John T. Baldwin jbaldwin at uic.edu
Tue Jul 1 09:09:41 EDT 2003

These are responses to some of Harvey's questions about the complexity 
of definable subsets of a strongly minimal set.

Friedman/ Baldwin earlier wrote.

>>> Also, I gather that T, a theory, is minimal (strongly minimal) if 
>>> every definable set (in every elementary extension) is finite or 
>>> cofinite.
>>> Obviously one can define a hierarchy here according to the 
>>> complexity of the definable set. Thus we have
>>> degree k minimal
>>> degree k strongly minimal
>> I think this is a misreading.  I mean definable subset of the model 
>> in the language of the model.  Complexity questions about the 
>> recursiveness of the
>> model raise an entirely different set of issues.
> I was not directly talking about the recursiveness of the model, but 
> rather the sizes of the finite sets involved as a function of the 
> definitions.
> For example, for algebraically closed fields, one has something like a 
> double exponential bound, I gather.
> In o-minimality, something I know better, there is the famous question 
> about whether one can have transexponential behavior in an o-minimal 
> expansion of the field of real numbers.

Some questions of this sort have been explored, especially in the 
totally categorical case in the Cherlin, Hrushshovski, Lachlan line.

>>> Do we in fact get a hierarchy? If we want, say, only degree 100, 
>>> then presumably there is no shortage of finitely axiomatizable 
>>> examples? And what happens if we study that?
> I mean that we weaken the notion of (strongly) minimal theory to apply 
> only to formulas with, say, a certain number of quantifiers. What 
> happens? Anything interesting? Are there examples that create a 
> hierarchy here? 

In general the quantifer complexity of  definable sets can be 
arbitrarily high (even among aleph_0 categorical almost strongly minimal 
sets.)  Holland for the
strongly minimal case and (Baldwin-Holland) for rank >1 have shown that 
the Hrushovski construction (of aleph_1 categorical theories gices model 
-every formual equivalent to an existential. ) This was some what 
surprising because one of the interesting features of the Hrushovski 
construction is that in general
(stable or infinite rank omega stable cases) it on gives every formula 
is equivalent to a Boolean combination of existentials.  Also the 
Baldwin Holland result in the
case of expanding a strongly minimal set requries a GEOMTERIC 
COMPLEXITY  hypothesis on the strongly minimal set.

In constrast, Goncharov, Harizanov, Laskowski, Lempp,
and McCoy \cite{5rec} show that if a strongly minimal set has
TRIVIAL geometry then an extension by naming (possibly) infinitely
many constants is model complete.

>>> Also, what can one say about effectiveness in the sense that, given 
>>> a definition, what is a bound on the size of the extension or 
>>> coextension, as a function of the complexity of the definition? In 
>>> the paradigm case of algebraically closed fields, this is a 
>>> manageable function. Otherwise?
You can do anything you want --- Just take the theory of a 1-1 bijective 
function and fix the size of cycles arbitrarily.  Then you can ask 
qeustions relative to
the complexity of the theory but ....


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