[FOM] Model theory and foundations

John T. Baldwin jbaldwin at uic.edu
Tue Jul 22 14:05:32 EDT 2003

Alasdair Urquhart wrote:

>I'd like to addd a small observation to the debate
>between John Baldwin, Steve Simpson and Harvey Friedman.
>Nonstandard analysis is of clear foundational interest
>and relevance, and unequivocally has its roots in
>model theory.  
>A surprising thing about nonstandard analysis is that
>it exploits only very elementary model theory, 
>specifically compactness and ultraproducts.  This leads
>me to wonder just what foundational ideas we might
>not discover by exploiting the great sophistication
>of recent model theory.  
I think this is a half-truth.  Much of the work in nonstandard analysis 
depends only compactness and ultraproducts.
However, the use of `saturation principles' is formally stronger.  The 
intrdcution of the following paper is particulary usefuL

 On the strength of nonstandard analysis (with W. Henson), J. Symb. 
Logic, vol. 51 (1986), pp. 377-386.

The strength of nonstandard methods in arithmetic (with W. Henson and M. 
Kaufmann), J. Symb. Logic, 49 (1984), pp. 1039-1058.

It is also the case that recent work by Iovino makes a close link 
between intermediate stability theoretic analysis (Morley seqences, 
semidefinable types)
and serous Banach space theory -spreading models.  (A note on this is on 
my to-do list).


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