[FOM] Dependence relations in model theory

[John T. Baldwin]

Dependence relations

This note is a short survey of the role of dependence relations in
model theory. It continues the series I have been asked to write
discussing various fundamental concepts of model theory.  I will
discuss 3 developments. A) the major generalization of Van der
Waerdens notion of independence by Shelah, which is a leit-motif
of first-order stability theory; B) the development of geometric
stability theory as the intense investigation of one aspect of A);
and C) the variations on A) and B) by Shelah and Zilber which are
appropriate for infinitary logic.

I conclude with a few more `philosophical' remarks.

A) The relation a is in cl(X) between elements and subsets of a
set D is called an `algebraic dependence relation on D if it is

1) transitive: cl(cl(X) = cl(X)

2) reflexive:  a in cl({a})

3) monotonic: a in cl(X) and X subset Y implies a in cl(Y)

4) exchange: a in cl(Xb) and a not in cl(X) implies b in cl(Xa)

5) finite character: a in cl(X) iff a in cl(X_0) some finite
subset X_0 of X.


In this setting X contained in D is independent if each x in X is
not in cl(X-{x}).  An independent set X is a basis for D if cl(X)
= D.

The properties of linear independence and algebraic independence
were isolated by Van der Waerden (I think in the 30's).  The
investigation of closure systems continues as a minor area--
sometimes called matroid theory.  Much of the effort goes to the
representation of abstract closure systems as vector spaces or
fields or .....  There was (is?) also active development of these
notions in universal algebra.


Marsh noticed that defining a in acl(X) if a is in a
 FINITE set which is first order definable with parameters from X,
 then on a strongly minimal structure, one has a closure system in
 the above sense.  Baldwin and Lachlan generalized this by letting
 D be any strongly minimal set in a model.

 The key point is that if there is a closure system on D then D is
 determined up to isomorphism by the cardinality of a basis (a
 maximal set of independent points.

 Shelah discovered the notion of forking provides a relation
 on a model M of a stable theory such that for any subset B of M
 tp(a/BX) forks  over B  is a relation_B which satisfies all
 the axioms except transitivity 1).

The big surprise here is that `exchange' is global; unlike the
prototypic notions the difficulty is to establish transitivity.


 This leads in two directions.

 a) Investigate models using the global forking notion.  This is
 one important technique in stability (and later simplicity)
 theory.

 b)  Forking determines a family of closure system in the Van der
 Waerden sense (on strongly minimal sets or more generally on
 realizations of regular types).   The connections between these
 various `local' dimensions is provided by Shelah's
 notion of orthogonality. `Geometric' properties of the different regular
 sets play an important role.  In general, Shelah needed only the
 trivial/nontrivial dichotomy for his investigations.  In B) below
I mention other aspects of the classification.


 Forking is defined combinatorially: tp(a/BX) forks  over B if
 there is a formula phi(v,xbar_0) and an infinite set xbar_0, xbar_1 of
 indiscernibles over B such that the conjunction phi(v,xbar_i) is
 inconistent but phi(a,xbar) holds.  In omega stable theories
 tp(a/BX) forks  over B iff  Morley rank of a/BX equals Morley rank a/B.
 Further specializing one
 gets the classical notions of closure and dimension in
 algebracially closed fields, vector spaces, differential fields
 etc etc.

 B) Here one follows up the Baldwin-Lachlan idea and finds strongly 
minimal sets
 (so closure systems) and
 investigates them more intensively.  I have reported the
 distinctions between trivial, modular (vector-space like) , and non-modular
 earlier in this series.  These distinctions play an important role in
 pure model theoretic considerations: spectrum problem (Shelah, 
Hrushovski, Laskowski, Hart)
 Vaught's conjecture (Buechler, Newelski), finding classical groups in 
abstract structures
 (Zilber, Hrushovski).  Interesting directions in algebra have
 come from using this classification in the study of differential
 fields, compact complex manifolds, diophantine geometry (Hrushovski,
 Bouscaren, Scanlon, Moosa, Pillay, Zilber....)

 There has been some development and I expect more in finding
 further classification of strongly minimal sets. For example,  a closure
 system is
 k-psuedoprojective if a in cl(Xb) implies a in cl(X_0b) where
 |X_0| =k and X_0 subset cl(X).  Fields are not k-psuedoprojective
 for any k; vector spaces (more generally modular strongly minimal
 sets) are 1-pseudoprojective.  Hrushovski proved a
 strongly minimal set  which is not 2-pseudoprojective iterprets a field
 and thus is not  k-psuedoprojective for any k. Buechler and
 Hrushovski this to prove there are no strictly stable theory which
 are unidimensional  such that all the closure systems have the
 same dimension.  (unidimensional stable theories are superstable)

Lately Holland and I have used an other property, exact
k-independence, to establish model completeness results.

C) As noted earlier in the series to study complex exponentiation,
Zilber was forced to formulate a notion of dependence in
infinitary logic.  The natural idea is `a in cl(X)' if a in is a
COUNTABLE set L omega_1 omega definable with parameters from X.
Certain technicalities intrude.   This is a special case of a
notion defined for a wider class of infinitary logics by Shelah in
the 80's and clarified by Lessmann lately.

References:  books by Baldwin (esp, Chap 2), Buechler, and Pillay
and Shelah; web pages of Shelah and Zilber


CONCLUSION:  Perhaps it was to avoid wrangling about the meaning
of `foundations' that led Tarski to his less than  mellifluous
phrase `methodology of the deductive sciences'.  In this spirit,
let me point out that it is already an astonishing convergence of
diverse methods when Morley's combinatorial definition of rank
yields the same object as Krull dimension (defined in terms of
chains of prime ideals) for the rank of a variety in algebaically
closed field.  The unifying role of `forking' as providing a
universal definition for notions of dependence  in the study of
diophantine geometry, algebraic geometry, vector space,
differential fields,compact complex manifolds etc.  is an
important contribution of model theory to the `methodology of
mathematics'.