  On the more-purely philosophical side, many authors have criticized Putnam's assumption that simply satisfying a first-order formalization of our theoretical constraints is enough to make a model “intended.” So, for instance, Hacking has argued that we should really be committed to a second-order formulation of set theory and that Putnam's key theorem doesn't apply to such formulations (Hacking 1983). Others have argued that the intended model for set theory needs to be transitive and that, once again, there's no reason to believe that the model produced by Putnam's theorem is transitive (Bays 2001). Finally, and as mentioned in the last paragraph, several authors have suggested that an intended model for set theory should at least be well-founded, but there's no reason to think that Putnam's own model is well-founded (Bellotti 2005).