FOM: Call for help from number theorists

Joe Shipman shipman at
Thu Apr 8 09:31:34 EDT 1999

Friedman claims an expert he knows says that the results of Wiles do not
depend on Universes.  McLarty points out that following bibliographic
references leads to work of Grothendieck and others that uses
Universes.  McLarty also remarked (privately) that Deligne used
Grothendieck's SGA 4 overtly in proving the Weil conjectures.

It is possible for Friedman and McLarty to both be right, if the results
cited don't depend on Universes even if other results in the cited
references do.  The way to settle this is not to simply follow
bibliographic chains but rather to look at chains of actual precisely
formulated mathematical statements.  Can McLarty or anyone who agrees
with him provide a chain of statements S_1,S_2,...,S_n such that the
following is true?

S_1 = the Universes assumption

S_i is used in the proof of S_i+1 (citation required!)

S_n = FLT or a Weil conjecture (funny how these are still best known by
the names of their conjecturers rather than by their provers Wiles and
Deligne; thankfully it was not Wiles who proved Weil's conjectures or
the confusion would be intolerable)

If this is possible, then Wiles or Deligne owes us an argument that the
Universe assumption is removable.

If this is not possible because any such chain must pass through
unpublished "folklore", Wiles or Deligne owes us a more carefully
documented proof.

If this is not possible because Universes were not actually needed,
McLarty is wrong.

I don't have the expertise to resolve this.  Number theorists please

-- Joe Shipman

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