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Hilbert's Axioms

The 21 assumptions which underlie the Geometry published in Hilbert's classic text Grundlagen der Geometrie. The eight Incidence Axioms concern collinearity and intersection and include the first of Euclid's Postulates. The four Ordering Axioms concern the arrangement of points, the five Congruence Axioms concern geometric equivalence, and the three Continuity Axioms concern continuity. There is also a single parallel axiom equivalent to Euclid's Parallel Postulate.

See also Congruence Axioms, Continuity Axioms, Incidence Axioms, Ordering Axioms, Parallel Postulate


Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980.

Iyanaga, S. and Kawada, Y. (Eds.). ``Hilbert's System of Axioms.'' §163B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 544-545, 1980.

© 1996-9 Eric W. Weisstein