  Putnam's argument has received a number of kinds of criticism in the literature. On the technical front, Bays has argued that Putnam's use of the downward Löwenheim-Skolem theorem is illegitimate, since standard systems of set theory do not allow us to apply this theorem to a proper class like L. Indeed, even if we leave the details of Putnam's proofs aside, Gödelean considerations show that Putnam's theorem cannot be proved in ZFC at all (since the theorem entails ZFC's consistency). Of course, if Putnam is willing to use a stronger background theory to prove his theorem—e.g., ZFC + “there exists an inaccessible cardinal”—then he can evade these kinds of criticisms. But in this case, it's unclear why the model which results from Putnam's theorem should still be thought to satisfy our theoretical constraints. After all, anyone who accepts the new axioms used in Putnam's revised proof will have theoretical constraints which go somewhat beyond ZFC + V=L—e.g., their theoretical constraints might well include the axiom “there exists an inaccessible cardinal.” See Bays 2001 for Bays' original formulation of this objection; see Velleman 1998 and Gaifman 2004 for some alternate formulations; see Bellotti 2005 and Bays 2007b for critical discussion; and see chapter 3 (esp. § 3.3.3) of Hafner 2005 for discussion of a similar point concerning Putnam's use of transitivity.