Four point geometry satisfies the Elliptic Parallel Postulat

Four point geometry satisfies the Elliptic Parallel Postulate.

Solution

The four point geometry has the following axioms.

Elliptic Parallel Postulate: Any two lines intersect in at least one point.

From these properties of a sphere, we see that in order to formulate a consistent axiomatic system, several of the axioms from a neutral geometry need to be dropped or modified, whether using either Hilbert\'s or Birkhoff\'s axioms. The incidence axiom that \"any two points determine a unique line,\" needs to be modified to read \"any two points determine at least one line.\" Hilbert\'s Axioms of Order (betweenness of points) may be replaced with axioms of separation that give the properties of how points of a line separate each other. (For a listing of separation axioms see Euclidean and Non-Euclidean Geometries Development and History by Greenberg.) With these modifications made to the axiom system, the Elliptic Parallel Postulate may be added to form a consistent system. Often spherical geometry is called double elliptic geometry, since two distinct lines intersect in two points.

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