[FOM] model theory culture

[Mitchell Spector]

Harvey Friedman wrote:
> http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/l1.pdf
>
> contains some very interesting and perceptive remarks by well known
> core mathematician Kazhdan concerning the experience mathematicians
> generally have with model theory.
>
> ...


One thing that struck me in this interesting set of notes was Kazhdan's remark:

"On the other hand, the Model theory is concentrated on gap between an abstract definition and a 
concrete construction.  Let T be a complete model [sic]. On the first glance one should not 
distinguish between different models of T, since all the results which are true in one model of T 
are true in any other model."

(It looks like "complete model" above should be "complete theory".  I'll proceed under the 
assumption that that's what was intended.)

To me this misses, or at least bypasses, a key point.  His statement that anything true in one model 
of T is true in all models of T only applies to *first-order* statements.


He does go on to say:
"One of main observations of the Model theory says that our decision to ignore the existence of 
differences between models is too hasty. Different models of complete theories are of different 
flavors and support different intuitions."

However, this makes it all sound very nebulous and subjective.  Different models of complete 
theories are usually different in specific ways that can be easily expressed, but not by first-order 
statements.



Maybe the thesis in the first sentence quoted above should be replaced by something like: "Model 
theory is concentrated on the gap between what is expressible in first-order logic about a structure 
and what is not."  That may not mesh as well with the overall philosophical point he's making about 
the concrete and the abstract in mathematics, but I think it's a more accurate description of model 
theory.

I should add that model theory also discusses logics other than first-order logic, but the same 
delineation as above applies: the tension is between properties that are expressible in whatever 
logic you are studying and properties that are not expressible in that way.

Mitchell Spector