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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 
$y_1,\ldots,y_n$ are distinct solutions, then 
$\mathrm{tr.deg}(\mathbb{C}(t)(y_1,y'_1,\ldots,y_n,y'_n)/\mathbb{C}(t))=2n$. (This was proved by Nishioka for the single equation 
$P_{{\rm I}}$.) For generic Painlevé III and VI, we have a slightly weaker result: 
$\omega $-categoricity (in the sense of model theory) of the solution space, as described below. The results confirm old beliefs about the Painlevé transcendents.
