What is an Axiomatic System?
- Ku, Yin Bon (Albert)
Axiomatic method (公理系統) – a method of proving that results are correct. To use axiomatic method, the following requirements must be satisfied:
- Acceptance of certain statements called “axioms (公理)”, or “postulates” without further justification.
- Agreement on how and when one statement “follows logically” from another, i.e., agreement on certain rules of reasoning.
Every term used in an axiomatic system must be well-defined. There must be some initial terms that do not need to be defined. We called them undefined terms.
A very simple kind of geometry
Now we consider the following undefined terms: "Point", "line" and "lies on" We can't describe directly what a point or a line is. Their meanings are manifested through the following axioms that tell us how we can use those terms: Axiom 1: For any two distinct points, they lie on a unique line. Axiom 2: For any line, there exist at least two distinct points lying on it. Axiom 3: There exist three distinct points such that they do not lie on a line. This is already an axiomatic system of geometry, which is a simplified version of the geometry we learn in high school.
3-point and 4-point geometry
Task 1: Suppose you are given 3 distinct points. How can you draw the lines such that all the axioms are satisfied? Is there only one way to draw such lines? Task 2: Suppose you are given 4 distinct points. How can you draw the lines such that all the axioms are satisfied? Is there only one way to draw such lines?
Can you "prove" the following theorem using the given axioms? Theorem: If two distinct lines intersect at a point i.e. the point lies on both lines, then they intersect at a unique point.