[FOM] Weaker foundations for FLT
colin.mclarty at case.edu
Wed Mar 1 18:16:08 EST 2006
We now have thoughts about proving FLT in ZF, 3rd order PA, and EFA.
Harvey well points to some particulars of my strategy for Grothendieck
techniques in gneral into ZF. Taking those into account I think I can
replace universes by V(lambda) where lambda are suitable ordinals (all
provably existing in ZF), and "suitable" means that V(lambda) supports
a certain amount of second order replacement, which is adapted to work
for a given site (or set of sites). I will look at it.
The question of ZFC helps me to focus something I keep trying to say
more clearly: Grothendieck's use of universes is not to gain vast
generality or to MAXIMIZE in Pen Maddy's sense or to have large
cardinals. It is just a quick and dirty way to get enough elbow
room. So relativizing to constructible sets is fine. It loses
nothing that we want. So we cannot need the Axiom of Choice. We can
just do it all in constructibles. I have not looked into how often
choice is even ostensibly used in SGA style proofs. It certainly is
> As the cited book shows the Wiles-Taylor proof is a truly
>enormous beast. The work naturally seems to be carried out in third
>number theory, and to compress it into PA seems a heroic if not
Okay, with this nuance: Look at Washington's article on Galois
cohomology in the book. It introduces group cohomology as a derived
functor cohomology, which on the face of it uses a universe. But the
point of the article is to say that we really only need the 0- 1- and
2- dimensional cohomology groups of some structures coming from
arithmetic. These have explicit constructions in some low-order PA
(indeed the constructions were known to Emmy Noether in the 1930's
before group cohomology existed). I will take Solovay's word that it
is 3rd order, I have not looked into it.
This is a strategy of ``work-arounds.'' Wherever Wiles uses a theorem
from Galois cohomology we need to find a work-around getting the
specific information he needs without the general theory. This can
surely be done, but of course the result will be specific to FLT.
Other theorems would need other work-arounds.
I suspect this is the main reason that algebraic geometers have not
worked harder on our problem--they know that easy work-arounds exist
and will be uninteresting in themselves.
As to a proof in EFA, I am no expert but I think it would take real
new insights if it exists at all. Maybe really good insights. I
think I agree with Solovay here? I do not at all doubt the theorem is
provable in EFA. I just have no idea. But the current proof will
probably not compress that far without being fundamentally reconceived.
There has lately been progress on the algebraic geometry side, which I
just noticed. Angelo Vistoli has substantially re-written SGA 1 in
the terms of SGA 4, as Grothendieck said someone should do when he
first published SGA 1. This is in an AMS book FUNDAMENTAL ALGEBRAIC
GEOMETRY published last Fall and just arrived in my university library
this week. Others in the book rewrite theorems from the rest of
Grothendieck's algebraic geometry.
This does not go directly to the logical question but is part of an
overall clarification that will help clarify the logic in the long
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