Hilbert's axioms
Hilbert's axioms are a formal system for Euclidean geometry that organizes assumptions about incidence, order, congruence, parallels, and continuity.
Euclidean geometry: The geometry of points, lines, planes, distances, and angles governed by Euclid's parallel postulate. Hilbert's system formalizes this geometry while making its assumptions explicit.
Axiom: A basic statement accepted as a starting assumption within a formal theory. Each group supplies assumptions from which geometric theorems can be proved.
David Hilbert: A German mathematician whose work shaped geometry, number theory, mathematical logic, and mathematical physics. Hilbert presented the axiom system in his 1899 book on the foundations of geometry.
Non-Euclidean geometry: Geometries that reject or modify Euclid's parallel postulate while retaining other geometric principles. Changing the parallel axiom yields geometries outside Hilbert's Euclidean system.
Incidence geometry: A study of points, lines, and higher-dimensional spaces connected by incidence relations. Hilbert's incidence axioms specify which points lie on which lines and planes.
Formal system: A collection of symbols, formation rules, axioms, and inference rules for deriving statements. Hilbert's axioms function as the starting statements of a geometric formal system.
Foundations of Geometry: David Hilbert's 1899 book establishing a systematic axiomatic treatment of Euclidean geometry. This book introduced the axioms and investigated their logical relations.
Hyperbolic geometry: A geometry in which multiple lines through a point can be parallel to a given line. Its alternative parallel behavior directly contrasts with Hilbert's Euclidean parallel axiom.
Betweenness: A relation that states when one point lies between two other points on a line. Order axioms use betweenness to describe how points are arranged along lines.
Independence (mathematical logic): A statement is independent of an axiom set when neither it nor its negation is derivable from that set. Independence tests ask whether a Hilbert axiom contributes an assumption not implied by the others.