Hilbert's system of axioms of Euclidean geometry

David Hilbert uses for his axiomatic foundation of Euclidean geometry (in three-dimensional space) "three different systems of things", namely points , lines and planes , and "three basic relationships", namely lie , between and congruent . As a formalist , Hilbert makes no assumptions about the nature of these “things” or their “relationships” . They are only defined implicitly , namely through their connection in a system of axioms .

Hilbert is said to have once said that instead of “points, straight lines and levels” one could also say “tables, chairs and beer mugs ” at any time ; all that matters is that the axioms are fulfilled. However, he went to great lengths to ensure that his “tables, chairs and beer mugs” fulfilled all the laws that the geometers of the previous two thousand years discovered for “points, straight lines and planes”. The strength of the axiomatic approach is not that it disregards reality. But it allows, by changing the axioms and analyzing their connection, to examine the logical structure that this reality follows in a way that was previously unthinkable.

Absolute geometry can be based on a system of axioms that is weakened compared to Hilbert's system and has no axiom of parallels : There are either no parallels there ( elliptical geometry ) or any number of parallels ( hyperbolic geometry ) through a point outside a straight line . The hyperbolic geometry fulfills Hilbert's axiom groups I-III and V, the elliptical geometry I, II and V and a weaker version of the congruence axioms (III).

The axioms

To this end, Hilbert links "things" and "relationships" through 21 axioms in five groups:

Axioms of Linkage (or Incidence), Group I.

With these axioms, the term lie should be implicitly defined. Hilbert uses the term determine or belong together and a number of other ways of speaking: goes through , connects and , lies on , is a point from , on there is the point etc. Today in mathematics one speaks of incidence : " incised " (formally: ). ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle PIg}$

• I.1. Two different points and always define a straight line . ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle g}$
• I.2. Any two different points on a straight line determine this straight line.
• I.3. On a straight line there are always at least two points, in a plane there are always at least three points that are not on a straight line.
• I.4. Three points that are not on the same straight line always define a plane. ${\displaystyle P,Q,R}$
• I.5. Any three points on a plane that are not on the same straight line determine this plane.
• I.6. If two points and a straight line lie in one plane , each point lies in . ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle g}$ ${\displaystyle \alpha }$ ${\displaystyle g}$ ${\displaystyle \alpha }$
• I.7. If two levels and have one point in common, they have at least one other point in common. ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle P}$ ${\displaystyle Q}$
• I.8. There are at least four non-plane points.

Axioms 1–3 are called planar axioms of group I and axioms 4–8 are called spatial axioms of group I.

From these axioms alone it can be concluded, for example,

• that two different straight lines intersect at exactly one point or not at all,
• that two different planes intersect in exactly one straight line or not at all,
• that a plane and a straight line not lying in it intersect at exactly one point or not at all,
• that a straight line and a point not lying on it determine a plane,
• that two different, intersecting straight lines define a plane.

Axioms of Arrangement (Group II)

With these, the term between is defined as a relationship between three points. If three points are said to lie between the other two, this always means that they are different points and that they lie on a straight line. With this assumption, the following axioms can be formulated very briefly:

• II.1. If lies between and , then also lies between and . ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle A}$
• II.2. For two points and there is always at least one point that lies between and and at least one point so that lies between and . ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle D}$
• II.3. Under any three points on a straight line there is always one and only one point that lies between the other two.

On the basis of these axioms, what a segment is can be defined : the set of all points that lie between and . (The segments and are identical according to this definition.) The term segment is needed to formulate the following axiom: ${\displaystyle AB}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle AB}$ ${\displaystyle BA}$

• II.4. Let there be three points not in a straight line and a straight line in the plane that does not meet any of these three points; if the straight line then goes through a point on the line , it certainly also goes either through a point on the line or through a point on the line . ${\displaystyle A,B,C}$ ${\displaystyle a}$ ${\displaystyle ABC}$ ${\displaystyle a}$ ${\displaystyle AB}$ ${\displaystyle BC}$ ${\displaystyle AC}$

This axiom is also called Pasch's axiom ; it has a special significance in the history of science since it does not appear in Euclid .

From the axioms of the connection (incidence) and the arrangement it already follows that between two given points of a straight line there are always an infinite number of further points, that the points of a straight line are therefore dense . Furthermore, it can be shown that every straight line as a set of points can be ordered in exactly two ways , so that a point lies between the points and exactly if or is. ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A<C<B}$ ${\displaystyle B<C<A}$

Next it can be concluded that each line (and each located in one plane and not self-tapping broken line ) divides a plane into two areas. In the same way, each level divides the room into two areas.

See also: order and division of pages

Axioms of Congruence (Group III)

The third group of axioms defines the term congruent as a relationship between distances and between angles . Another name for this is the same or (for routes) the same length . Hilbert uses this as a symbol . ${\displaystyle \equiv }$

• III.1. If and are two points on a straight line and there is also a point on the same or another straight line , one can always find a point on a given side of the straight line from , so that the segment of the line is congruent (or equal ), in signs : . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle a}$ ${\displaystyle A^{\prime }}$ ${\displaystyle a^{\prime }}$ ${\displaystyle a^{\prime }}$ ${\displaystyle A^{\prime }}$ ${\displaystyle B^{\prime }}$ ${\displaystyle AB}$ ${\displaystyle A^{\prime }B^{\prime }}$ ${\displaystyle AB\equiv A^{\prime }B^{\prime }}$

Any distance can therefore be plotted from any point. That this removal is unambiguous can be proven from the totality of axioms I - III, as well as that AB ≡ AB and that it always follows from ( reflexivity and symmetry ). ${\displaystyle AB\equiv A^{\prime }B^{\prime }}$ ${\displaystyle A^{\prime }B^{\prime }\equiv AB}$

• III.2. If a route is congruent to two other routes , these are also congruent to one another ; more formal: if and , so is . ${\displaystyle AB\equiv A^{\prime }B^{\prime }}$ ${\displaystyle AB\equiv A^{\prime \prime }B^{\prime \prime }}$ ${\displaystyle A^{\prime }B^{\prime }\equiv A^{\prime \prime }B^{\prime \prime }}$

So it is required that the congruence relation is transitive . It is therefore an equivalence relation .

• III.3. Let there be and two lines without common points on the straight line and further and two lines on the same or another straight line likewise without common points; if then and , so is always . ${\displaystyle AB}$ ${\displaystyle BC}$ ${\displaystyle a}$ ${\displaystyle A^{\prime }B^{\prime }}$ ${\displaystyle B^{\prime }C^{\prime }}$ ${\displaystyle a^{\prime }}$ ${\displaystyle AB\equiv A^{\prime }B^{\prime }}$ ${\displaystyle BC\equiv B^{\prime }C^{\prime }}$ ${\displaystyle AC\equiv A^{\prime }C^{\prime }}$

When combining (adding) lines, the congruence is retained.

An angle is now defined as a disordered (!) Pair of half-straight lines that start from a common point and do not belong to the same straight line. ( So there is no distinction between and ; according to this definition there are neither re-obtuse nor elongated angles.) It can also be defined what the interior of an angle is: These are all those points of the plane spanned by and that with together on the same side of and with lying together on the same side of . An angle always comprises less than one half-plane. ${\displaystyle S}$ ${\displaystyle \sphericalangle (h,g)}$ ${\displaystyle \sphericalangle (g,h)}$ ${\displaystyle \sphericalangle (g,h)}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle h}$

• III.4. Let there be an angle  in a plane and a straight line in a plane , as well as a certain side of on . It means a half-ray of the straight line ; then there is one and only one half-ray in the plane , so that the angle is  congruent (or equal ) to the angle  and at the same time all inner points of the angle  lie on the given side of . In sign: . Every angle is congruent to itself, that is, it is always . ${\displaystyle \sphericalangle (h,g)}$ ${\displaystyle \alpha }$ ${\displaystyle a^{\prime }}$ ${\displaystyle \alpha ^{\prime }}$ ${\displaystyle a^{\prime }}$ ${\displaystyle \alpha ^{\prime }}$ ${\displaystyle h^{\prime }}$ ${\displaystyle a^{\prime }}$ ${\displaystyle \alpha ^{\prime }}$ ${\displaystyle g^{\prime }}$ ${\displaystyle \sphericalangle (h,g)}$ ${\displaystyle \sphericalangle (h',g')}$ ${\displaystyle \sphericalangle (h',g')}$ ${\displaystyle a^{\prime }}$
  ${\displaystyle \sphericalangle (h,g)\equiv \sphericalangle (h',g')}$
  ${\displaystyle \sphericalangle (h,g)\equiv \sphericalangle (h,g)}$

In short, this means: any angle can be plotted in a given plane to a given half-ray to a given side of this half-ray in a clearly determined way.

It is noticeable that the uniqueness of the construction and the self- congruence here (in contrast to the congruence of lines) must be axiomatically determined.

• III.5. Out and follows . ${\displaystyle \sphericalangle (h,g)\equiv \sphericalangle (h',g')}$ ${\displaystyle \sphericalangle (h,g)\equiv \sphericalangle (h'',g'')}$ ${\displaystyle \sphericalangle (h',g')\equiv \sphericalangle (h'',g'')}$

From this axiom it follows with self-congruence that the congruence for angles is a transitive and symmetric relation.

After defining in an obvious way what is to be understood by, the last axiom of congruence can also be formulated: ${\displaystyle \sphericalangle ABC}$

• III.6. If for two triangles and the congruences ${\displaystyle ABC}$ ${\displaystyle A^{\prime }B^{\prime }C^{\prime }}$
  ${\displaystyle AB\equiv A'B',\quad AC\equiv A'C'\quad {\text{und}}\quad \sphericalangle BAC\equiv \sphericalangle B'A'C'}$

hold, then the congruences are always

${\displaystyle \sphericalangle ABC\equiv \sphericalangle A'B'C'\quad {\text{und}}\quad \sphericalangle ACB\equiv \sphericalangle A'C'B'}$

Fulfills.

It is here to congruence " PBUH ", sets the Hilbert as an axiom. Euclid formulated a "proof" (I L. 1) for this, against which Peletarius first expressed reservations in 1557 . Hilbert has shown that this sentence, or at least its essential content, is indispensable as an axiom.

The other congruence theorems can be proven from this, as can the addability of angles. A relationship can be defined at angles that is compatible with congruence. ${\displaystyle <}$

Hilbert also defines the term secondary angle in an obvious manner, and the term right angle as an angle that is congruent with its secondary angle.

It can then be shown that all right angles are congruent to one another. Euclid had put this - probably unnecessarily - as an axiom.

See also: Congruence and Pre-Euclidean Plane

Axiom of Parallels (Group IV)

• IV. ( Also Euclidean axiom.) Let be an arbitrary straight line and a point outside of . Then there is at most one straight line in the plane defined through and which runs through and does not intersect. ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle g^{\prime }}$ ${\displaystyle P}$ ${\displaystyle g}$

That there is at least one such straight line follows from axioms I - III and directly from the theorem of the external angle derived from them . This single straight line is called the parallel to through . ${\displaystyle g^{\prime }}$ ${\displaystyle g}$ ${\displaystyle P}$

This axiom, with its presuppositions and consequences, is probably the most discussed subject in geometry. See also : Parallels problem

As an axiom equivalent to the axiom of parallels, Hilbert states:

If two straight lines do not intersect a third straight line , although this is in the same plane with them, they also do not intersect one another. ${\displaystyle a,b}$ ${\displaystyle c}$

Furthermore, it follows from axioms I-IV that the sum of the angles in the triangle is two rights . This angle sum theorem only becomes an equivalent to the axiom of parallels if one includes Archimedes' axiom (V.1).

Under these conditions, the axiom can also be formulated as equivalent (compare Saccheri-Viereck ):

There is a rectangle !

Axioms of Continuity (Group V)

• V.1. (Axiom of Measurement or Archimedes' Axiom ) . Are and any routes, so there are a number such that the -times cascading removal of the track from from to by previous field beam over the point leading out. ${\displaystyle AB}$ ${\displaystyle CD}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle CD}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$

Even the smallest route can, if you put them together often enough, to surpass any route, no matter how large . One could also say: there are no “infinitely small” or “infinitely large” stretches; the natural numbers are sufficient to make all routes comparable (in the sense of greater, smaller, equal). ${\displaystyle CD}$ ${\displaystyle AB}$

• V.2. (Axiom of (linear) completeness) No further points can be added to the points of a straight line, if their arrangement and congruence relationships are preserved, without the relationships existing under the previous elements, the basic properties of the linear ones following from axioms I-III Arrangement and congruence or the axiom V.1 is violated.

The Euclidean geometry is therefore the largest possible geometry that corresponds to the preceding axioms. It is therefore complete in the same way that real numbers are complete . Therefore the analytical geometry of can also be used as a model for the Euclidean geometry. ${\displaystyle \mathbb {R} ^{3}}$

This becomes clearer in the " Completeness clause " - following from V.2 :

• The elements (points, straight lines and planes) of the geometry form a system which, if all axioms are maintained, is no longer capable of being extended by additional points, straight lines and / or planes.

Without the Archimedean axiom, this requirement cannot be met. Rather, any geometry that corresponds to axioms I – IV, but not V.1, can be expanded with additional elements. Non-standard systems then arise .

On the other hand, the axiom of completeness V.2 is indispensable; it cannot be derived from axioms I – V.1. Nevertheless, a large part of Euclidean geometry can be developed without Axiom V.2.

See also: Euclidean body

Consistency and independence

Relative consistency

Hilbert also proved that his system of axioms is free of contradictions if one assumes that the real numbers can be justified without contradiction.

As mentioned, the analytical geometry of the , i.e. the set of all triples of real numbers, together with the usual definitions for lines and planes as linear point sets, i.e. as secondary classes of one or two-dimensional subspaces, serves as a model for the axiom system . The incidence in this model is set-theoretic containment, and two lines are congruent if they have the same length in the sense of the Euclidean distance . ${\displaystyle \mathbb {R} ^{3}}$

Independence of the axioms from one another

Hilbert's declared aim was to build up his system of axioms in such a way that the axioms are logically independent of one another , so that none is dispensable because it can be proven from the others.

This can easily be shown for the axioms of groups I and II among themselves; likewise the axioms of group III are mutually independent. The aim is to show that the axioms of groups III, IV and V are independent of the others, as well as the independence of V.1 and V.2.

The proof procedure basically consists in specifying a model (or, in Hilbert's words: “a system of things”) to which all axioms apply, with the exception of axiom A. Obviously, such a model could not exist if A would be a logical consequence of the remaining axioms.

In this way Hilbert et al. a. that the axiom III.5 (the congruence theorem "sws") is indispensable.

The independence of the parallel axiom s IV results from the proof of the existence of non-Euclidean geometries , the independence of the Archimedean axiom s V.1 from the existence of nonstandard systems , and the independence of the completeness axiom s V.2 z. B. from the existence of an analytic geometry over the field of real algebraic numbers . (→ see also Euclidean body )

It can be shown that a geometry which fulfills these axioms is uniquely determined except for isomorphism: In the language of linear algebra, the following applies to this geometry:

A geometry that fulfills Hilbert's system of axioms is an affine space whose vector space of displacements is a three-dimensional Euclidean vector space , i.e. isomorphic to with a scalar product . ${\displaystyle (\mathbb {R} ^{3},\langle \cdot ,\cdot \rangle )}$

literature

• David Hilbert: Fundamentals of Geometry . 14th edition. Teubner, Stuttgart 1999, ISBN 3-519-00237-X ( online copy of the 1903 edition [accessed on June 9, 2013] first edition: 1899).
• Benno Klotzek: Euclidean and non-Euclidean elementary geometries . 1st edition. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0 .

Individual evidence

1. ↑ Susanne Müller-Philipp, Hans-Joachim Gorski: Guide to Geometry: For students of teaching posts . Vieweg + Teubner 2008, ISBN 978-3-8348-0097-8 , p. 67 ( excerpt from the Google book search)
2. ^ Hans Wußing : 6000 years of mathematics: A cultural-historical journey through time. From Euler to the present . Springer 2008, ISBN 9783540773139 , p. 174 ( excerpt from Google book search)
3. ↑ Klotzek (2001)
4. ↑ Jacobus Peletarius: In Euclidis Elementa Geometrica Demonstrationum Libri sex. J. Tornaesius; G. Gazeius: Lugduni 1557