  Quite early on Quine added classes to NF to obtain a theory of sets-with-classes known as ML. In the ZF context, adding classes is a natural thing to do, for it enables one to reduce the infinite replacement scheme to a single set-existence axiom: “the image of a set in a class is a set” if augmented with suitable class-existence axioms. However, none of the axioms of NF refer to classes in the way the replacement scheme of ZF does, so there is nothing for the class existence axioms to do. For this reason ML is nowadays regarded as a pointless syntactic complication of NF with no new mathematics, and it is not the subject of any research. At best the classes can be a harmless halo of epiphenomena (Wang [1950]). However, at worst – if the set existence axioms are allowed to have bound class variables (as in the 1940 first edition of Quine [1951a]) – then the class of total orders that are genuinely (“externally”) wellordered (no subclasses without least members) becomes a set and we have the Burali-Forti paradox (Rosser [1942]). The rumours still occasionally surfacing among mathematicians of the inconsistency of NF probably have their origin in a misreporting of this result, but the result itself concerns ML only so it has no bearing on the consistency question for NF. If NF is inconsistent it is for totally different reasons. There is recent work suggesting that NF is consistent but it has not yet been confirmed. A paper containing an alleged proof of Con(NF) is working its way through the refereeing process as this entry is being revised in 2018.