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Incidence Proposition List

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.

Proposition (Point Uniqueness) If and are distinct lines that are not parallel, then and have a unique point in common.

Proposition (Non-Concurrent Lines) There exist three distinct lines that are not concurrent.

Proposition (Point Not On Line) For every line there is at least one point not lying on it.

Proposition (Line Missing Point) For every point there is at least one line not passing through it.

Proposition (Lines Through Point) For every point there exist at least two lines through

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