FOM: foundationS of mathematics
jbaldwin at math.uic.edu
Sun Oct 26 10:02:15 EST 1997
First couple of paragraphs are reprise for context.
>One should understand mathematical
>structures in terms of the relationships amongst the objects
>of the structure with our regard to their internal properties.
>Thus we study the reals as a complete ordered field, not
>Cauchy sequences of equivalence classes of integers.
>So I suppose the question to Steve is whether any analysis
>beginning at this level could be viewed as foundational?
Of course. But I have always had trouble doing any systematic analysis
beginning at this level. The usual setup in Foundations of Mathematics
is of course quite different as you know. But it's KING until you
figure out how to get going on a comparably smooth and systematic basis
with what you suggest. John - tell me about your results in this
Baldwin now replies
Of course I agree with Dave that FOM is the only successful systematic
attempt at foundations (that I know of). At
you can find my only attempt to write my ideas on the subject down.
I avoid calling my analysis foundations to evade the terminolical
discussion that has sometimes paralyzed this list.
After some histrionics about what I am not doing, 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.
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
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.
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.
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