Hilbert's axioms [SEP] creator
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book "Grundlagen der Geometrie" (tr. "The Foundations of Geometry") as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
The axioms.
Hilbert's axiom system is constructed with six primitive notions: three primitive terms:
- point;
- line;
- plane;
have different signs.
- "F" is the intersection of its Euclidean closures.
Models of geometry.
Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry . The coordinate geometry given by formula_14 for formula_1 a Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. However, in general this geometry need not satisfy all Hilbert's axioms unless the field "F" has extra properties: for example, if