Hilbert’s axioms

Hilbert (1899) proposed additional axioms in order to establish Euclidean geometry as a rigorous mathematical theory.

Recall Euclid’s proof of Proposition I.1:

• take distinct points \(A\) and \(B\);
• the circles centered at \(A\) and \(B\) of radius \(AB\) intersect at a point \(C\);
• then \(ABC\) is an equilateral triangle.

Why do the circles intersect? For example, in the model where the points are the integers and there is only one line, namely the whole set itself, circles may fail to intersect.

So Euclid’s proof relies on several hidden assumptions. Hilbert’s goal was to introduce a new set of axioms making Euclid’s Elements fully rigorous.

Goals for today

Today we will:

(a)

re-introduce axiomatic theories and models;

(b)

introduce the axioms of incidence;

(c)

introduce the Euclidean, elliptic, and hyperbolic parallel properties;

(d)

study various models of incidence geometry;

(e)

define isomorphisms of models.

1. Axioms of incidence and models

To introduce the axioms of incidence, we use three primitive terms:

• points,
• lines,
• incident.

(I1) For any two distinct points \(A\) and \(B\), there exists a unique line \(\ell \) incident with both \(A\) and \(B\).

(I2) For every line \(\ell \), there exist at least two distinct points incident with \(\ell \).

(I3) There exist three distinct points with the property that no line is incident with all three of them. Such points are called non-collinear.

What is a model?

A model for an axiomatic theory containing the primitive words points, lines, and incidence can be rigorously defined as:

• a set \(P\) of points,
• a set \(L\) of lines,
• an incidence subset \(R \subseteq P \times L\).

Remark. The role of the incidence subset is to tell us which points and lines are incident. In set theory, \(R\) is called a relation.

Notation. If \(p\in P\) and \(\ell \in L\) are a point and a line, we write

(read “\(p\) is incident with \(\ell \)”) when

An incidence geometry is any model for which axioms I1, I2, and I3 are satisfied.

Remark. Points, lines, and incidence can be anything.

Example 1

Let

and let

Equivalently, we may say:

Then I1 and I2 are satisfied, but I3 is not.

Example 2

Let

Define incidence by declaring that

Question: are I1, I2, and I3 satisfied?

Example 3

Let

So there is one line and no incidences.

Example 4

Let

Define incidence by

In this example, incidence is given by the usual set-theoretic notion of belonging. For example,

This example may be represented by a triangle diagram: bold dots for points and strokes for lines.

Question: is \((P,L,R)\) an incidence geometry? In other words, does it satisfy I1, I2, and I3?

Activity 1

Show that, assuming only axioms I1 and I3, the following properties hold.

(i)

There exist three distinct and non-concurrent lines. In other words, there exist distinct lines \(\ell ,\ell ',\ell ''\) such that there is no point \(P\) satisfying

\[ P\, I\, \ell ,\qquad P\, I\, \ell ',\qquad P\, I\, \ell ''. \]

(ii)

If \(\ell \) and \(\ell '\) are lines and \(P\neq Q\) are distinct points such that

\[ P\, I\, \ell ,\qquad P\, I\, \ell ',\qquad Q\, I\, \ell ,\qquad Q\, I\, \ell ', \]

then

\[ \ell =\ell '. \]

(iii)

For every line \(\ell \), there exists at least one point not lying on it (that is, not incident with it).

2. Parallel properties

Definition. We say that an incidence geometry satisfies:

(i)

the Euclidean parallel property if, for every line \(\ell \in L\) and every point \(P\in P\) such that \(P \mathrel {\not \!I}\ \ell \), there exists a unique line \(\ell '\) such that

\[ P\, I\, \ell ' \qquad \text {and}\qquad \ell \parallel \ell '; \]

(ii)

the elliptic parallel property if there exist no parallel lines;

(iii)

the hyperbolic parallel property if, for every line \(\ell \) and every point \(P\) such that \(P \mathrel {\not \!I} \ell \), there exist at least two distinct lines parallel to \(\ell \) and incident with \(P\).

Activity 2

(i)

In Example 4, determine whether the Euclidean, elliptic, or hyperbolic parallel property is satisfied. Or maybe none of these are satisfied.

(ii)

Consider the following model:

• \(P=\{\text {points on a sheet of paper}\}\),
• \(L=\{\text {circles of positive radius}\}\),
• \(p\, I\, \ell \) if \(p\in \ell \) (standard incidence).

Is this an incidence geometry? What parallel property is satisfied?

3. Isomorphisms between models

Example 5

Let

a set with three elements, and let

the set of all subsets of \(L\) with two elements.

Define incidence by

Here lines can be subsets of points. For example,

since \(a\in \{a,b\}\) and \(b\in \{a,b\}\). But

since \(c\notin \{a,b\}\).

This example also has a triangle diagram. Examples 4 and 5 look very similar. We say that they are isomorphic.

Definition. Two models \((P,L,R)\) and \((P',L',R')\) are isomorphic if we can find:

• a bijective map \(f:P\to P'\),
• a bijective map \(g:L\to L'\),

such that for every \(p\in P\) and \(\ell \in L\), we have

In other words, two models are isomorphic if we can map their points and lines while preserving incidence.

Example

Consider Examples 4 and 5 and define

and

Then \(f\) and \(g\) are bijective and preserve incidence. Therefore Examples 4 and 5 are isomorphic.