Hilbert's Axioms for Geometry

# Hilbert's Axioms

## Undefined Terms

• Points
• Lines
• Planes
• Lie on, contains
• Between
• Congruent

## Axioms

1. Axioms of Incidence
Postulate I.1.
For every two points A, B there exists a line a that contains each of the points A, B.
Postulate I.2.
For every two points A, B there exists no more than one line that contains each of the points A, B.
Postulate I.3.
There exists at least two points on a line. There exist at least three points that do not lie on a line.
Postulate I.4.
For any three points A, B, C that do not lie on the same line there exists a plane α that contains each of the points A, B, C. For every plane there exists a point which it contains.
Postulate I.5.
For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C.
Postulate I.6.
If two points A, B of a line a lie in a plane α then every point of a lies in the plane α.
Postulate I.7.
If two planes α, β have a point A in common then they have at least one more point B in common.
Postulate I.8.
There exist at least four points which do not lie in a plane.
2. Axioms of Order
Postulate II.1.
If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line, and B then also lies between C and A.
Postulate II.2.
For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B.
Postulate II.3.
Of any three points on a line there exists no more than one that lies between the other two.
Postulate II.4.
Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet 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 of the segment AC, or through a point of the segment BC.
3. Axioms of Congruence
Postulate III.1.
If A, B are two points on a line a, and A' is a point on the same or on another line a' then it is always possible to find a point B' on a given side of the line a' such that AB and A'B' are congruent.
Postulate III.2.
If a segment A'B' and a segment A"B" are congruent to the same segment AB, then segments A'B' and A"B" are congruent to each other.
Postulate III.3.
On a line a, let AB and BC be two segments which, except for B, have no points in common. Furthermore, on the same or another line a', let A'B' and B'C' be two segments which, except for B', have no points in common. In that case if ABA'B' and BCB'C', then ACA'C'.
Postulate III.4.
If ∠ABC is an angle and if B'C' is a ray, then there is exactly one ray B'A' on each "side" of line B'C' such that A'B'C'≅∠ABC. Furthermore, every angle is congruent to itself.
Postulate III.5. (SAS)
If for two triangles ABC and A'B'C' the congruences ABA'B', ACA'C' and BAC ≈ ∠B'A'C' are valid, then the congruence ABC ≈ ∠A'B'C' is also satisfied.
4. Axiom of Parallels
Postulate IV.1.
Let a be any line and A a point not on it. Then there is at most one line in the plane that contains a and A that passes through A and does not intersect a.
5. Axioms of Continuity
Postulate V.1. (Archimedes Axiom)
If AB and CD are any segments, then there exists a number n such that n copies of CD constructed contiguously from A along the ray AB willl pass beyond the point B.
Postulate V.2. (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 (Axioms I-III and V-1) is impossible.

### Sources:

1. Foundations of Geometry, D. Hilbert (trans. L. Unger), Open Court Publ.
2. Roads to Geometry, E. Wallace & S. West, Prentice-Hall (Sect 2.4 & App. B)

R. I. Campbell, campbell@math.umbc.edu
10 Feb, 2002