Hilbert's axioms

David Hilbert's axioms are a set of 21 assumptions designed to form the foundation for a modern treatment of Euclidean geometry. The axioms were originally published in Grundlagen der Geometrie (Foundations of Geometry) in 1899.

Postulates

I. Axioms of Incidence

Postulate I.1

For any two points A, B, there exists a line a that contains each of the points A, B

Postulate I.2

For any two points A, B there exists one and only one line containing both A and B

Postulate I.3

There exist at least two points on any given line. There exist at least three points that do not lie on a given line

Postulate I.4

For a set of three points {A, B, C} that do not lie on the same line, there exists a plane α that contains each of the points in the set. For every plane there exists at least one point which it contains.

Postulate I.5

For a set of three points {A, B, C} that do not all lie on the same line, there exists only one plane that contains each of the points in the set.

Postulate I.6

If two points {A, B} of a line, a, lie in a plane, α, then every point in a lies in α

Postulate I.7

If two planes {α, β} have a point A in common, then they have at least one other point, B, in common

Postulate 1.8

There exist at least four points which do not lie in a plane

II. Axioms of Order

Postulate II.1

If a point B lies between points A and C, then the points {A, B, C} are three distinct points on the same line and B lies between C and A

Postulate II.2

Given two points {A, C}, a point B exists on the line AC such that C lies between A and B

Postulate II.3

Given any three points {A, B, C} of a line, there exists one and only one point in the set that lies betwen the other two points in the set

Postulate II.4

Given three points {A, B, C} that do not lie on a line and given a line, a, that lies in the plane ABC which does not not intersect any of the points A, B, C: if the line a passes through a point of the segment AB, it also passes through a point in the segment AC or through a point in the segment BC

III. Axioms of Congruence

Postulate III.1

Given two points {A, B<i>} on a line <i>a and given a point A' on a or another line a', there exists a point B' on a side of the line a' such that AB<math>\cong<math>A'B' are congruent

Postulate III.2

Given segments A'B' and AB such that both are congruent to the same segment AB, then A'B'<math>\cong<math>AB

Postulate III.3

Given a line a with segments AB and BC such that the point B is the only intersection of the two points and on the same line or a line a' with segments A'B'<i> and <i>B'C' such that the point B' is the only intersection: if AB<math>\cong<math>A'B' and BC<math>\cong<math>B'C' then AC<math>\cong<math>A'C'

Postulate III.4

If <math>\angle<math>ABC is an angle and B'C' is a ray, then there is one and only one ray B'A' on each side of the line B'C'<i> such that <math>\angle<math><i>A'B'C'<math>\cong<math><math>\angle<math>ABC Corollary: Every angle is congruent to itself

Postulate III.5

Given two triangles ABC and A'B'C' with congruences such that AB<math>\cong<math>A'B', AC<math>\cong<math><i>A'C' and <math>\angle<math>BAC<math>\cong<math><math>\angle<math><i>B'A'C' then <math>\angle<math>ABC<math>\cong<math><math>\angle<math>A'B'C'.

IV. Axiom of Parallels

Postulate IV.1

Given a line a and a point A not on a, there is at most one line in the plane that contains a and A that passes through A and does not intersect </i>a</i>

V. Axioms of Continuity

Postulate V.1 (Archimedes Axiom)

Given segments AB and CD, there exists n copies of CD constructed contiguously from A along the ray AB will pass beyond the point B

Postulate V.2 (Line Completeness)

There exists no extension of a set of points on a line with order and congruence relations that would preserve the relations existing among the original elements as well as preserving line order and congruence, i.e., Axioms I-III and V.1.

References

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

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