Proving Infinitely Many Points on a Line in Geometry

[Lee33]

Homework Statement

Prove that a line in a metric geometry has infinitely many points.2. The attempt at a solution

I can't use any real analysis, like completeness. I can only use geometry to prove this, specifically distances and rulers.

Intituvely I understand why. Any segment with at least two points has infinitely many points, because, intuitively, given any two distinct points, there is a third one, distinct from both of them and so on. But how can I prove this formally?

 

[Office_Shredder]

What is your definition of a "metric geometry"?

 

[pasmith]

And, for that matter, your definition of "line"?

 

[Lee33]

Sorry for that:

Metric geometry: An incidence geometry $\{P, L\}$, where $P$ is the set of points, $L$ set of lines, together with a distance function $d$ satisfies if ever line $l\in L$ has a ruler. In this case we say $M = \{P,L,d\}$ is a metric geometry.

Line: for line I will define it as an incidence geometry. If every two distinct points in $L$ lie on a unique line and there exist three points $a,b,c\in L$ which do not lie all on one line. If $\{P,L\}$ is an incidence geometry and $p,q\in P$, then the unique line $l$ on which both $p,q$ lie will be written as $l=\vec{pq}$.