[FOM] What is the current state of the research about proving FLT?

[Colin McLarty]

My proof that finite order arithmetic suffices in on the arXiv as
arXiv:1102.1773v4 <https://arxiv.org/abs/1102.1773v4>

That proof does not involve any change to the structure of Wiles's argument
.  It involves no changes to anything in the Grothendieck apparatus (SGA
and EGA) except a redefinition of "universe."

As to

2) What is the current state of *knowledge* (not beliefs!)
>    about what is needed to prove FLT (assuming that it can
>    indeed be proved in acceptable mathematics)?
>

I fear there is some disagreement over what constitutes knowledge.  I would
say Angus Macintyre's 2011 article"The Impact of Goedel's Incompleteness
Theorems on Mathematics," in the book _Kurt {Godel and the Foundations of
Mathematics: Horizons of Truth_ shows it is known that many results like
the ones used in proving FLT are provable in PA.  I claim it remains a
valuable project (and a large one, as Macintyre has emphasized) to actually
show FLT is provable in PA or even just to get it clearly into second order
arithmetic.

Personally I expect that when (and if) this can be done, it will show FLT
is provable in EFA.

best, Colin




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