We prove that if
$y''=f(y,y',t,\alpha ,\beta ,\ldots)$
is a generic Painlevé equation from among the classes II, IV and V, and if
are distinct solutions, then
. (This was proved by Nishioka for the single equation
.) For generic Painlevé III and VI, we have a slightly weaker result:
-categoricity (in the sense of model theory) of the solution space, as described below. The results confirm old beliefs about the Painlevé transcendents.