Euclid's work in
geometry set new
standards for mathematical
rigor, and for centuries was the
paragon that all other
proofs were compared to. Even so, Euclid's proofs are teeming with unstated
assumptions that no modern mathematician could get away with. As
Bertrand Russell put it:
Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates. A vastly greater number of axioms, which Euclid
unconsciously employs, are required for the proof of his propositions.
Lewis Carroll made the same point more amusingly when he created a
proof that all triangles are isosceles. The proof is just as rigorous as any of Euclid's; it just happens that some of its unstated assumptions are false.
So, what exactly are the "vastly greater number of axioms"? David Hilbert provided an answer in his 1899 book Grundlagen der Geometrie (Foundations of Geometry). The complete set, which includes all five of Euclid's axioms or their equivalents, can be seen below.
It should be noted that these axioms describe three dimensional Euclidean geometry. For two dimensional geometry, some of the axioms can be simplified or omitted. In four or more dimensions, only one of the axioms is false: the seventh axiom of incidence, that two planes with one point in common must have another point in common.
Alfred Tarski later came up with another axiomatization of geometry, but his is not an extension of Euclid's axioms. Rather, it starts over from first order logic, with the point as its only primitive object.
Axioms of incidence:
1. For every two points A, B there exists a line a that contains each of
the points A, B.
2. For every two points A, B there exists no more than one line that
contains each of the points A, B.
3. There exist at least two points on a line. There exist at least
three points that do not lie on a line.
4. For any three points A, B, C that do not lie on the same line there
exists a plane alpha that contains each of the points A, B, C. For every
plane there exists a point which it contains.
5. For any three points A, 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
points A, B, C.
6. If two points A, B of a line a lie in a plane alpha then every point
of a lies in the plane alpha.
7. If two planes alpha, beta have a point A in common then they have at
least one more point B in common.
8. There exist at least four points which do not lie in a plane.
Axioms of order:
1. If a point B lies between a point A and a point C then the points A,
B, C are three distinct points of a line, and B then also lies between C
and A.
2. For two points A and C, there always exists at least one point B on
the line AC such that C lies between A and B.
3. Of any three points on a line there exists no more than one that lies
between the other two.
4. Let A, B, C be three points that do not lie on a line and let a be a
line in the plane ABC which does not meet any of the points A, B, C. If
the line a passes through a point of the segment AB, it also passes
through a point of the segment AC, or through a point of the segment BC.
Axioms of congruence:
1. If A, B are two points on a line a, and A' is a point on the same or
on another line a' then it is always possible to find a point B' on a
given side of the line a' through A' such that the segment AB is
congruent or equal to the segment A'B'.
2. If a segment A'B' and a segment A''B'', are congruent to the same
segment AB, then the segment A'B' is also congruent to the segment
A''B'', or briefly, if two segments are congruent to a third one they are
congruent to each other.
3. On the line a let AB and BC be two segments which except for B 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 is congruent to A'B' and BC is congruent to
B'C', then AC is congruent to A'C'.
4. Let theta be an angle in the plane alpha given by the rays h and k
and a' a line in a plane alpha' and let a definite side of a' in
alpha' be given. Let h' be a ray on the line a' that emanates from the
point O'. Then there exists in the plane alpha' one and only one ray k'
such that the angle that is congruent or equal to the angle theta' given
by the rays h' and k' and at the same time all interior points of the
angle theta' lie on the given side of a'. Every angle is congruent to itself, i.e., theta
(is congruent to) theta is always true.
5. If for two triangles ABC and A'B'C' the congruences AB (is congruent
to) A'B', AC (is congruent to) A'C', (the angle) BAC (is congruent to)
(the angle) B'A'C', then the congruence (the angle) ABC (is congruent to)
(the angle) A'B'C' is also satisfied.
Axiom of parallels:
1. Let a be any line and A a point not on it. Then there is at most one
line in the plane, determined by a and A, that passes through A and does
not intersect a.
Axioms of continuity:
1. If AB and CD are any segments then there exists a number n such that n
segments BC constructed contiguously from A, along the ray from A through
B, will pass beyond the point B.
2. 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 elements as well as the fundamental properties of line order and
congruence that follows from all the previous axioms but the axiom of
parallels is impossible.