The first is required when you want to later refer to an individual element from the set.
"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.
If this is a commonly used notation in the paper, then it would make sense to state once up front "here's how we refer to sets and their members". DRY and all that.
versus
"Consider the set P of providers and the set C of customers"
The former is about 2x the length of the latter. This is the verbosity I read into the original comment.