Incidence

• Axioms:
  • $I_1$, Foreach line, $r$, exists a point on the line - and a point off the line: \[ \forall r \in \mathcal{R}: \qquad \exists P \in \mathcal{P} \quad\wedge\quad \exists P \notin \mathcal{P} \]
  • $I_2$, two points defines a line (uniquely): \[ \forall P \neq Q \in \mathcal{P}: \qquad \exists ! r \in \mathcal{R}: \qquad P, Q \in r \]
  • $I_3$: Exists two (distinct) Points. \[ \exists P,Q \in \mathcal{P}: ~p \neq Q \]
• Theorems:
  • $\mathcal{P} \neq \emptyset$
  • $ \exists P,Q,R \in \mathcal{P}$ distintos.
  • $ \exists r,s,t \in \mathcal{R}$ distintos.
• Models:
  • 3 Points and 3 Lines.
  • 4 Points and 6 Lines.