During the introduction of the axiomatic system in our MATH 10 discussions, I found it complicated and I was sort of in a state of pure blankness because I don’t remember having lessons about the topic during high school. After a number of examples and through Dr. Jom’s thorough explanations, I was able to keep up with the lesson.
It was fun to finally be able to prove theorems after numerous attempts at understanding the confusing statements. It was also satisfying to write that QED at the end of the sentence.
Additionally, just recently, Dr. Babbiera tackled the relation of fantasy stories to mathematics and axiomatic systems in a lecture. As an avid fantasy reader, the relation between the two that were proposed during the discussion was very interesting for me.
What if mathematics really is just fantasy, and fantasy is mathematics?
Is it a theorem or a theory? Were you also a victim of the wrong use of these words during high school? Here’s a quick guide to the components of an axiomatic system:
To show more about the process of an axiomatic system, here’s an example:
PROVING AN AXIOMATIC SYTEM.
Example. Consider the following axiom set.
Axiom 1. Every star has at least two paths.
Axiom 2. Every path has at least two stars.
Axiom 3. There exists at least one star.
- What are the undefined terms in this axiom set?
The undefined terms are star, path, and has. Note that star and path are elements, and has is a relation since it indicates some relationship between star and path.
- Prove Theorem 1. There exists at least one path.
Note that Axiom 3 guarantees the existence of an star, but no axiom explicitly states that there is a path. We need to prove the theorem to prove the existence of a path.
Proof. By Axiom 3, there exists an star. Now since each star must have at least two paths by Axiom 1, there exists at least one path. //
- What is the minimum number of paths? Prove.
The minimum number of paths is two.
Proof. By Axiom 3, there exists an star, call it S1. Then by Axiom 1, S1 must have two paths call them P1 and P2. Hence, there are at least two paths.
We form a model that shows it is possible to have exactly two paths, which demonstrates that the minimum number of paths is two. By Axiom 2, P1 must have an star other than S1, call it S2. We form a model where S1 and S2 both are assigned to P1 and P2, then we have exactly two paths.
We show the model satisfies all three axioms. Axiom 1 is satisfied, since S1 and S2 each have both P1 and P2. Axiom 2 is satisfied since P1 and P2 each have both S1 and S2. Axiom 3 is satisfied, since we have two stars. Hence, the minimum number of paths is two QED.
For more information about axiomatic systems: (http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxSysWorksheet.htm)