[FOM] dependence relations - formal theory

John T. Baldwin jbaldwin at uic.edu
Tue Jul 22 09:18:41 EDT 2003

Friedman writes:

What I have in mind is this. Some clear general theory of "dependence 
relations". A clear set of examples of "dependence relations" in diverse 
areas of mathematics. Even better would be to include some outside 
mathematics. A clean theorem to the effect that these notions of 
dependence can be uniquely defined from certain simple parameters 
clearly associated with the mathematical context. Perhaps a theorem to 
the effect that under suitably general conditions, there is always a 
unique good notion of dependence.

Baldwin replies.

 I gave a first order formulation of dependence theory inFirst order 
theories of abstract dependence relations,Ann. Pure Appl. Logic, 26 
(1984), 215-243.  This includes a  list of  mathematical examples and 
even a bit of recursion theory

The formalism is to add n+1 -ary relations: R_n(x,ybar) means x is in 
the closure of ybar.  Now the Van der Waerden axioms are first order.The 
notions of modularity, triviality, k-psuedoprojectivity, m-Pappian
and exact k-independence are examples of sentences showing the Van der 
Warden axioms are far from complete.

A natural notion is to try to find natural completions of these axioms. 
 Since the coordinazation theorem for projective planes shows much of 
the first
order theory of the underlying field is coded in the dependence 
structure on subspaces of vector spaces, this line doesn't seem too 
promising. That is,
there are going to be many completions.  There has been a scattering of 
work as I indicate above (some from the matroic side) that produce 
natural first order properties.  But I don't think anyone has attempted 
a classification as Friedman proposes.

In some sense the work has gone forward as a study of `combinatorial 
geometries' -- which is an equivalent formulation.

Friedman also writes:

Incidentally, I note that Baldwin writes above:

> The
> investigation of closure systems continues as a minor area--
> sometimes called matroid theory.

By this standard, would Baldwin regard model theory as a minor area? 
What kind of "minor" is used here?

I mean this in a sociological sense.  My impression is that there are 
less than 50 people working in matroid theory (could be way off and
happy to get a correction).  This is about the number that were working 
in stability theory in the early 80's when I wrote the above article
and is the number of people who wrote articles involving  the Hrushovski 
construction in the  last 10 years.  
But I am not trying to compare model theory and matroid theory.  Either 
is a minor area where logic and combinatorics are major areas.

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