We apply and extend the priority algorithm framework introduced by Borodin, Nielsen, and Rackoff to define “greedy-like” algorithms for the (uncapacitated) facility location problems and set cover problems. These problems have been the focus of extensive research from the point of view of approximation algorithms and for both problems greedy-like algorithms have been proposed and analyzed. The priority algorithm definitions are general enough to capture a broad class of algorithms that can be characterized as “greedy-like” while still possible to derive non-trivial lower bounds on the approximability of the problems by algorithms in such a class. Our results are orthogonal to complexity considerations, and hence apply to algorithms that are not necessarily polynomial time.