FOM: Urbana Thoughts
John Baldwin
jbaldwin at math.uic.edu
Sun Jun 11 11:25:19 EDT 2000
This is a response to Harvey's note of June 10 on the Urbana
meeting. He referred
to comments I (Baldwin) made during the final session of the Urbana
meeting.
I think my attempts to draw careful lines in a brief
comment at Urbana were inadequate. Let me try again.
internal/intrinsic: Those aspects of a subject which are
developed by reflection on its own practice.
external/extrinsic: Those aspects of a subject which are
developed to solve problems posed in another area.
In this sense I view many of the `algebraic developments in
model theory' as intrinsic. In particular, the Zil'ber
conjecture that every strongly minimal set (i.e. every
basic building block of an aleph-one categorical structure)
is `set-like', `vector-space like' or `field-like' as
intrinsic. Hrushovski's refutation is certainly intrinsic;
the applications of the method of his construction to
projective planes, generalized n-gons, and 0-1 laws
are a bit more questionable. But except in the case of 0-1
laws the questions were asked by model theorists.
Let me clarify the intrinsic nature of the Zil'ber conjecture
a bit more starkly. It can be viewed as an assertion that
most of the model theory of aleph-1 categorical structures was
simply a different way to approach algebraic geometry. Fortunately,
from my personal standpoint, this turned out to be false.
A clear move to the extrinsic side comes with the Hrushovski
work on Mordell-Lang. But note that this is a result of
determining the sense in which the Zil'ber conjecture was
inexact: the failure of first order definablity theory to
distinguish between positive and arbitrary sentences. (Pillay
described this as logic speaking of all isomorphisms and
algebraic geometers speaking of rational isomorphisms; I hope
I am not misquoting.)
A little thought shows that it easy to get carried away by
forcing such a classification on results. The following
theorem of Hrushovski is a key development
in what I would have expected Harvey to call `pure model
theory' -although `stability theory' was curiously missing from his list.
There is no uni-dimensional strictly stable first order theory.
The proof relies on the interpretation of an algebraically
closed field and information about the transitivity properties
of actions of algebraic groups. Both hypothesis and conclusion
are concerned with arbitrary structures.
This and a wealth of other examples show that the interaction
between various areas of mathematics is much more complicated
that the discussion in Urbana really touched on.
I did not intend to prejudge the possibility of success of
Boolean Relation Theory (despite some cynicism) but to
claim that if it succeeded it would be developing a new,
interesting its own right, area -- not meeting the
`extrinsic' demand of answering the questions of other
mathematicians.
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