Hilbert's axioms
David Hilbert used for its axiomatic foundations of Euclidean geometry ( in three dimensions ) " three different systems of things ", namely points, lines and planes, and " three basic relations ", that lie between and congruent. About the nature of these "things" and their "relationships " Hilbert makes no assumptions as formalist. They are only defined implicitly, namely by its link in a system of axioms.
Hilbert is reported to have said, you could take " points, lines and planes " at any time say " tables, chairs, and beer mugs "; it 'll only ensure that the axioms are satisfied. However, he has spared no effort to ensure that its "Tables, chairs and beer mugs " fulfill all the laws that have found the surveyor of the previous two thousand years for " points, lines and planes ." The strength of the axiomatic approach is not that it disregards the reality. It allows, however, by modifying the axioms and analysis of the connection between the logical structure that follows this reality, to be screened in a previously conceivable manner.
On one over the Hilbert system weakened axiom system without parallel axiom can be the absolute geometry reasons: there are either no parallels ( elliptic geometry) or through a point outside a line any number of parallels ( hyperbolic geometry). The hyperbolic geometry satisfies Hilbert's axiom groups I- III and V, the elliptical geometry I, II and V and a weaker version of the congruence (III).
- 2.1 Relative consistency
- 2.2 Independence of axioms with each other
The axioms
For this purpose, the associated Hilbert "things" and " relations " by 20 axioms into five groups:
Axioms of logic (or incidence), Group I
With these axioms, the term should be defined implicitly lie. Hilbert used here define the concept of or belong together and a number of other ways of speaking: g goes through P, g connects P and Q, P lies on g, P is a point of g, to g, there is a point P, etc. Today it is called in mathematics of incidence: "P incised g" (formally: Pig).
- I.1. Two distinct points P and Q determine always a straight line g
- I.2. Any two distinct points determine a straight line this straight.
- I.3. On a straight line, there is always at least two points in a plane, there are always at least three points not located on a straight line.
- I.4. Three are not on a same straight line and the points P, Q, R always define a plane.
- I.5. Any three points on a plane which does not lie on the same straight line, to determine the layer.
- I.6. If two points P and Q of a line lie in a plane α g, so each point of g lies in α.
- I.7. If two planes α and β a point P in common, so they have in common at least one additional point Q.
- I.8. There are at least four not in a plane located points.
Axioms 1-3 are called planar axioms of group I and axioms 4-8 spatial axioms of group I.
From these axioms alone can be deduced, for example,
Axioms of arrangement (group II)
With this term is defined as a relationship between three points. Being told by three points, that the one is between the other two, so this always expressed that there are different points, and that they lie on a straight line. Under this condition can be the following axioms formulate very short:
- II.1. When B is between A and C, then B is between C and A.
- II.2. There are two points A and C is always at least a point B which is located between A and C, and at a point D so that C is between A and D.
- II.3. Among any three points on a line, there is always one and only one point, which lies between the other two.
On the basis of these axioms can be defined, which is a distance AB: The set of all points that lie between A and B. ( The segments AB and BA are the same according to this definition. ) The concept of distance is needed to formulate the following axiom:
- II.4. Let A, B, C three not located in a straight line and a point is a straight line in the plane ABC, which meets any of these three points; if then the line goes through a point of a line segment AB, so she goes certainly also by either a point of the segment BC or a point of the segment AC.
From the axioms of logic (incidence) and the arrangement already follows that between two given points on a line always infinitely many other points are that the points on a line so are dense in itself.
Furthermore, it can be shown that each can be arranged precisely as point set exactly two ways, so that a point C if and only lies between points A and B, if A
Next it can be concluded that each line ( and each located in a plane and not self-intersecting polyline ) divides a plane into two areas.
Also, each level separates the room into two areas.
See also: Order and pagination
The third group of axioms defines the term as a congruent relationship between distances and angles between.
Another name for this is equal to or (with distances) of equal length.
As a sign of this Hilbert ≡ used.
From any point so any route can be removed.
That this erosion is clearly can be from all the axioms I - III show, as well, that AB ≡ AB and that of AB ≡ A'B ' always A'B' ≡ AB follows ( reflexivity and symmetry).
It is therefore requested that the congruence relation is transitive.
Thus, it is an equivalence relation.
When joining (addition ) of routes, the congruence is therefore retained.
An angle is now defined as a disordered (!) Pair of half-lines emanating from a common point S and not belong to the same straight line.
( Intermediate and therefore is not distinguished even there according to this definition neither dull nor straight angles. ) It can also be defined, which is the interior of an angle: These are all the points of the of g and h plane spanned, with
h together on the same side of g and g are together on the same side of H.
An angle has always less than a half-plane.
In short, this means: Each angle can be removed in a given plane at a given half- ray after a given page this half line on a uniquely determined way.
It is striking that the uniqueness of the design and the Selbstkongruenz must be (as opposed to the congruence of lines ) defined here axiomatic.
For this axiom follows with the Selbstkongruenz that the congruence of angles is transitive and symmetric relation.
Having been defined in an obvious way, what is meant by, also the last Kongruenzaxiom can be formulated:
This is about the congruence theorem " sws ", the Hilbert recognized as an axiom.
Euclid formulated for this purpose a "proof" (I L. 1), against the Peletarius 1557 concerns first set out.
Hilbert has shown that this set, or at least its essential content, as an axiom is essential.
The remaining congruence theorems can be proved from this, as well as the additivity of angles.
It can be a relationship at angles define, which is compatible with the congruence.
Next define the Hilbert term supplementary angles in an obvious way, and the concept of right angle as an angle that is congruent with its supplementary angle.
It can be shown then that all right angles are congruent to each other.
Euclid had this - probably unnecessarily - set as an axiom.
See also: congruence and präeuklidische level
That there are at least a straight line such, it follows from the axioms I - III and directly from the derived therefrom sentence from the outer angle.
This single line g 'is the parallel to g by P.
This axiom with its premises and conclusions is probably the most talked about subject of geometry.
See also: Parallel Problem
As an equivalent to the axiom of parallels axiom Hilbert states:
It also follows from the axioms I-IV, that the sum of angles in a triangle is two right angles.
An equivalent to the parallel postulate is this angle sum set only if one may call in the Archimedean axiom ( V.1).
Under these conditions, the axiom can also be formulated as equivalent (compare Saccheri quadrilateral ):
With every little track CD it is therefore possible, if you just often enough together sets that surpass even the great distance AB.
One could also say: There is no " infinitely small " or " infinitely large " distances;
the natural numbers are sufficient to make all routes comparable (in terms of greater, less, equal).
Euclidean geometry is therefore the maximum possible geometry which corresponds to the preceding axioms.
It is thus completely in the same sense as real numbers is complete.
Therefore, the analytic geometry can use the model for Euclidean geometry.
More clearly, this is still in the - from V.2 following - " completeness theorem ":
Without the Archimedean axiom, this requirement is not met.
Instead, you can choose any geometry corresponding to the axioms I-IV, but not V.1, expanded even further with additional elements.
Are then formed non-standard systems.
On the other hand, the completeness axiom V.2 is essential, it can not be inferred from the axioms I- V.1.
Nevertheless, can be a big part of Euclidean geometry without the axiom V.2 develop.
See also: Euclidean body
Hilbert also proved that his axiom system is consistent, assuming that can be justified, the real numbers consistent.
As a model for the axiom system is then used, as mentioned, the analytical geometry, that is the set of triples of real numbers, along with the usual definition of lines and planes as linear sets of points, that is, as a coset on or two-dimensional sub-spaces.
The incidence in this model is the set-theoretic containment and two lines are congruent if they have the same length in terms of the Euclidean distance.
The stated goal of Hilbert was to build his system of axioms so that the axioms are logically independent of each other, so that none is unnecessary because it can be proved from the other.
For the axioms of group I and II each other it is easy to demonstrate this;
as the axioms of group III are mutually independent.
It is all about showing that the axioms of groups III, IV and V are independent from the other, as well as to the independence of V.1 and V.2.
The proof method basically consists in a model (or, with Hilbert's words: " a system of things " ) indicate for which all axioms except the detected as independent axiom A. Apparently there could not be such a model, if A
a logical consequence of the other axioms would be.
In this way, Hilbert shows, among other things, that the Axiom III.5 ( the congruence theorem " PBUH " ) is indispensable.
The independence of the parallel axiom IV follows from the proof of the existence of non-Euclidean geometries, the independence of the Archimedean axiom V.1 from the existence of non-standard systems, and the independence of the completeness axiom V.2 eg
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 satisfies these axioms is uniquely determined up to isomorphism: In the language of linear algebra is valid for this geometry:
Axioms of congruence ( group III)
Axiom of parallels (Group IV)
Axioms of continuity ( group V)
Consistency and independence
Relative consistency
Independence of the axioms with each other