Math 402         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 two (distinct) points.
    2. There is at most one straight line between two (distinct) points.
  2. The "ruler axiom":
    1. You can travel any distance along a straight line in either direction.
    2. If you travel along any straight line, you will never go over the same point more than once.
  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 "plane separation axiom":
    5. If you cut the surface along a straight line, you get two (non-empty) pieces so that:
    1. If two points lie in difference picces, the line segment between them crosses the line.
    2. If two points lie in the same piece, the line segment between them does not cross the line
  5. The "mirror axiom":
    6. There is a global reflection through every straight line.