[FOM] nonstandard analysis
John T. Baldwin
jbaldwin at uic.edu
Tue Jul 29 22:05:03 EDT 2003
Friedman writes in a reply to Urquart:
What does finite nonstandard analysis look like? Obviously this is an
absurdity if taken literally. Perhaps one can use ideas surrounding
nonstandard analysis to create new interesting and important finite
mathematics.
end Friedman:
As an example of type 2 nonstandard analysis in Friedmans classification:
(Nonstandard analysis as a development in standard mathematics which
purports to simplify proofs in standard mathematics, replacing them with
simpler proofs in standard mathematics.)
I want to point to the exhaustive analysis of finite structure: Annals of Math Studies 152
Finite Structures with few types by Cherlin and Hrushovski.
I won't at this time attempt to summarise the book but point to page 11:
'We will make use of nonstardard terminology as a convenient way of dealing with `large' integers.
(See Fried and Jarden, Field Arithmetic)... The method is based on the idea of replacing the standard
model of set theory in which one usually works by a proper elementary extension, the "enlargement" in which
there are "new" (hence, infinite) integers.' Cherlin and Hrushovski
I write only to point out two examples of this usage; Friedman points out the iversion of Gromov's theorem
on groups of polynomial growth proved by Wilkie and Van den Dries. Benedikt uses the techniques described above routinely
in the study of database models. I am sure there are other examples.
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