Why must there be an infinite number of lines in an absolute geometry?

Question

Why must there be an infinite number of lines in an absolute geometry? I see that there must be an infinite number of points pretty trivially due to the protractor postulate, and there are an infinite number of real numbers. Hence, theres an infinite number of points, but why must there be an infinite number of lines? Does it follow from the existence axiom where two distinct points completely determine the existence of a line between these two points?

Answer 1

Pick one point $P$. Through $P$ and each other point $Q$ there's a line, $\ell_Q$, by axiom 1.

Case 1: There are infinitely many $\ell_Q$, and we're done.

Case 2: There are finitely many lines $\ell_X$. In that case, one of them -- say $\ell_Q$ -- must contain infinitely many points $R_1, R_2, \ldots$. (Because we know that the geometry has infinitely many points, and each point $X \ne P$ is on $\ell_X$.) Now let $S$ be a point not on $\ell_Q$. Then the lines $SR_i$ are all distinct. (For if $SR_1$ and $SR_2$ shared a point $R_k$ for some $k \ne 1, 2$, we'd have $R_1$ and $R_k$ on both $SR_1$ and on $\ell_Q$, a contradiction.) Hence in this case there are infinitely many lines as well.