# How do you show that the three incidence axioms are independent of each other.

How do you show that the three incidence axioms:

Incidence Axiom 1. For every pair of distinct points P and Q there is exactly one line l such that P and Q lie on l.

Incidence Axiom 2. For every line l there exist at least two distinct points P and Q such that both P and Q lie on l.

Incidence Axiom 3. There exist three points that do not all lie on any one line.

are independent of each other (i.e it is impossible to prove any one of them from the other two) by inventing a nontrivial interpretation for each pair of incidence axioms, in which those axioms are satisfied but the third axiom is not.

• Systematically construct incidence relations between a very small set of lines and points. Check the three incidence axioms. There are eight possible ways that they are or are not satisfied. – Somos Feb 25 at 5:25

1) There are two points $$P$$ and $$Q$$ and one line $$\{PQ\}$$. Axioms 1 and 2 are satisfied but 3 is not.
2) There are three points $$P$$, $$Q$$, $$R$$ and one line $$\{PQ\}$$. Axioms 2 and 3 are satisfied but 1 is not.
1) There are two points $$P$$ and $$Q$$ and two lines $$\{PQ\}$$ and $$\{P\}$$. Axioms 1 and 3 are satisfied but 2 is not.