### Math 302         The Axioms

An axiom means "A proposition that commends itself to general acceptance; a well established or universally-conceded 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.
1. The "incidence axiom"
1. There is at least one straight line between any two points.
2. There is at most one straight line between two points.
2. The "ruler axiom"
1. You can travel an infinite distance along a straight line in either direction.
2. As you travel along a straight line, you never pass over the same point twice.
3. The "protractor axiom"
1. There is at least one straight line through any point in any direction.
2. There is at most one straight line through any point in any direction.
4. The "half-plane" axiom
1. If you cut the surface along a straight line, you get two pieces.
2. Every straight line segment that connects two points on one of the pieces is contained entirely in that piece.
5. The "mirror axiom"
1. There is a global reflection through every straight line.