Axiom:Hilbert's Axioms

Definition

Hilbert's Axioms are a series of axioms whose purpose is to provide a rigorous basis for the definition of Euclidean geometry.

In the following the primitive objects are:

points (denoted by $A, B, C \ldots$)
lines (denoted by $a, b, c \ldots$)
planes (denoted by $\alpha, \beta, \gamma \ldots$)

the composite objects are

segment
ray
angle

the primitive relations are:

distinct (binary)(defined for all objects)
betweenness (ternary)(point between two other points)
lies on (binary)(point lies on line, point lies on ray, segment lies on line)
congruence (binary)(defined for segments and angles).

the composite relations are:

collinear ($n$-ary)(defined for points
coplanar ($n$-ary)(defined for points)
insideness (binary)(point between segment endpoints)
intersection (binary)(line, ray, segment)

Collinearity

Let $A, B, \ldots$ be distinct points.

Let $a$ be a line.

Suppose $A, B, \ldots$ lie on $a$.

Then $A, B, \ldots$ are said to be collinear.

Coplanarity

Let $A, B, \ldots$ be distinct points.

Let $\alpha$ be a plane.

Suppose $A, B, \ldots$ lie on $a$.

Then $A, B, \ldots$ are said to be coplanar.

Segment

Let $A$, $B$ be distinct points.

Then by segment we mean all the points which are same as $A$ or $B$ or are between $A$ and $B$.

Congruence of Segments

Let $\overline {AB}$, $\overline {A'B'}$ be distinct segments.

Let $\overline {AB}, \overline {A'B'}$ be congruent.

Then we denote this by $\overline{AB} \cong \overline{A'B'}$.

Ray

Let $O$, $A$ be distinct points.

Let $P$ be a point such that $P = O$ or $P$ is between $O$ and $B$ or $B$ is between $O$ and $P$.

The the collection of all such $P$ is called the ray $\buildrel \to \over {O A}$.

Intersection

Let $a$ and $b$ be distinct extended structures like segments, rays or lines with their types not necessarily matching.

Suppose there is a point $P$ that lies both on $a$ and $b$.

Then we say that $a$ and $b$ intersect.

Interior Point of Ray

Let $\buildrel \to \over {A B}$ be a ray.

Suppose $p$ is a point such that $p$ lies on $\buildrel \to \over {A B}$ and $p \ne A$.

Then $p$ is called the interior point (of $\buildrel \to \over {A B}$).

Interior Point of Angle

Let $\angle AOB$ be an angle.

Let $A'$ be a point on the ray $\vec {OA}$.

Let $B'$ be a point on the ray $\vec {OB}$.

Let $P$ be a point between $A'$ and $B'$.

Then $P$ is called an interior point of $\angle AOB$.

Angle

Let $A$, $O$, $B$ be noncolinear distinct points.

Let $\buildrel \to \over {O A}$ and $\buildrel \to \over {O B}$ be distinct rays.

Then the collection of $\buildrel \to \over {O A}$ and $\buildrel \to \over {O B}$ is called the angle $\angle AOB$.

Congruence of Angles

Let $\angle ABC$ and $\angle A'B'C'$ be distinct angles.

Suppose $\angle ABC$ and $\angle A'B'C'$ are congruent.

Then we denote this by $\angle ABC \cong \angle A'B'C'$.

Points on the Same Side

Let $l$ be a line.

Let $A$, $B$ be distinct points.

Suppose $A$ and $B$ do not lie on $l$.

Let $AB$ be a segment.

Suppose $AB$ does not intersect $l$.

Then we say that $A$ and $B$ are on the same side of $l$.

Side of Line

Let $l$ be a line on the plane $\alpha$.

All points on $\alpha$ lying on the same side of $l$ are called the side (of $l$).

Parallel lines

Let $l, l'$ be distinct lines.

Suppose no point lies both on both $l$ and $l'$.

Then $l$ and $l'$ are said to be parallel.

Incidence axiom 1

Let $A, B$ be two distinct points.

Then there is only one line $a$ such that $A, B$ lie on $a$.

Incidence axiom 2

Let $a$ be a line.

Then there are two distinct points $A$ and $B$ which lie on $a$.

Incidence axiom 3

Let $A, B, C$ be three distinct noncollinear points.

Then there is only one plane $\alpha$ such that $A, B, C$ lie on $\alpha$.

Incidence axiom 4

Let $\alpha$ be a plane.

Then there are three noncollinear distinct points $A, B, C$ which lie on $\alpha$.

Incidence axiom 5

Let $A, B$ be two distinct points.

Let $a$ be the line on which $A, B$ lie.

Suppose $A, B$ lie on the plane $\alpha$.

Then every point of the line $a$ lies on $\alpha$.

Incidence axiom 6

Let $\alpha, \beta$ be distinct planes.

Suppose the point $A$ lies on $\alpha$ and $\beta$.

Then there is another point $B$ which lies on $\alpha$ and $\beta$.

Incidence axiom 7

Let $\alpha$ be a plane.

Then there are four distinct points which are not coplanar.

Order axiom 1

Let $A$, $B$, $C$ be points.

Suppose $B$ lies between $A$ and $C$

Then $A$, $B$, $C$ are distinct and collinear, and $B$ lies between $C$ and $A$.

Order axiom 2

Let $a$ be a line.

Let $A$, $C$ be distinct points on $a$.

Then there are points $B, D$ on $a$ such that $C$ lies between $A$ and $B$, and $C$ lies between $A$ and $D$.

Order axiom 3

Let $A$, $B$, $C$ be distinct collinear points.

Then no more than one of them lies between the other two.

Order axiom 4 (Axiom of Pasch)

Let $A$, $B$, $C$ be non-collinear points.

Let $a$ a line such that $A$, $B$, $C$ do not lie on $a$.

Suppose there is a point $D$ that lies on $a$ and is in the segment $AB$.

Then there is a point $E$ which lies on $a$ and is in $AC$ or $BC$.

Congruence Axiom 1

Let $l, l'$ be same or distinct lines.

Let $A, B$ be distinct points on $l$.

Suppose $A'$ is a point on $l'$.

Then there are two and only two points $B'$ and $C'$ on $l'$ such that all the following conditions hold:

$A'$ is between $B'$ and $C'$;
$\overline {AB} \cong \overline {A'B'}$;
$\overline {AB} \cong \overline {C'A'}$.

Congruence Axiom 2

Let $\overline {AB}, \overline {A'B'}, \overline { {A'}'{B'}'}$ be segments.

Suppose $\overline {A'B'} \cong \overline {AB}$ and $\overline { {A'}'{B'}'} \cong \overline {AB}$.

Then $\overline {A'B'} \cong \overline { {A'}'{B'}'}$

Congruence Axiom 3

Let $l, l'$ be distinct lines.

Let $\overline{AB}$, $\overline{BC}$ be segments on $l$ with a commont point $B$.

Let $\overline{A'B'}$, $\overline{B'C'}$ be segments both either on $l$ or $l'$ with a common point $B'$:

Suppose $\overline {AB} \cong \overline{A'B'}$ and $\overline{BC} \cong \overline{B'C'}$.

Then $\overline{AC} \cong \overline{A'C'}$.

Congruence Axiom 4

Let $\angle AOB$ be an angle on the plane $\alpha$.

Let $l'$ be a line on the plane $\alpha'$.

Let $\vec{O'A'}$ be a ray.

Suppose $\vec{O'A'}$ lies on $l'$.

Then there is only one ray $\vec{O'B'}$ such that $\angle AOB \cong \angle A'O'B'$ and all interior points of $\angle A'O'B'$ lie on the given side of $l'$.

Congruence Axiom 5

Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles.

Suppose:

$\overline{AB} \cong \overline{A'B'}$
$\overline{AC} \cong \overline {A'C'}$
$\angle BAC \cong \angle B'A'C'$

Then $\angle ABC \cong \angle A'B'C'$ and $\angle ACB \cong \angle A'C'B'$.

Congruence Axiom 6

Let $\angle AOB, \angle A'O'B', \angle {A'}'{O'}'{B'}'$ be angles.

Suppose $\angle AOB \cong \angle A'O'B'$ and $\angle AOB \cong \angle {A'}'{O'}'{B'}'$.

Then $\angle A'O'B' \cong \angle {A'}'{O'}'{B'}'$.

Euclidean Parallel Postulate

Let $l$ be a line on the plane $\alpha$.

Let $A$ be a point that does not lie on $l$.

Then there exists a unique line $l'$ on $\alpha$ such that $A$ lies on $l'$ and $l'$ is parallel to $l$.

Axiom of Continuity (Archimedes Axiom)

Let $\vec{AB}$ be a ray.

Let $A_1$ be a point between $A$ and $B$.

Let $A_2$ be a point such that $A_1$ lies between $A$ and $A_2$.

For $i\in\N_{>2}$ let $A_i$ be a new point such that $A_{i - 1}$ lies between $A_{i - 2}$ and $A_i$.

Suppose $AA_1 \cong A_1 A_2 \cong \ldots \cong A_i A_{i+1} \cong \ldots$.

Then there is $n \in \N_{>0}$ such that $B$ lies between $A_{n - 1}$ and $A_n$.

Axiom of Completeness

There are no other primitive objects that could be added to points, lines and planes while satisfying all of the axioms above.

Source of Name

This entry was named for David Hilbert.