Question

Why gauss said the set of points on a line segment is finite

Answer 1

First of all, a Line segment is not a set of finite points (whatever that might mean). It is a piece of a line that includes all of the points in the line between two distinct points. One can make a one-to-one mapping between the interior points of a line segment (the points other than its end points) and an entire line, thereby demonstrating that a line segment has an infinity of points in it like the entire line. 

Now, take a circle, open it somewhere and flatten it to make a line segment. It's becoming clear that it has the same number of points as a line. (One could also make a one-to-one mapping between the points in a circle a the points on a line to demonstrate this.) Notice that there are an infinite number of pairs of points through the centre of the circle, each of which is a line of symmetry.