In a forcing-based consistency proof of a proposition P, you start with a model of ZFC (or a sufficiently large subset of ZFC, depending on the exact formalism being used), approximate a witness to P (e.g. a bijection between a set of reals and aleph_2) within the starting model, and then apply the forcing machinery to construct a new model of ZFC where P holds. Unlike the diagonalization argument, it's inherently metamathematical.