[FOM] Proving FLT in PA
joeshipman at aol.com
Tue Feb 21 21:32:25 EST 2006
Tim , one place you've seen it raised is on the FOM, by me, in
Friedman, in this post,
reports that, according to an expert source, the use of Grothendieck
Universes (that is, infinitely many inaccessible cardinals) is not
necessary in Wiles's proof of FLT and all the top number theorists know
how to eliminate the assumption.
However, there are many other theorems of number theory and algenraic
geometry proved that use the full machinery Grothendieck and others
developed, and nobody ever gave a satisfactory answer to my repeated
query "WHERE is the metatheorem that allows us to tell when the
'Universes' assumption can be eliminated?"
I think this ought to be considered scandalous.
The 1993 newsgroup discussions you link to seem to be more worried
about eliminating AC from Wiles's proof than about eliminating the
Universes assumption, but that's just silly, because in this case there
IS a metatheorem, due to Godel, that any arithmetical statement
provable from ZFC (or even from ZFC + V=L) is provable in ZF.
Obviously, one can't simply say "statements of a certain form which are
provable with Grothendieck Universes are theorems of ZFC", because
Con(ZFC) is of a very simple logical form and follows almost trivially
from the Universes assumption. What I would expect to see is a
metatheorem of the form "pi^0_1 statements proved using etale
cohomological facts X Y and Z applied to sets of type A B and C are
theorems of ZFC". That is, even if X Y and Z need a strong assumption
in order to be proven in their full generality, the assumptions can be
eliminated for applications of a certain type.
-- Joe Shipman
>Here's one issue I've seen raised. The fundamental facts of etale
>cohomology are developed in SGA, specifically SGA4, and Grothendieck
>universes are (apparently---I've never studied SGA4) freely assumed
>The experts seem to feel that for a "concrete application" like FLT,
>would be routine to eliminate any use of Grothendieck universes. In
>particular, there's probably no interesting math involved in the
>elimination process. However, unless someone actually goes to the
>of doing the necessary bookkeeping, there remains a question mark.
>Unfortunately, because of the highly technical nature of the subject,
>doing the bookkeeping would seem to require an inordinate amount of
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