[FOM] Model theory and foundations

[Alasdair Urquhart]

John Baldwin is quite right to call attention
to the fact that saturation principles are central
in recent nonstandard analysis, and also to point
out the brilliant work of Ward Henson, Matt Kaufmann
and Jerry Keisler on the strength of nonstandard 
analysis.  In my own defence, I should say that 
saturation properties of ultrapowers are a standard
part of basic model theory.  But he is right
if by "theory of ultraproducts" you mean only
the part that depends on the original basic
theorem of Los.

Incidentally, the Henson/Keisler paper is truly 
marvellous in explaining why things like the 
construction of Loeb measures (that depend on
saturation properties) are so powerful.  The
system of nonstandard analysis with countable
saturation gives us the power of third order
arithmetic!  

Contrary to a universally accepted opinion, there
are results in areas like stochastic processes
that can ONLY be proved by nonstandard means,
and are unprovable by standard methods.  
This surprising fact is explained in the work
of Fajardo and Keisler.  

I am increasingly of the opinion that nonstandard
analysis is one of the big foundational discoveries
of the 20th century.  Model theorists need to promote
their own foundational successes.