I am trying to relearn analytic geometry, and have encountered a problem. My question is simple. Are the lines and planes in Hilbert's axioms of connection infinite? That is, are the lines infinitely long, and the planes go to infinity in all directions?
The reason I am asking is due to the second axiom; If the points A and B give us a line a, and the points A and C give us line a, then the points B and C gives us line a as well.
I initially thought a line given by A and B would be the straight line going from A to B.
Yes, "line" means "line extending infinitely in both directions"; the term for the part of the line just joining $A$ and $B$ is "line segment."
(Also, note that the second axiom requires $B\not=C$, to avoid triviality. :P)