findings correctly.
Basic terms for defining Euclidean space are "point", "straight line" and "plane" as well as the term "congruence". The complete and contradiction-free axiom system of Euclid-Hilbertian geometry includes the following axioms of incidence (connection), arrangement, parallelism and continuity - math problem solver , whose independence Hilbert was able to prove:
- Axioms of incidence (connection)
- Axioms of arrangement
- Axioms of congruence
- Axiom of parallels - Euclidean axiom of parallels
- Axiom of continuity - Archimedes' axiom
- Axioms of motion
The absence of contradictions as well as the completeness and independence of this system of axioms can be proven, but this is complicated due to the large number of axioms - https://domyhomework.club/analogy-homework/ . For example, the independence of the parallel axiom has only been shown with the development of non-Euclidean geometries.
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