Hilbert's axioms

David Hilbert's axioms are a set of 20 assumptions (originally 21) 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.

Contents

1 Postulates 1.1 I. Axioms of incidence 1.1.1 Postulate I.1 1.1.2 Postulate I.2 1.1.3 Postulate I.3 1.1.4 Postulate I.4 1.1.5 Postulate I.5 1.1.6 Postulate I.6 1.1.7 Postulate I.7 1.1.8 Postulate 1.8 1.2 II. Axioms of order 1.2.1 Postulate II.1 1.2.2 Postulate II.2 1.2.3 Postulate II.3 1.2.4 Postulate II.4 1.3 III. Axioms of congruence 1.3.1 Postulate III.1 1.3.2 Postulate III.2 1.3.3 Postulate III.3 1.3.4 Postulate III.4 1.3.5 Postulate III.5 1.4 IV. Axiom of parallels 1.4.1 Postulate IV.1 1.5 V. Axioms of continuity 1.5.1 Postulate V.1 (Archimedes axiom) 1.5.2 Postulate V.2 (line completeness) 2 References 3 External links

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 no more than one line containing both A and B

Postulate I.3

There exists at least two points on any given line. There exists 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, one and only one of the points is between the other two

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 but does not contain any of the points A, B, C: if the line a passes through a point of the segment AB, then 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 A"B" such that both are congruent to the same segment AB, then A'B' <math>\cong<math>A"B"

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 a number n such that 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

External links

• Math Department at the UMBC (http://www.math.umbc.edu/~campbell/Math306Spr02/Axioms/Hilbert.html)
• Mathworld (http://mathworld.wolfram.com/HilbertsAxioms.html)fr:Axiomes de Hilbert de la géométrie euclidienne

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