The consistency of NF in Lean?
Timothy Y. Chow
tchow at math.princeton.edu
Mon Feb 21 12:35:26 EST 2022
FOM readers may be interested to know that I asked Randall Holmes about
the prospects for formalizing his proof of the consistency of NF using the
proof assistant Lean, in order to address lingering doubts about the
correctness of the proof. Holmes responded by saying that he had just
been thinking the same thing himself. Some preliminary discussions have
begun on the online Lean forum. If you already have a Zulip account then
you can find the discussion in the #maths stream in the topic on "The
consistency of NF". Otherwise, you can view the discussion so far here:
https://leanprover-community.github.io/archive/stream/116395-maths/topic/The.20consistency.20of.20NF.html
Whether this project will come to fruition remains to be seen, but the
preliminary indications are promising. Holmes has indicated that he is
willing to take the lead on the project, subject to the constraints
imposed by his other commitments. Mario Carneiro has already taken a huge
step by attempting to formalize the statement of Holmes's main theory
(although Carneiro has made it clear that he can devote only a limited
amount of time to this project going forward). Thomas Forster has also
indicated that he would like to help, not by directly working in Lean, but
for example by supervising interested students.
Forster, by the way, has said that he understands the general structure of
the proof and believes that it is probably correct, but that because the
details are so complicated, and have never been fully nailed down in
writing, he cannot personally vouch for its correctness. So there do
remain some legitimate doubts. I am hoping that this project (which I
have suggested be called "Conifer" since it sounds a bit like Con(NF))
will settle the matter one way or the other.
Tim
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