"Consider the set P of providers" means when you eventually refer to p_2, you'll have to note that you mean an element (the second, in some sense) of P. That moved the verbosity around rather than eliminating it.

To me it seems like a much bigger error that "Consider the set P {p_1, p_2, ... p_N} representing providers and the set c {c_1, c_2, ... c_N} representing customers" states outright that the sets are equal in size. I would expect C to be much larger than P.