John T. Baldwin jbaldwin at uic.edu
Sat Jun 21 09:46:29 EDT 2003

The study of categoricity dates (at least) from the work of
Huntington around the turn of the 20th century.  While a second
order sentence can be categorical -- have exactly one infinite
model, the Lowenheim Skolem theorem shows that a first order
sentence which has an infinite model has one in every infinite
cardinality.  Understanding the nature of  sentences or theories
which are categorical in kappa (have only one model of cardinality
kappa) has been a major theme in model theory since the 50's. This
study evolved through the methods of stability theory into
understanding which models admit a nice structure theory (in the
sense of an algebraist).  This structure theory is built around a
dimension theory which is incompatible with Godelian phenomena, so
categorical structures are `tame' in the sense of Van den Dries. I
may say more about this in a later note.  The tameness
observation, although implicit since at least the 70's, has not
played a significant explicit role in the development.

Any aleph one categorical theory is determined by its so-called
strongly minimal sets. In this note I will sketch some background
information about strongly minimal sets.

Goal:  While the goals of this subject have evolved over time, a
starting place is:  The known examples of theories categorical in
an uncountable power: vector spaces, algebraically closed fields
are extremely well-behaved mathematical structures.  Is there a
logical explanation for this good behavior?

What does well behaved mean? one model in each uncountable power;
aleph_0 countable models, a nice dimension theory, a `geometry' on
the definable sets ... (Decidability does not follow -- we can
easily code undecidable phenomena into the axioms.)

One explanation for all these phenomena is that we are dealing
with strongly minimal sets --  but that explanation is too


Los-Vaught test:  If a theory is categorical in power then it is

Morley's Theorem: A(countable) theory is categorical in one
uncountable power iff it is categorical in all uncountable powers.

the `little Vaught conjecture': an aleph_1 categorical theory has
either 1 or aleph_0 countable models.

Morley's conjecture:  No aleph_1 categorical theory is finitely

 Origins of strong minimality:

The notion of strongly minimal was invented to solve the little
Vaught conjecture. The investigation of the geometry of strongly
minimal sets arose to settle Morley's conjecture (for theories
which are also aleph_0 categorical).

Marsh proved the little Vaught conjecture under the hypothesis
that the universe is strongly minimal.

Baldwin-Lachlan  proved that strongly minimal sets `controlled'
the model of an aleph one categorical theory thus establishing the
full `little Vaught conjecture'.   Here is more detail on the rest
of the material describe in the abstract.

Basic properties of strong minimality:

A set is minimal (strongly minimal) if every definable set (in
every elementary extension) is finite or cofinite. Equivalenlty,k
the set has Morley rank and degree 1. This condition can be
thought of as a particular way to eliminate the `there exists
infinitely many quantifier'; strong minimality implies there
exists infinitely many x phi(x,ybar) iff exists k x phi(x,y) where
k is chosen uniformly in y. (This observation is implicit in
Baldwin-Lachlan and was surely known to Morley.)

minimal = strongly minimal follows from T does not have the finite
cover property (which can be seen as `uniform elimination of the
Ramsey quantifier' -- see Baldwin-Kueker; the Ramsey quantifier
(R\xbar \phi(\xbar) holds if there is an infinite homogeneous set
for \phi ). The standard example of a minimal but not strongly
minimal set is an equivalence relation with one class of each
finite cardinality.

Model theorists generalize the field theoretic notion of algebraic
closure by saying a is in the algebraic closure of X in M if M
models phi(a,xbar) for some formula with parameters from X that
has only finitely many solutions.  It is easy to see that in a
strongly minimal set, algebraic closure has the properties of a
closure relation studied in linear algebra (or of transcendence
degree). (An account of generalized closure relations will be
another post.) Thus there is a natural notion of dimension,
invariance of basis number etc. and this implies that every
strongly minimal theory is categorical in all uncountable powers.

Listing the strongly minimal sets:

The standard strongly minimal models are (Z,S), vector spaces over
any fixed field, algebraically closed fields.  Zilber conjectured
that in some sense this was all.  The Hrushovski construction
refutes that conjecture.

Zilber proved that every strongly minimal set that is
aleph_0-categorical is essentially a vector space over a finite
field.  (The `essentially' hides some rather interesting technical
mathematics of Cherlin-Harrington-Lachlan, Ahlbrandt-Ziegler,
Pillay-Hodges which involves both cohomology and
well-quasi-orderings.)  This characterization was an essential
tool is show there is no finitely axiomatizable totally
categorical theory.  The characterization was also derived
(independently by Cherlin and Mills) from the classification of
finite simple groups.  Zilber's approach led to a new proof of
work on 2-transitive permutation groups.

Strong minimality and categoricity in power:

Baldwin-Lachlan showed that strongly minimal sets were the
building blocks of aleph one categorical theories in the sense
that every model  M of T is minimal prime over a strongly minimal
subset D. (I.e prime: every elementary  embedding of D into a
model N of T extends to an elementary  embedding of M into N and
mnimal: there is no submodel between D and M.)

Structures which are the algebraic closure of a strongly minimal
set are called almost strongly minimal.  The simplest example of
an aleph-one categorical theory which is not almost strongly
minimal is the direct sum of countably many copies of the cyclic
group of order 4.

Any two strongly minimal sets in an aleph-one categorical theory
are linked by a definable finite-finite relation.  This is a
special case of Shelah's notion of non-orthogonality. Zilber
discovered that in the aleph-one categorical setting it always
gave rise to `1inking groups'. These groups (which have a natural
action on the structure) are definable.  Moreover, in natural
cases the groups which arise natural. E.g. in algebraically closed
fields the linking groups are algebraic groups.

This led to the Cherlin-Zilber conjecture: every simple group of
finite Morley rank is algebraic over an algebraically closed
field. There is an enormous literature on this subject.  Much of
it       ( Borovik,Cherlin, Nesin, Tuna, Poizat, Atinel, Jaligot
....) attempts to prove a version of the classification of finite
simple groups with `finite rank' replacing `finite'.

Strong minimality and o-minimality:

Another way to express `strongly minimal' is to say that every
definable subset is definable in the language of pure equality.
(Definable always allows parameters.)  This is a nice property of
the complex field.  The real field has a nice analog. Every
definable subset is definable in the language of order.  This
notion is called o-minimality.

In some ways 0-minimality is better behaved.  In particular, if a
structure is o-minimal, so is every elementary extension.
Moreover, while the Zilber conjecture on types of strongly minimal
sets  was refuted, the corresponding result for o-minimal sets was
proved by Peterzil and Starchenko.  I.e. very roughly: every
o-minimal set is an expansion of a trivial set, an abelian group
or a real closed field: "A trichotomy theorem for o-minimal
structures", Proc. London Math. Soc. (3) 77 (1998), pp. 481-523.
Peterzil and Starchenko

Current work:

In his work establishing that no theory categorical in every
uncountable power is finitely axiomatizable, Zilber
`characterized' (almost) the strongly minimal sets categorical in
all powers and found that they were `natural' mathematical
objects.  This led to his conjecture that all strongly minimal
sets are natural.  Investigation of this conjecture has resulted
in relaxation of `strongly minimal' to `quasiminimal excellent'
and open questions about the meaning of natural.

The last sentence is expounded in my survey on quasiminimal
excellence and on Zilber's home page.

Other References:  Standard texts on stability theory, Hodges' or
Marker's Model Theory, Zilber's web page. There are a number of
expository papers on my web page
(www.math.uic.edu\~jbaldwin\model.html ) with bibliographies.

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