[FOM] Rigorous Foundations of Model Theory?

[John Baldwin]

On Mon, 5 Oct 2009, Rex Butler wrote:

> As an enthusiast of Foundations of Mathematics, I have yet to see a
> clear cut declaration of what exactly the rigorous foundations of
> Model Theory are.

As phrased, I think this is a non-issue.  Since Tarski, the basic notions 
of model theory are fully formalized in set theory.  Many authors write as 
mathematicians and don't bother to write what is obvious to them (this 
can be formalized in ZFC). Some, as 
Butler reports, are sloppy.

For clear examples, see Marker or Hodges or Chang-Keisler.


This set theory is usually ZFC.

There are some interesting works in two directions. In a 2009 JSL article 
Shelah reprove Morley's theorem without choice.  There is other work (e.g. 
Agatha Walczak-Typke) who have worked on model theory without choice. 
Choice is relevant both for calibrating compactness and for defining 
cardinals.

Many results in infinitary logic rely on some extensions of ZFC.  A 
surprizing amount can be done using only 2^{aleph_n} < 2^{aleph_(n+1)}. 
But stronger axioms including large cardinals appear in the literature.


John T. Baldwin
Emeritus Professor
Department of Mathematics, Statistics, and Computer
Science
jbaldwin at uic.edu
312-413-2149
Room 613 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607