# How Hilbert's axioms of incidence ensure the existence of a space?

My book asks two questions.

Any of the axioms of incidence ensures the existence of space?

I know that

• For every two points A and B there exists a line a that contains them both.

• For every two points there exists no more than one line that contains them both

ensure the existence of lines.

• There exist at least two points on a line. There exist at least three points that do not lie on the same line.

• For every three points A, B, C not situated on the same line there exists a plane α that contains all of them.

• For every three points A, B, C which do not lie in the same line, there exists no more than one plane that contains them all.

ensure the existence of planes.

However, there is no axiom that says something like

• $n$ lines/planes/whatsoever determine a space.

The closest thing is.

• There exist at least four points not lying in a plane.

But this doesn't refer explicitly to space, as the other axioms, so that makes me doubt.

The other questions is related

Prove that space contains at least 4 points

I know that I have to use the axioms that states

• There exist at least four points not lying in a plane.

But then again nothing in my book says that 4 points determine a space, it just says that there are 4 points not lying in a plane, but it doesn't even mention space as a primitive concept.

I identify that the main problem is that I don't understand how Hilbert's axioms determine a space without actually mentioning it. Or at least I think that's the problem.

What do you think?

• This doesn't seem possible to answer without a definition of "a space." Certainly, since there are four points, not all on the same plane, and any three points determine a plane, we know there is more than one plane. But does that constitute a "space?" – saulspatz Jun 22 '18 at 18:16
• They don't include a definition of space in my book. Do you know if, maybe, in Hilbert's original publication he defines it? – Daniel Bonilla Jaramillo Jun 22 '18 at 18:24
• I don't know. I think there's a problem with the question, not with Hilbert. I think that the remarks you have made are very apposite. If this is a homework problem, I think you should write up an answer along the lines of your question. Include the point that the homework problem does not describe what a "space" is. – saulspatz Jun 22 '18 at 18:29

## 1 Answer

There are no axioms for space : space is the domain of the interpretation of the theory, that has three "sorts" of objects : points, lines, planes.

See David Hilbert, The Foundations of Geometry (1899), Eng.transl.1902, page 2 :

Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters $A, B, C,\ldots$; those of the second, we will call straight lines and designate them by the letters $a, b, c,\ldots$; and those of the third system, we will call planes and designate them by the Greek letters $\alpha, \beta, \gamma, \ldots$

The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space.

Space is used in Ax.I,7.

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

Compare with a more modern version of Hilbert's axioms for plane geometry : William Richter, A MINIMAL VERSION OF HILBERT’S AXIOMS FOR PLANE GEOMETRY :

We are given a set which is called a plane. Elements of this set are called points. We are given certain subsets of a plane called lines. There are axioms that a plane and its subset lines must satisfy [...]