Reference for the notion of "dimension" in incidence geometry? Consider an incidence geometry as in an elementary course on geometry. The german wikipedia 
https://de.wikipedia.org/wiki/Inzidenzgeometrie
mentions a well defined notion of "dimension". Does anybody know of a good reference?
The literature given at the end of the wiki-page seems to deal only with the well known case of affine and projective geometry. 
 A: You mention in a comment that you are primarily interested in a precise statement. I believe the Wikipedia article has one, so I'll translate that to English and add a bit of notation.
The incidence geometry is defined by a set of points $P$, a set of lines (or blocks) $B$, and an incidence relation $I\subset P\times B$. Each pair of distinct points is incident with exactly one line:
$$\forall p,q\in P:p\neq q\rightarrow\dot\exists g\in B:\bigl((p,g)\in I\wedge(q,g)\in I\bigr)$$
A subset $M\subset P$ is called a linear set iff
$$\forall p,q\in M,r\in P,g\in B:\bigl((p,g)\in I\wedge(q,g)\in I\wedge(r,g)\in I\bigr)\rightarrow r\in M$$
So for every two points in $M$, if they are incident with a line $g$, then any (other) point $r$ incident with that same line has to be in $M$ as well.
The linear hull $\langle M\rangle$ of a set $M\subset P$ is the intersection of all linear sets containing $M$. In (ugly-looking) formula:
$$\langle M\rangle:=\bigcap_{\substack{N\subset P\\M\subset N\\N\text{ is linear set}}}N$$
If $M$ is already a linear set, $\langle M\rangle$ is just $M$ itself. Otherwise you can picture that for every pair of points in the set, you add all the points on the lines joining them, repeatedly until you reached a fixed point.
Now $M\subset P$ is a basis of $P$ iff $\langle M\rangle=P$ and there is no smaller set with that property, i.e.
$$\langle M\rangle=P\wedge \neg\exists N\subset P:\left(\lvert N\rvert<\lvert M\rvert\wedge\langle N\rangle=P\right)$$
The dimension of the geometry is the size of the basis minus one, i.e. $\lvert M\rvert-1$.

Is it: Two sets of independent generators have the same cardinality?

Not necessarily, which I believe is the reason why the definition of the basis explicitly requires it to be of minimal cardinality.
I'm not sure what you mean by “independent”, so the subsequent text here is just ignoring that word. You can add arbitrary additional points to any generater of the whole geometry, and again end up with a generator. I guess this much is fairly intuitive, and shows that generators can come in different cardinalities.
The question becomes more interesting if you consider “local minima”: start with a generator of $P$, then remove points until you can remove no more points without loosing that generator property. In formula, look for an $M$ such that
$$\langle M\rangle=P\wedge \neg\exists N\subsetneq M:\langle N\rangle=P$$
In a way, such a subset would be locally minimal: you could not make it smaller just by removing points. The interesting question here would be, would each such locally minimal generator be globally minimal and hence a basis?
I will see whether I can come up with an example where this is not the case, or an idea of why this will always be the case.
