2nd Stage, Lecture 2: The Finite Geometry

An axiom is called independent if it cannot be proven from the other
axioms.

Example:
Consider Axiom 1 from the Committee system. Let’s omit it and see what kind of model we can come up with.
Members ={ Ali, Abbas, Ahmed, Huda, Zainab, Sara}.
Committees:
{Ali, Abbas, Zainab }
{Ali, Huda }
{Abbas, Huda, Ahmed }
{Ahmed, Sara }
{ Sara, Zainab}.
Notice that, it can found a model where Axiom 1 is not true; we have committees that do not have exactly three members. Since all of the other axioms are true in this model, then so is any statement that we could prove using those axioms. But, since Axiom 1 is not true, it follows that Axiom 1 is not provable from the other axioms. Thus, Axiom 1 is independent.

2.2 Consistency
If there is a model for an axiomatic system, then the system is called consistent.
Otherwise, the system is inconsistent. In order to prove that a system is consistent, all we need to do is come up with a model: a definition of the undefined terms where the axioms are all true. In order to prove that a system is inconsistent, we have to prove that no such model exists.

Example 2.2.1. The following axiomatic system is not consistent
Undefined Terms: boys, girls
A1. There are exactly 2 boys.
A2. There are exactly 3 girls.
A3. Each boy likes exactly 2 girls.
A4. No two boys like the same girl.

2.3 Completeness
An axiomatic system is complete if every true statement can be proven from the axioms.

Example 2.3.1. Twin Primes Conjecture: There are an infinite number of pairs of primes whose difference is 2.
Some examples of “twin” primes are 3 and 5, 5 and 7, 11 and 13, 101 and 103, etc. Computers have found very large pairs of twin primes, but so far no one has been able to prove this theorem. It is possible that a proof will never be found.