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I just noticed that Randall Holmes claims to have a proof of the consistency of NF. Does anyone know more about this?
There was some discussion of the result on the FOM mailing list in 2013, then again in July last year.
See also this MO question.
This has been going on for ages. My guess is that a portion of the difficulty may be arising from the transtranslation from an underlying structural set theory construction to a material interpretation, without never seeing the structural version.
Thanks! I added a comment to New Foundations.
Where is it proved that $NFSet$ is not Cartesian closed?
From McClarty’s article, he is assuming that a pairing operation gives rise to a cartesian product as well as a category with arrows the entire functional relations using said pairing operation. Why not consider the pairing operation as defining a non-cartesian monoidal product and aim for just a closed monoidal category? I guess dropping Cartesianness violates assumption 1 in McClarty’s proof and assumption 2 in Forster’s proof, so neither apply to show the internal hom can’t exist.
Clearly we then have to work harder to get existence of limits of various sorts. Also, some binary products may exist, if not all of them.
EDIT: Hmm, the recent talk “A Monoidal Closed Category in NF” by Gorbow [1] seems to establish this result for a slightly larger category of relations, with $Hom(A,B)$ relations on $A\cup B$ (I guess) restricting to functions $A\to B$ as relations from $A$ to $B$.
[1] This seems to be his university page, but not sure how up to date this is. It seems he is now in private industry
I added the references Zhen Lin supplied to the page NF. It might be worth typing up a sketch of at least McLarty’s proof, since it is very short.
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