Pasch's axiom

Pasch is perhaps best remembered for Pasch's axiom:

Hilbert uses Pasch's axiom in his axiomatic treatment of Euclidean geometry.

Pasch's axiom is distinct from Pasch's theorem which is a statement about the order of points on a line.

Pasch's axiom should not be confused with the Veblen-Young axiom, which may be stated as:

Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.

Given the remaining axioms in Hilbert's system, it can be shown that Pasch's axiom is logically equivalent to the plane separation axiom.

Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (such as the introduction of Pasch's axiom of Euclidean geometry).

In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them.

In other treatments of elementary geometry, using different sets of axioms, Pasch's axiom can be proved as a theorem; it is a consequence of the plane separation axiom when that is taken as one of the axioms.