Section: Axioms of Connection and Their Consequences

As mentioned above, David Hilbert (1862 - 1943) proposed an axiomatic system dealing with geometrical objects like "points", "straight lines", and "planes", detaching them from their specific meaning. Hilbert wanted to avoid any attempt to define exactly what a point, a straight line, or a plane is. He rather focussed on the relationships between these objects. The following definitions and axioms demonstrate this approach.

  1. Definition: Points, Straight Lines, and Planes
  2. Definition: "Lies on" Relation
  3. Axiom: Axioms of Connection
  4. Proposition: Common Points of Two Distinct Straight Lines in a Plane
  5. Proposition: Common Points of a Plane and a Straight Line Not in the Plane
  6. Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line
  7. Proposition: Plane Determined by two Crossing Straight Lines

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  2. Berchtold, Florian: "Geometrie", Springer Spektrum, 2017
  3. Klotzek, B.: "Geometrie", Studienbücherei, 1971
  4. Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903