# What's the need for Hilbert's 7th axiom of incidence?

If two planes α, β have a point A in common, then they have at least a second point B in common.

I perfectly understand the axiom, but i don't see why it's necessary, and it's kind of counterintuitive for me.

When I think of two planes, I can think of these ways they could intersect:

The two implied by this axiom, and that are shown in my book But I could also think of two planes just "barely" touching in one point, kinda' like this I know that this is an axiomatic system, so this is just a "a rule of the game", and that it doesn't want to describe what we understand as physical flat surfaces, or what I think of a plane. However, I feel a little werwerfasdre about why this axiomatic system does not contemplate a case in which two planes intersect or touch in just one point. Also, I wonder why this axiom correctly describes our idealization of planes in order to study them, and what was the reason for Hilbert (hence, probably Euclid) to contemplate just two cases in which planes intersect. Could Euclidean Geometry be studied without this axiom?

Thanks in advance

• Planes are not parallelograms; they are infinitely extended. – Mauro ALLEGRANZA Jun 13 '18 at 20:18
• Two planes could intersect in a single point in four dimensions, so it sounds like this is intended to restrict space to three dimensions. – saulspatz Jun 13 '18 at 20:59
• Indeed, 4D Euclidean geometry is a model of all of Hilbert's axioms except this one. Equally valid but weirder is 3D Euclidean geometry with an extra "plane" added containing the origin but no other point. Both are ruled out by this axiom. – Misha Lavrov Jun 13 '18 at 21:41

## 1 Answer

Actually, at least in this version of Hilbert's book the 7th axiom of incidence / connection is stated as

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.

so I'll asume you are refering to the axiom I, 6. And it's necessary since in this book plane means a "straight infinitely extensible surface" (whatever that means axiomatically). Also, as you stated, you must interpretate this axiom as meaning

If two planes intersect, then they have a line in common.

Since two points define a line. And for the part of necessary for it to be an Euclidean Geometry, as far as I am concerned, both Tarski's and Birkhoff's axiomatization of the Euclidean Geometry lack the use of this particular axiom, but utilize other axioms to "use its place", some of them regarding mathematical analysis, so it's not estrictly necessary, but in Hilbert's axiom I would say yes.

Finally, I would like to show you another kind of "planes" in a non-euclidean space that satisfy their intersection being a single point:

Two "planes" intersecting at one single point

Where in this non-euclidean geometry, planes would be pretty much like a 3d parabola, where they'd been explained by the formula $$z=a((x-u)^2+(y-v)^2)+w$$, where $$a\in\mathbb{R}$$ and $$(u,v,w)$$ is your "starting point". Similarly lines are "slices" of this surfaces.

• A slightly more familiar object satisfying all of Hilbert's axioms except this one is four-dimensional Euclidean space $\mathbb R^4$. – Misha Lavrov Feb 21 at 4:24