There are two axioms:
- There are two different points $A, B$, and two different lines $\ell_1, \ell_2$ in such way that A, B are located on both of this lines $(A,B \in \ell_1$ and $A,B \in \ell_2)$.
- For each line $\ell$ there exists a point $P$ which is not on $\ell$ $(P \notin \ell)$.
The question is: Prove that the system is not categorical.
My way of answer: Firstly, I've drawn this axiomatic system, and as I understand from the meaning of "Categorical axiomatic system" it means that there isn't a different axiomatic system. Did I understand correctly or do I miss anything? System attached:
Your understanding of "categorical axiomatic system" is flawed. What you need to find is two ways to interpret these axioms that not equivalent to each other in some way that is involved with the axioms.
For example of how do it wrong: As seen in your drawing, you could say for one model the "lines" are circles, and for the other model, the "lines" are parabolas. These models would be different, but not in any way that impacts the axioms. Anything that is true about $\ell_1, \ell_2, A, B, P_1, P_2$ in one is also true in the other. (One of the lines has an additional point $C$ on it, but $C$ doesn't have any place in the axioms, so it makes no difference.)
You have one model for the axioms. Can you think of another model for the axioms that is fundamentally different from this one?
Hint: