# Axiom:Hilbert's Axioms

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## Definition

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

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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.

## Sources

- 1902: David Hilbert:
*The Foundations of Geometry*(translated by E.J. Townsend): Chapter $1$: The Five Groups of Axioms - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Euclidean geometry** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Hilbert's axioms** - 2003: L.I. Meikle and J.D. Fleuriot:
*Formalizing Hilbert's Grundlagen in Isabelle/Isar*(*TPHOLs 2003***Vol. 2758**: pp. 319 – 334) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Euclidean geometry** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Hilbert's axioms** - 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Euclidean and Non-Euclidean Geometries