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The ruler postulate is defined as the following

For any line $l$ and any two distinct points $O$ and $P$ on $l$, there exists a bijection $c : l \rightarrow \mathbb{R}$ such that the following holds:

  1. $c(O)=0$ and $c(P) \geq 0$.
  2. $d(A,B)=|c(A)-c(B)|$, for all points $A$ and $B$ on $l$

The way I understand the postulate is that you can have the points of a line be placed in such a way such that you have a bijection between the points of the line and the real numbers. And the distance between two points equals the absolute value of the difference of the corresponding numbers. I want to use the first fact more to help show infinite length. Now can I use the fact that you have a bijection with the real numbers that I can then imply that since the real numbers are infinite then thus the line must be infinite. However I only know that holds for sure with finite sets(that is the cardinality is the same). Now my question is if I have a bijection with a infinite set then is the other set therefore also infinite, and would this help to prove my problem? (It seems it have to be in order to satisfy subjectivity)

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    If you have a bijection $X \to Y$, then $X$ is finite if and only if $Y$ is finite. As a consequence, $l$ must be an infinite set. – 2017-02-01

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