I'm trying to learn the concept of axiomatic systems. I found an example online which is as follows:
Let's consider the following example of an axiomatic system.
Axiom 1. Every ant has at least two paths.
Axiom 2. Every path has at least two ants.
Axiom 3. There exists at least one ant.
Then, the following theorem is proved under this axiomatic system.
Theorem 1. There exists at least one path.
Proof. By Axiom 3, there exists an ant. Now since each ant must have at least two paths by Axiom 1, there exists at least one path.
My observation:
If Axiom 3 is true, then it is possible to have a situation where there is only one ant. If there is only one ant, then no path can exist because according to Axiom 2, every path should have at least two ants.
Is it not necessary to modify Axiom 3 to, There exist at least two ants, to make Theorem 1 true?
No. Axiom 3 states that “There exists at least one ant.” You cannot deduce from this assertion that it is possible to have a situation where there is only one ant. And, in fact, it follows from the other axioms that this cannot happen.