FOM: report from expert

Harvey Friedman friedman at
Tue Apr 6 12:37:01 EDT 1999

[Summary: the postings of McLarty on Grothendieck and the use of abstract
categorical methods in number theory are grossly misleading, according to
my expert source].

I have been told that there is absolutely no trace back from the references
used in the body of the Wiles paper to Universes (of Grothendieck). And
that in any serious use of category theory in number theory, one must
control the categories involved in order to accomplish anything, and a tiny
part of that control is knowing that the cardinality of the categories is
small. In the case of Wiles, the structures involved are all finite. With
some stretching of interpretation, maybe something is countably infinite.

In particular, I have been told that it is grossly misleading to consider
any connection whatsoever between Universes and serious number theory of
the integers, rationals, or finite degree extensions of the rationals.

I did ask the following question, which is very relevant to f.o.m. What if
Wiles proved FLT using Universes. Would Wiles get credit for having
established the truth of FLT?

The answer is that he would definitely get credit for having established
the truth of FLT. However, there would be considerable interest paid to the
problem of removing Universes from the proof. In fact, in making the proof
as "normal" as we now know it can be made.

Thus in a sense, "a proper class of strongly inaccessible cardinals" - what
Grothendieck used in his Universe work - is a kind of gray area for
mathematics. It appears that down to earth theorems established with their
use will be generally regarded as having been "proved." However, this is
not quite a counterexample to my claim that ZFC is the gold standard for
publication in the Annals of Mathematics. In my opinion, the Editorial
Board and/or the referee would require that the use of Universes be
explicitly mentioned.

In any case, the acceptance of Universes does show that it is possible to
get the mathematical community to accept at least some large cardinal
hypotheses. Probably much more convincing would be needed for, say, Mahlo

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