Appendix A - Hilbert's Axioms for Euclidean Geometry
Mathematics is a game played according to certain rules with meaningless marks on paper.
—David Hilbert (1862–1943)
Introductory Note. Hilbert's Axiom set is an example of what is called a synthetic geometry. A synthetic geometry has betweenness and congruence as undefined terms, properties of congruence are given in the axioms, does not have axioms for distance and angle measure. However, Axiom V.1 is related to measurement of length by giving a method of comparing two arbitrary segments.
Undefined Terms point, line, plane, lie, between, and congruence.
(Some web browsers display some characters incorrectly, an angle shows as ∠, congruent shows as ≅, and the Greek characters alpha and beta show as α, β.)
Group I. Axioms of Incidence
I.1. For every two points A, B, there exists a line m that contains each of the points A, B.
I.2. For every two points A, B, there is not more than one line m that contain each of the points A,B.
I.3. There exist at least two points on a line. There exist at least three points that do not lie on a line.
I.4. For any three points A, B, C that do not lie on the same line, there exists a plane a that contains each of the points A, B, C. For every plane there exists a point which it contains.
I.5. For any three points A, B, C that do not lie on the same line, there exists no more than one plane that contains each of the three points A, B, C.
I.6. If two points A, B of a line m lie in a plane α, then every point of m lies in the plane α.
I.7. If two planes α, β have a point A in common, then they have at least one more point B in common.
I.8. There exist at least four points which do not lie in a plane.
Group II. Axioms of Order
II.1. If point B lies between points A and C, then A, B, C are three distinct points of a line, and B also lies between C and A.
II.2. For any two distinct points A and C, there exists at least one point B on the line AC such that C lies between A and B.
II.3. Of any three points on a line there exists no more than one that lies between the other two.
II.4. Let A, B, C be three points that do not lie on a line and let m be a line in the plane ABC which does not meet any of the points A, B, C. If the line m passes through a point of the segment AB, it also passes through a point of the segment AC or segment BC.
Group III. Axioms of Congruence
III.1. If A, B are two points on a line m, and A' is a point on the same or on another line m' then it is always possible to find a point B' on a given side of the line m' through A' such that the segment AB is congruent to the segment A'B'. In symbols AB ≅ A'B'.
III.2. If two segments are congruent to a third one, they are congruent to each other.
III.3. On the line m let AB and BC be two segments which except for B have no point in common. Furthermore, on the same or on another line m' let A'B' and B'C' be two segments which except for B' also have no point in common. In that case, if AB ≅ A'B' and BC ≅ B'C', then AC ≅ A'C'.
III.4. Let ∠(h,k) be an angle in a plane α and m' a line in a plane α' and let a definite side of m' in α' be given. Let h' be a ray on the line m' that emanates from the point O'. Then there exists in the plane α' one and only one ray k' such that the angle ∠(h,k) is congruent to the angle ∠(h',k') and at the same time all interior points of the angle ∠(h',k') lie on the given side of m'. Symbolically ∠(h, k) ≅ ∠(h',k'). Every angle is congruent to itself.
III.5. If for two triangles ABC and A'B'C' the congruences AB ≅ A'B', AC ≅ A'C', ∠BAC ≅ ∠B'A'C' hold, then the congruence ∠ABC ≅ ∠A'B'C' is also satisfied.
Group IV. Axiom of Parallels
IV. Let m be any line and A be a point not on it. Then there is at most one line in the plane, determined by m and A, that passes through A and does not intersect m.
Group V. Axioms of Continuity
V.1. (Axiom of measure or Archimedes' Axiom) If AB and CD are any segments, then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B.
V.2. (Axiom of line completeness) An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and form V.1 is impossible.
Hilbert, David, Foundations of Geometry (Grundlagen der Geometrie), Second English Edition trans. by Unger,L. LaSalle: Open Court Publishing Company, 1971 (1899).
© Copyright 2005, 2006 - Timothy Peil