FOM: From philosophy to `applied' model theory

[Stephen G Simpson]

John Baldwin Wed, 21 Jun 2000 09:15:58 -0500 (CDT) writes:

 > In his thesis, Morley posed the question.  Can an aleph-1 categorical
 > theory be finitely axiomatizable?
 > 
 > This question has some philosophical content.

I am wondering, what exactly is the philosophical content of Morley's
question?  See below.

Baldwin continues:

 > Bill Tait put it, do we know all the ways a single sentence can
 > demand infinity: discrete linear order, dense linear; a pairing
 > function.  (I think these basically remain the only examples.)

This question attributed to Tait sounds as if it could be quite
interesting from a philosophical standpoint.  It reminds me of some
old results characterizing the prenex classes in pure predicate
calculus which contain sentences having only infinite models.  But I
don't remember a reference for this work.  Perhaps Church's book?
Anyway, so far as I remember, this old work went under the name
"axioms of infinity" and was well-motivated in terms of philosophical
issues concerning the role of infinity in f.o.m.  Tait's question
suggests a natural refinement: to classify all possible "axioms of
infinity".  Does anyone have a conjecture here?  Perhaps something
along the lines of: any finitely axiomatizable theory with only
infinite models must interpret one of a small finite number of such
theories.  Or, is this picture too simple?

But in any case, the assumption of aleph_1 categoricity appears to
make Morley's question much more specialized and less philosophical
than Tait's.  Such an assumption appears to remove most of the
f.o.m. content, because, as is well known, theories which arise in
f.o.m. -- theories such as ZFC, 2nd order arithmetic, PA, PRA, bounded
arithmetic, etc -- are necessarily very far away from being aleph_1
categorical.  Not only are such theories incomplete (because they are
subject to the G"odel incompleteness phenomenon), but they do not even
have any aleph_1 categorical completions.  And even Tarski's
first-order Euclidean geometry, although complete, is still very far
from being aleph_1 categorical.

Or, am I missing something?  Is Baldwin saying that the restriction to
aleph_1 categorical theories is somehow natural from the viewpoint of
the philosophical motivation of Tait's question, in terms of "axioms
of infinity"?  I don't see why, but ....

Of course I agree with Baldwin's other point, to the effect that
Morley's question has led to a lot of interesting work in pure model
theory, with interesting connections to group theory, etc.  From the
pure model theory standpoint, it is extremely impressive.

-- Steve