[FOM] Foundations and Model Theory I.

[John T. Baldwin]

Harvey made a number of interesting comments on my last posting.  In a 
effort at user friendliness I am going to
give a series of responses to various of his points.

I.  Friedman writes:

I am of course quite interested in seeing model theory take a more 
foundational viewpoint, and in fact the explicit form of the upward 
Lowenheim Skolem theorem that I have been talking about is an example of 
model theory taking a more foundational viewpoint. However, you have 
taken the position that this is no longer model theory. So you must have 
a quite different view of what "foundational model theory" should look 
like.

Baldwin replies:  

A.  The explicit form of the upward Lowenheim Skolem theorem morphed 
into a model theoretic question. Characterize those theories which
interpret every finitely axiomatized theory.  This is a rather natural 
question that I have no idea of how to tackle.

B.  What I said was that the explicit representation of models (i.e what 
is their set theoretic structure) was not model theory but set theory.
Better would be to say some amalgam of model theory and set theory,.

Arguing against myself, I gave two examples of what I consider natural 
model theoretic questions that were answered using concrete representations
a) the Whitehead problem.  b) The proof that no L(Q) sentence (fo logic 
+ there exist uncountably many) can have exactly one model of power Aleph_1.
Despite the fact that Harvey publicized the last question when I first 
ask it, now he rejects both questions.  (Maybe I misled him by talking 
about the infinitary
aspect of b).

Friedman wrote later:
By the way, an underdeveloped subject you would reject as not model 
theory, is that of Borel algebraically closed fields, and Borel real 
closed fields. I.e., the uncountable ones that can be rather explicitly 
defined.


These are interesting subjects -- just not `model theory full stop'.
By the way I appreciate the result of Schmerl that Steve Simpson posted. 
 This perhaps emphasizes that this attempt to lay out tight boundaries 
on subjects
is not really a fruitful endeavor.