In Hilbert's Foundations of Geometry he gives his axioms for geometry. Group I, called the axioms of incidence, details the way points, lines and planes exist. The last two axioms of the group are
I, 7. If two planes $\alpha$, $\beta$ have a point $A$ in common then they have at least one more point $B$ in common
I, 8. There exists at least four points which do not lie in a plane
In the next paragraph he explains that "Axiom I, 7 expresses the fact that space has no more than three dimensions, whereas Axiom I, 8 expresses the fact that space has no less than three dimensions." Effectively saying that space has exactly three dimensions.
How do these two axioms intuitively express these two statements?
Best Answer
In a space with at least four dimensions, you can exhibit two planes which meet at a single point. This axiom precludes that from happening.
If every set of four points lies in a plane, then all points lie in a single plane. This axiom precludes that from happening.