I'm back to basics as I'm preparing a science test as an adult student and I'm struggling to understand the significance of some of Hilbert's axioms.
Specifically, the third Incidence axiom tells us that:
There exist at least two points on a line. There exist at least three points that do not lie on the same line.
My gripe is with the second sentence. I understand you can identify a line with two points but I'm brought to think here that, since there are infinite points on a plane, well of course there exist at least three points that do not lie on the same line. You can actually find any number of points that do not lie on the same line.
Likewise, the eight Incidence axiom says:
There exist at least four points not lying in a plane.
Again, I realise that a plane can be identified with three points, but there are any number of points outside of a plane.
What am I not undestanding? Why make these statements?