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for Euclidean geometry

A system of axioms first proposed by D. Hilbert in 1899, and subsequently amended and made more precise by him.

Hilbert's system of axioms

The primary (primitive undefined) objects are $points$, (straight) $lines$, $planes$.

The relations between these objects are belonging to, being between, and being congruent to.

The nature of both the primary objects and the relations between those objects are arbitrary as long as the objects and the relations satisfy the axioms.

Hilbert's system contains 20 axioms, which are subdivided into five groups.

Group I: Axioms of Incidence or Connection

This group comprises 8 axioms describing the relation belonging to.

$\mathbf{I}_1$. For any two points there exists a straight line passing through them.

$\mathbf{I}_2$. There exists only one straight line passing through any two distinct points.

$\mathbf{I}_3$. At least two points lie on any straight line. There exist at least three points not lying on the same straight line.

$\mathbf{I}_4$. There exists a plane passing through any three points not lying on the same straight line. At least one point lies on any given plane.

$\mathbf{I}_5$. There exists only one plane passing through any three points not lying on the same straight line.

$\mathbf{I}_6$. If two points $A$ and $B$ of a straight line $a$ lie in a plane $\alpha$, then all points of $a$ lie in $\alpha$.

$\mathbf{I}_7$. If two planes have one point in common, then they have at least one more point in common.

$\mathbf{I}_8$. There exist at least four points not lying in the same plane.

Group II: Axioms of Order

This group comprises four axioms describing the relation being between.

$\mathbf{II}_1$. If a point $B$ lies between a point $A$ and a point $C$, then $A$, $B$, and $C$ are distinct points on the same straight line and $B$ also lies between $C$ and $A$.

Definitions:

• The line segment $AC$ is the set of points $A$, $C$, and all points lying between $A$ and $C$.
• Points $A$ and $C$ are the endpoints of the segment
• Point $B$ and all other points between $A$ and $C$ are interior points of the segment

$\mathbf{II}_2$. For any two points $A$ and $B$ on the straight line $AB$, there exists at least one point $C$ such that the point $B$ lies between $A$ and $C$.

$\mathbf{II}_3$.Out of any three points on the same straight line there exists not more than one point lying between the other two.

$\mathbf{II}_4$.(Pasch's axiom). Let $A$, $B$, and $C$ be three points not lying on the same straight line,

and let $a$ be a straight line in the plane $ABC$ not passing through any of the points $A$, $B$, or $C$.

Then, if the straight line $a$ passes through an interior point of the segment $AB$,

it also passes through an interior point of the segment $AC$ or through an interior point of the segment $BC$.

Group III: Axioms of Congruence

This group comprises five axioms describing the relation "being congruent to" (Hilbert denoted this relation by the symbol $\equiv$).

$\mathbf{III}_1$. Given a segment $AB$ and a ray $OX$, there exists a point $B’$ on $OX$ such that the segment $AB$ is congruent to the segment $OB’$, i.e. $AB \equiv OB’$.

$\mathbf{III}_2$. If $A’B’ \equiv AB$ and $A’’B’’ \equiv AB$, then $A’B’ \equiv A’’B’’$.

$\mathbf{III}_3$. Let $AB$ and $BC$ be two segments on a straight line without common interior points,

and let $A’B’$ and $B’C’$ be two segments on the same or on a different straight line, also without any common interior points.

If $AB \equiv A’B’$ and $BC \equiv B’C’$, then $AC \equiv A’C’$.

$\mathbf{III}_4$. Let there be given an angle $\langle AOB$, a ray $O’A’$ and a half-plane $\Pi$ bounded by the straight line $O’A’$.

Then $\Pi$ contains one and only one ray $O’B’$ such that $\langle AOB \equiv \langle A’O’B’$. Moreover, every angle is congruent to itself.

$\mathbf{III}_5$. If for two triangles $ABC$ and $A’B’C’$ one has $AB \equiv A’B’$, $AC \equiv A’C’$, $\langle BAC \equiv \langle B’A’C’$, then $\langle ABC \equiv \langle A’B’C’$.

Group IV: Axioms of Continuity

This group comprises two continuity axioms.

$\mathbf{IV}_1$. (Archimedes' axiom). Let $AB$ and $CD$ be two arbitrary segments.

Then the straight line $AB$ contains a finite set of points $A_1,\dotsc, A_n$

such that the point $A_1$ lies between $A$ and $A_2$, the point $A_2$ lies between $A_1$ and $A_3$, etc.,

and such that the segments $AA_1,\dotsc,A_{n-1}A_n$ are congruent to the segment $CD$,

and $B$ lies between $A$ and $A_n$.

$\mathbf{IV}_2$. (Cantor's axiom). Let there be given, on any straight line $a$, an infinite sequence of segments $A_1B_1, A_2B_2,\dotsc,$ which satisfies two conditions:

1. each segment in the sequence forms a part of the segment which precedes it;
2. for each preassigned segment $CD$ it is possible to find a natural number $n$ such that $A_nB_n < CD$.

Then $a$ contains a point $M$ belonging to all the segments of this sequence.

Group V: Axiom of Parallelism

This group comprises one axiom about parallels.

$\mathbf{V}_1$. Let there be given a straight line $a$ and a point $A$ not on that straight line.

Then there exists not more than one straight line passing through $A$ not intersecting $a$ and lying in the plane defined by $a$ and $A$.

Hilbert’s system and Euclid’s Elements

Hilbert's system of axioms was the first fairly rigorous foundation of Euclidean geometry.

All elements (terms, axioms, and postulates) of Euclidean geometry that are not explicitly stated in Hilbert’s system can be defined by or derived from the basic elements (objects, relations, and axioms) of his system.

Similarly, all the propositions, theorems, and constructions of Euclidean geometry not specifically stated in Hilbert’s system are logically deducible from his axioms, or from statements which are deducible from these axioms.

Metamathematics of Hilbert’s system

Hilbert's system of axioms is complete.

Further, if the arithmetic of real numbers is consistent, then Hilbert’s system is consistent.

The Axiom of Parallelism is independent of the other axioms, shown by the following:

replacing the Axiom of Parallelism by its negation yields a new system of axioms (the system of axioms of Lobachevskii geometry) that is also consistent

Other axioms of this system are also demonstrably independent of one other.

Historical note

In Hilbert’s original (German) system, the axioms were grouped differently than shown above:

• Group IV contained the Axiom of Parallelism
• Group V contained a single Axiom of Continuity -- Archimemes’ Axiom

Shortly afterwards, in translations (French/English) of his original system, Hilbert added a second Axiom of Continuity -- an Axiom of Completeness of his own devising. In subsequent editions and translations, the Axiom of Completeness has been based on various definitions of the real numbers. The axiom shown above is based on Cantor’s definition.

Primary sources

• Hilbert, D. (1899). "Grundlagen der Geometrie". [Reprint (1968) Teubner.]

References

• Aleksandrov, A.D. "Foundations of geometry," Siberian Mathematical Journal, July 1987, Vol. 28, Issue 4, pp 523-539. [Trans. Sibirskii Matematicheskii Zhurnal, Vol. 28, No. 4, pp. 9–28, July–August, 1987.]

• Efimov, N.V. (1960). "Höhere Geometrie", Deutsch. Verlag Wissenschaft. [Translated from Russian.]

• Forder, H.G. "Foundations of Euclidean geometry". [Reprint (1958) Dover.]