friedman at math.ohio-state.edu
Tue Oct 28 03:00:31 EST 1997
>Of course I agree with Dave that FOM is the only successful systematic
>attempt at foundations (that I know of).
I would like to hear from people who disagree with this assessment,
including maybe Jon Barwise.
> I try to
>analyze one use of `finite' in algebra. Specifically, I discuss the
>use of the descending chain condition in ring theory and group theory.
>I note that in several situations (most well-known is the Wedderburn
>theory) the method succeeds because (?) the apparent second order
>condition dcc on ideal needs only be used as dcc on uniformly first
>order definable ideal. I.e. it is a property of the ring being stable
>in the model theoretic sense.
Whenever I hear about the descending chain condition, my hormones jiggle
and I get interested in the associated ordinal. In the case of the Hilbert
basis theorem for polynomial rings over fields in finitely many variables,
the associated ordinal is omega^omega, as is exploited by Steve Simpson in
his Reverse Mathematics study of the Hilbert basis theorem. The bigger the
ordinal, the more metamathematical power lurks under the surface. Any
interesting ordinals here, John?
>A more elaborate analysis concerns the lifting of the result `semisimple
>implies completely reducible' from finite to infinite groups. This
>has been done for algebraic groups. There are various approximations
>for stable groups all relying on replacing `arbitrary finite chains'
>by `uniformly definable finite chains'. None of them specialize to
>completely yield the result for algebraic groups. The grand version
>of this program would be `explain' algebraic geometry in terms of
>`finiteness conditions on definable sets'. Much of the Zil'ber Hrushovski
>work is relevant to such a `foundational study'.
>However, I have found no clear way to even formalize the questions involved
>here let alone the methods. That's why the paper has been sitting in
>the drawer for ten years.
Maybe you can give us a watered down special case, and ask us to come up
with a formalization there, as a start to get the ball rolling?
>Finally, it seems to me that the methods of say algebraic geometry are
>very different from those of real analysis and so it seems appropriate to
>use different logical tools for different investigations. If something
>sensible develops then we can try to move to a global analysis.
Maybe a crucial difference is "tame" versus "not tame." This is like "not
subject to the usual Godel phenomena" versus "subject to the usual Godel
phenomena." I like the idea that maybe there is a "foundations of tame
mathematics" or "tame foundations of mathematics." This would be very
interesting, if done systematically and smoothly. Of course, I am also
vitally interested in spotting the most fundamental situations in
mathematics where things are demonstrably not tame.
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