[FOM] Foundations and Model Theory III
John T. Baldwin
jbaldwin at uic.edu
Wed Jul 9 22:08:29 EDT 2003
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.
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