[FOM] Foundations and Model Theory III

[John T. Baldwin]

Friedman writes:

Of course, isn't even classical algebraic geometry full of non first 
order constructions? Or do these have interesting first order 
replacements in the hands of model theorists?

Baldwin writes

> Model
> theory studies both the comparisons between various systems (thus
> the stability hierarchy, o-minimality etc) and the analysis of
> particular systems - e.g. the intensive work on expansions of the
> real numbers or groups of finite Morley rank.

Friedman writes.

What does "systems" mean here? I don't think you just mean "first order 
systems in predicate calculus with equality". Even o-minimality cannot 
be reduced to a set of first order sentences.

Baldwin replies.
Model starts to be intersting with notion that are slightly beyond first 
order: omitting types, categoricity, saturation, homogeneity.  It is the 
interplay
between (first order) definability and these slightly more expressive 
notions that are central.

System here mean roughly theories (in the first order context).  We are 
now learning the imprortance of infinitary sentences, the logic of 
Banach spaces,
and even abstract elemenatary classes which go beyond the context of 
most work in the last quarter century.