A first step is to remove the indexes from the irrationals. In a argument about the supposed countability of the irrationals, to refer to them with indexes (e.g. p_i) looks like begging the question.
As it happens, I don't think the indexes do anything but add a layer of confusion. I think you can rename the various quantities without altering the meaning, i.e. rather than talk about irrationals p_i and p_h just use p and r (q is taken).