What is an Axiomatic System?
Introduction
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.
Undefined Terms
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?
A theorem
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.