In this section we reproduce Hilbert’s axioms for geometry, as given in Hilbert (1971). Hilbert divides his axioms into five different groups.
TABLE13 Hilbert’s Axioms of Incidence Axioms of Incidence
I, 1. For every two pointsA,B there exists a lineathat contains each of the pointsA,B.
I, 2. For every two pointsA,B there exists no more than one line that contains each of the pointsA, B.
I, 3. There exist at least two points on a line. There exist at least three points that do not lie on a line.
I, 4. For any three pointsA,B,Cthat do not lie on the same line there exists a planeαthat contains each of the pointsA,B, C. For every plane there exists a point which it contains. I, 5. For any three pointsA, B,C that do not lie on one and the
same line there exists no more than one plane that contains each of the three pointsA,B,C.
I, 6. If two pointsA,Bof a linealie in a planeαthen every point ofalies in the planeα.
I, 7. If two planesα,β have a pointAin common then they have at least one more pointB in common.
I, 8. There exist at least four points which do not lie in a plane.
TABLE14 Hilbert’s Axioms of Order
Axioms of Order
II, 1. If a pointBlies between a pointAand a pointCthen the points A,B, C are three distinct points of a line, andB then also lies betweenCandA.
II, 2. For two pointsAandC, there always exists at least one pointB on the line ACsuch thatC lies betweenAandB.
II, 3. Of any three points on a line there exists no more than one that lies between the other two.
II, 4. LetA,B,C be three points that do not lie on a line and letabe a line in the plane ABC which does not meet any of the points A,B,C. If the lineapasses through a point of the segmentAB, it also passes through a point of the segment AC, or through a point of the segmentBC.
TABLE15 Hilbert’s Axioms of Congruence
Axioms of Congruence
III, 1. IfA,Bare two points on a linea, andA′is a point on the same or
on another line a′ then it is always possible to find a pointB′on
a given side of the linea′throughA′such that the segmentABis
congruent or equal to the segmentA′B′. In symbols,AB∼=A′B′.
III, 2. If a segmentA′B′and a segmentA′′B′′are congruent to the same
segmentAB, then the segmentA′B′is also congruent to the seg-
ment A′′B′′, or briefly, if two segments are congruent to a third
one they are congruent to each other.
III, 3. On the linealetABandBCbe two segments which except forB have no point in common. Furthermore, on the same or on another line a′ let A′B′ and B′C′ be two segments which except for B′
also have no point in common. In that case, if AB ∼= A′B′ and
BC∼=B′C′, thenAC∼=A′C′.
III, 4. Let∠(h, k) be an angle in a planeα anda′ a line in a planeα‘
and let a definite side ofa′ inα′ be given. Leth′ be a ray on the
line a′ that emanates from the pointO′. Then there exists in the
plane α′ one and only one rayk′ such that the angle
∠(h, k) is congruent or equal to the angle ∠(h′, k′) and at the same time
all interior points of the angle ∠(h′, k′) lie on the given side of
a′. Symbolically,∠(h, k)∼=∠(h′, k′). Every angle is congruent to
itself, i. e.,∠(h, k)∼=∠(h, k).
III, 5. If for two trianglesABCandA′B′C′the congruencesAB∼=a′B′,
AC ∼= A′C′, ∠BAC ∼= ∠B′A′C′ hold, then the congruence
TABLE16 Hilbert’s Axiom of Parallels Axiom of Parallels
IV. Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by aand A, that passes throughA and does not intersecta.
TABLE17 Hilbert’s Axioms of Completeness Axioms of Completeness
V, 1. (Archimedes’ Axiom) If AB and CD are any segments then there exists a numbernsuch thatnsegmentsCDcon- structed contiguously fromA, along the ray fromAthrough B, will pass beyond the pointB.
V, 2. (Axiom of line completeness) An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original ele- ments as well as the fundamental properties of line order and congruence that follows from Axioms I–III, and from V, 1 is impossible.