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Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates. In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the Elements. For example, Euclid assumed ...
Supporting hyperplane theorem (convex geometry) Swan's theorem (module theory) Sylow theorems (group theory) Sylvester's determinant theorem (determinants) Sylvester's theorem (number theory) Sylvester pentahedral theorem (invariant theory) Sylvester's law of inertia (quadratic forms) Sylvester–Gallai theorem (plane geometry)
The various attempted proofs of the parallel postulate produced a long list of theorems that are equivalent to the parallel postulate. Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms.
It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines.
Boolean prime ideal theorem; ... Geometry. Parallel postulate; Birkhoff's axioms (4 axioms) Hilbert's axioms (20 axioms)
Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms". [5] In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of ...