Hilbert's axioms of geometry

Hilbert's axioms of geometry postulate relations between three kinds of objects: points, lines and planes.

The first group of axioms describes how points, lines and planes are connected. The first two axioms involve only points and lines:

I.1: Two distinct points \(A\) and \(B\) always completely determine a straight line \(a\).

We will write \(a = \langle A, B\rangle\) or \(a = \langle B, A\rangle\) for the line determined by points \(A\) and \(B\).

I.2: Any two distinct points of a straight line completely determine that line; that is if \(\langle A, B\rangle = a\) and \(\langle A, C\rangle = a\), where \(B \neq C\), then \(\langle B, C\rangle = a\).

The first two axioms together are sometimes called the plane axioms (of group I), because they only involve objects living in two dimensions. The remaining five axioms involve the relations of points and lines to planes and are called the space axioms (of group I):

I.3: Three points \(A\), \(B\) and \(C\) not on the same line always completely determine a plane \(\alpha\).

For a plane determined in this way we use the notation \(\alpha = \langle A, B, C \rangle\).

I.4: Any three points \(A\), \(B\), \(C\) of a plane \(\alpha\) which do not lie on the same line completely determine that plane.

I.5: If two points \(A\) and \(B\) of a straight line \(a\) lie in \(\alpha\), then every point of \(a\) lies in \(\alpha\).

I.6: If two planes \(\alpha\) and \(\beta\) have a point \(A\) in common, then they have at least a second point \(B\) in common.

I.7: Upon every straight line there exist at least two points, in every plane at least three points not lying on the same line, and in space there exist at least four points not lying on the same plane.

The axioms of order give meaning to the word between.

II.1: If \(A\), \(B\) and \(C\) are points on a line and \(B\) lies between \(A\) and \(C\), then \(B\) also lies between \(C\) and \(A\).

A system of two points \(A\) and \(B\) on a line is called a segment and written \(AB\) or \(BA\). The second axiom says that we can always find a point inside a line segment as well as a point lying outside that segment:

II.2: If \(A\) and \(C\) are two points on a line, then we can always find at least one point \(B\) between \(A\) and \(C\). We can also always find a point \(D\) such that \(C\) lies between \(A\) and \(D\).

The next axiom says that straight lines are not "circular":

II.3: Of any three points on a line, there is always one and only one that lies between the other two.

II.4: Any four points \(A\), \(B\), \(C\) and \(D\) can be arranged such that \(B\) lies between \(A\) and \(C\) and also between \(A\) and \(D\), and moreover that \(C\) lies between \(A\) and \(D\) and also between \(B\) and \(D\).

The last axiom of this group essentially says that a line either passes through two sides of a triangle or through none:

II.5: Let \(A\), \(B\) and \(C\) be three points not on the same line and let \(a\) be a line in the plane \(\langle A, B, C\rangle\) that does not contain any of these points. If \(a\) passes through a point of the segment \(AB\), then it also passes either through a point of the segment \(AC\) of of the segment \(BC\).

III: In a plane \(\alpha\) there can be drawn through any point \(A\) not lying on a line \(a\), one and only one line \(b\) which does not intersect \(a\).

This line \(b\) is called the parallel to \(a\) through the point \(A\).

There are two statements here, namely that a parallel line exists and that it is unique. It turns out that the existence of a parallel line can be derived using the axioms of the groups I, II and IV. The uniqueness however is independent.

The congruence axioms describe what it means to displace a segment or an angle (which will be defined below).

IV.1: Let \(A\) and \(B\) be two points on a line \(a\) and \(A'\) a point on another or the same line \(a'\). Then, on a given side of \(A'\) we can always find one and only one point \(B'\) such that the segment \(AB\) is congruent to the segment \(A'B'\). Every segment is congruent to itself.

The notation for two congruent segments \(AB\) and \(A'B'\) is \(AB \equiv A'B'\).

The next axiom says that the congruence relation between line segments is transitive:

IV.2: If \(AB \equiv A'B'\) and \(AB \equiv A''B''\), then \(A'B' \equiv A''B''\).

IV.3: Let \(AB\) and \(BC\) be two segments of a line \(a\) that only have the point \(B\) in common. Let \(A'B'\) and \(B'C'\) be two segments on another line \(a'\) that only have the point \(B\) in common. Then, if \(AB \equiv A'B'\) and \(BC \equiv B'C'\), we have \(AC = A'C'\).

Hilbert defines an angle \(\angle(k, \ell)\) as two half-rays \(k\) and \(\ell\) in the same plane \(\alpha\) eminating from the same point \(O\). The rays divide the plane in two regions. The interior of the angle \(\angle(k, \ell)\) is the region of the plane \(\alpha\) where any two points can be joined by a line segment that lies entirely in this region.

The following axiom says that given a half-ray in some plane, we can lay off a prescribed angle in one and only way upon a given side of the half-ray:

IV.4: Let an angle \(\angle(k, \ell)\) be given in the plane \(\alpha\) and let a straight line \(a'\) be given in a plane \(\alpha'\). Suppose also that in the plane \(\alpha'\) a definite side of the line \(a'\) is assigned. Denote by \(k'\) a half-ray of the line \(a'\) emanating from a point \(O'\) of this line. Then in the plane \(\alpha'\) there is one and only one half-ray \(\ell'\) such that the angle \(\angle(k, \ell)\) is congruent to the angle \(\angle(k', \ell')\) and at the same time all the interior points of \(\angle(k', \ell')\) lie on the given side of \(a'\). Every angle is congruent to itself.

The notation for congruent angles \(\angle(k, \ell)\) and \(\angle(k', \ell')\) is \(\angle(k, \ell) \equiv \angle(k', \ell')\).

IV.5: If \(\angle(k, \ell) \equiv \angle(k', \ell')\) and \(\angle(k, \ell) \equiv \angle(k'', \ell'')\), then \(\angle(k', \ell') \equiv \angle(k'', \ell'')\).

The last axiom is about congruent triangles. It essentially says that a triangle is fully determined by two sides and the angle in between.

IV.6: If in the triangles \(ABC\) and \(A'B'C'\) the congruences

\begin{align} AB \equiv A'B', \quad AC \equiv A'C', \quad \angle(B,A,C) \equiv \angle(B,A',C') \end{align}

hold, then also the following congruences hold:

\begin{align} \angle(A,B,C) \equiv \angle(A',B',C'), \quad \angle(A,C,B) \equiv \angle(A',C',B'). \end{align}

The Archimedean axiom excludes the possibility of infinitely small or large segments by demanding that any point \(B\) can be reached from a starting point \(A\) in a finite number of steps, no matter how small the step.

V: Let \(A_1\) be any point on a line between two arbitrary points \(A\) and \(B\). Take the points \(A_2, A_3, A_4, \ldots\) such that \(A_1\) lies between \(A\) and \(A_2\), \(A_2\) lies between \(A_1\) and \(A_3\), \(A_3\) lies between \(A_2\) and \(A_4\) and so on. Moreoever, let the segments

\begin{align} A A_1,\, A_1 A_2,\, A_2 A_3,\, A_3 A_4, \ldots \end{align}

be "equal" to each other. Then there exists a point \(A_n\) in this sequence of points such that \(B\) lies between \(A\) and \(A_n\).

Hilbert shows that the axioms are indeed compatible by constructing a geometry where all of them are fulfilled. His example is built on top of those algebraic numbers \(\Omega\) that can be obtained by starting with \(1\) and applying the operations addition, subtraction, multiplication, division and \(\omega \mapsto \sqrt{1 + \omega^2}\). A point is represented by a tuple \((x, y)\), a line by a homogeneous tuple \((u : v : w)\) and the point lies on the line if \(u x + v y + w = 0\).

The coordinates of points and lines obtained by constructing parallels or laying off segments and angles can all be obtained with the arithmetic operations above under which \(\Omega\) is closed. The axioms of order follow because \(\Omega\) is a subset of the real numbers which has an order. In the end all axioms are fulfilled. Any contradictions in the axiom system would have to arise in the arithmetic of the numbers in \(\Omega\), but this does not happen.

Hilbert also proves that the axioms are mutually independent, so none of them can be derived from the others. This allows one to remove individual axioms while the other ones remain valid. For example, removing axiom III, the parallel postulate, gives rise to non-Euclidean geometries such as spherical geometry.

• David Hilbert: The Foundations of Geometry