It is possible to prove that there are infinitely many points in space in Hilber'ts axiomatization of geometry?

Question

Dumb question, i know, there's no explicit axiom about it, but it's somehow possible? Thanks in advance.

Answer 1

If we work with the formulation of Hilbert's axioms as seen on Wikipedia, then the Incidence and Order axioms together (for a plane, so ignore I.4 through I.8 and assume for II.4 that all points are coplanar) are enough to show the existence of infinitely many points.

A rough sketch of the proof is this:

1. Given a line $\ell$, define the "on the same side of $\ell$" relation between points not on $\ell$ as follows: $A$ is on the same side of $\ell$ as $B$ if there is no point of $\ell$ between $A$ and $B$.
2. Using axiom II.2, we can find two points $A$, $B$ not on the same side of $\ell$. Using axiom II.4 (Pasch's axiom), we can show that the "on the same side" relation is an equivalence relation, and that all points are either on the same side of $\ell$ as $A$ or as $B$. We conclude the Plane separation theorem: a line separates the plane into two sides.
3. From here, the Line separation theorem follows: a point $X$ on a line $\ell$ separates the line into two rays. (Two points $A,B$ are in the same ray relative to $X$ if $A * B * X$ or $B * A * X$.)
4. Finally, we can use induction to find arbitrarily many points on a given line $\ell$. Start with two points $X_1, X_2$ (by axiom I.3) and repeatedly use axiom II.2 to find a point $X_i$ on $\ell$ such that $X_{i-2} * X_{i-1} * X_i$. By the line separation theorem relative to $X_{i-1}$, $X_i$ is a point distinct from $X_1, \dots, X_{i-2}$, because $X_i$ lies in a different ray from all of them.