[FOM] Etale cohomology in third order, how and why?

[Colin McLarty]

I will not linger on why number theorists want etale cohomology.
Number theorists write about that.  In short, etale maps are an
astonishingly simple, effective means to transfer intuitions about
smooth functions and continuity from differential geometry into number
theory.    As in differential geometry, deep theorems often have long
demanding proofs.  These would be undiscoverable in the first place
and incomprehensible even after discovery without a guiding intuition
(compare Poincare on intuition).

Here i want to explain why to use third order arithmetic.  Third order
is cryingly disproportionate to the strength of the particular
calculations at the base of the applications in number theory.  People
on FOM have given valuable hints and encouragement: Harvey's
suggestion to use ZF[n] underlies all of this.  And just now Andreas's
help
showed me how to understand ZF[1*].  Further comments are welcome.

In brief: third order (and not second) handles the usual lemmas of
etale cohomology, it suits the way number theorists think about it,
and it gets us within range of reverse mathematics.

A) Third order arithmetic is extremely well adapted to the general
reasoning of etale cohomology.  Basic technical lemmas have to be
carefully proved in this form, but then the working tools work almost
unchanged.  We largely limit the subject to Noetherian schemes, as in
fact number theorists often do in their own accounts.  But we place no
limit on their cardinality and no limit on which modules we use over
those schemes.   This requires the set theory I will call ZFG[1*].
That is ZF without powerset but with global choice and with an axiom
"every set has a set of all its countable subsets."  Notably,
countable sets have power sets.  The stronger axiom serves to
establish this lemma: for any countably generated module M and any
other module N (no matter how N is generated or what size it is) there
is a set of all linear maps from M to N.  The theory ZFG[1*} has no
more consistency strength than ZF[1].  Both are interpretable in
standard Z_3 third order arithmetic.

B) This reflects how number theorists think.   Important number
theorists say they can bound all of their work within very small
cardinals.  By interpreting the work at the level of Z_3 we can say
this is not only true in practice, or true of the important cases
(which, after all, are constantly changing with time).  It is true in
principle:  The entire etale cohomology of Noetherian schemes can be
developed supposing the continuum C exists and no larger cardinal.
For these purposes the continuum counts as very small.

C) The third order presentation puts us within striking distance of
reverse mathematics.  I personally believe Fermat's Last Theorem must
be provable in PA itself, as I have often said.  And if so, then
surely it is provable in some well motivated weaker fragment -- say
EFA as Harvey has conjectured.  The only currently promising strategy
to demonstrate either of these is to build on existing proofs of FLT.
And no matter how sure we are that the transfinite-order machinery
used in publications is somehow eliminable, we cannot hope to progress
to actual PA provability or EFA provability without knowing *how* the
transfinite machinery is to be eliminated.  (By transfinite order
machinery I mean paradigmatically the use of replacement along
uncountably iterated power sets, which books like Weibel's Homological
Algebra offer as a way to get by with less than universes.)

best, Colin