FOM: Nice idea, fwd from Feferman
Colin McLarty
cxm7 at po.cwru.edu
Wed Apr 14 14:15:07 EDT 1999
Sol Feferman asked me to forward this. It began when I asked him
about a friend of mine who suggested Feferman had once studied "universes"
defined as V(i) for i any limit ordinal. I will only add that I believe he
is right about the minimal proof theoretic strength needed for Wiles's proof.
>'s recollection is far from the idea. What is needed is that V(i)
>is formally treated as an elementary substructure of V. That can be done
>by adding an additional symbol for such a universe, and adding, formula by
>formula, the axioms that it forms an elementary substructure of V (with
>the membership relation). I did this in a paper "Set-theoretical
>foundations of category theory" in _Reports of the Midwest Category
>Seminar III, Lecture Notes in Math. vol. 106 (M. Barr, et al, eds.) 1969,
>pp. 201-247. I considered a few theorems from general category theory and
>showed how they can be reformulated here, using "small" to mean a member
>of such a "universe". Examples that go through are the Adjoint Functor
>Theorem and Yoneda's Lemma. A question was raised (I think by Isbell at
>the time) about some theorems like the Kan Extension Theorem, which on the
>face of it require the ordinal of the universe to be inaccessible. If one
>adds that assumption, the theory is conservative over ZFC plus a statement
>of existence of Mahlo cardinals of the first kind (very low large
>cardinals in today's pantheon of large cardinals).
> I gather this has been a matter for discussion recently on the fom list.
>Would you please communicate the above information to the interested
>parties there.
>
>Thanks,
>Sol Feferman
>
>PS. I also indicated in my article that usual homological algebra, as e.g.
>in MacLane's Homology book, can be formalized in the theory I
>described, with a "universe" symbol for the "small" sets, conservative
>over ZFC.
>
> I would be extremely skeptical as regards the possible necessity of
>universes in Wiles' proof. Indeed, it has been speculated that the proof
>can be formalized in a system conservative over Peano Arithmetic. I don't
>know what, if any, evidence for this there is. But it would be doubtful
>if his proof really needed much of impredicative set theory. Only a
>detailed analysis would settle this, though. That has nothing to do with
>what mathematicians feel is conceptually necessary to understand his
>proof.
>Dear Colin,
> The V(i) with i a limit ordinal (take it to be greater than omega in
>order to satisfy the axiom of infinity), do not in general satisfy the
>replacement axiom. That is one reason more is done. Another reason is
>that by making the set s = V(i) an elementary substructure of V and
>reading "small set" to mean--element of s--any result about small sets,
>small categories, etc. can be transferred to V, so that it becomes a
>result about sets, resp. categories in general. That is one way to show
>that the distinction between small and large categories or between
>categories in one universe and those in another universe, is not an
>essential distinction, though something like it seems to be needed for the
>current set-theoretical foundations of category theory. The use of the
>'s' theory I described aims to have it both ways, to have your cake and
>eat it too.
> Please pass on this clarification with your fom communications.
>
>Best,
>Sol
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