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Hilbert's Incidence Axioms

By David A. Smith

Hilbert partitioned his axioms for Euclidean geometry into five groups (connection, order, congruence, parallels, continuity) and his axioms of connection (or incidence) will shed some light on his undefined terms. Hilbert's axioms went a long way towards helping people understand that the axiomatic approach to mathematics was a viable and much needed mathematical enterprise in its own right. Many axiomatic systems have been created and even recreated for subjects that are widely understood in the mathematical community. Hilbert's axioms are close in spirit to Euclid's postulates because they do not associate real numbers with segments; i.e. there is no way to measure a line segment. This section details Hilbert's incidence axioms and illustrates some basic propositions that can be proven directly from these axioms.

Axiom (Incidence Axioms) The following axioms are called the Incidence Axioms.

    (i) (Line Uniqueness) Given two distinct points and there exists a unique line incident with and
    
    (ii) (Points On Line) For every line there exist at least two distinct points incident with
    
    (iii) (Non-Collinear Points) There exists three non-collinear points.

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