Let $A,B$ be sets non empty sets. Let say that if $p\in A$ then $p$ is said to be a point and if $l \in B$ then $l$ is said to be a line. Let $C$ be a set of the form $\{p,l \}$ with $p \in A$ and $l\in B$. We say that $p$ is incident with $l$ if $\{p,l \} \in C$ and also we say that $l$ is incident with $p$.
We say that $(A,B,C)$ satisfy the axioms of a projective plane if the following holds:
Axioms:
For every pair of distinct points there is a unique line incident to both.
For every pair of distinct lines there is a unique point incident to both.
There are four distinct points, where no three are incident to any line.
Let $V$ be a finite dimensional vector space over $\mathbb{F}_p$ of dimension $n$. Prove let $A$ be the set of one dimensional subspaces of $V$ and $B$ be the set of two dimensional subspaces of $V$. Let say that $p \in A$ is incident with $l\in B$ if $p \subset l$ (in particular is a subspace). I'm not sure if this is effectively a projective space. I think that the second hypothesis requires that $n=3$ (only by counting dimensions). Assume that $n=3$. How can I prove that the third hypothesis holds? (Clearly the first and the second holds in this case).
Thanks!
Your observation that we need $n=3$ for the second axiom to hold is correct (+1).
Assuming $n=3$, consider the vectors with coordinates $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, $(1,1,1)$. Any three of them span all of $V$ (check this). Therefore no 2D subspace contains any three of them. Nor does any 2D subspace contain three 1D-subspaces spanned by those points. This gives you the final axiom.