We have recently shown that there are no algebraic relations over C(t) between distinct solutions and their derivatives of a general Painlevé equation, culminating a line of work starting with Painlevé, Drach, and others. The proof requires an analysis and translation of the existing knowledge regarding "irreducibility" and algebraic solutions, with the model theory of dierentially closed fields. We explain some of these concepts and sketch the proof. (Joint work with my student Nagloo.)