Math 302 The Axioms
An axiom means "A proposition that commends itself to general acceptance;
a well established or universallyconceded principle..." (OED2). Often,
one assumes the following statements are true. However, they are not
true on every space. Therefore, we will check
if each statement is true on each space.

The "incidence axiom"

There is at least one straight line between any two points.

There is at most one straight line between two points.

The "ruler axiom"

You can travel an infinite distance along a straight line in either
direction.

As you travel along a straight line, you never pass over the same point
twice.

The "protractor axiom"

There is at least one straight line through
any point in any direction.

There is at most one straight line through any point in any direction.

The "halfplane" axiom

If you cut the surface along a straight line, you get two pieces.

Every straight line segment that connects two points on one of the pieces
is contained entirely in that piece.

The "mirror axiom"

There is a global reflection through every straight line.