FOM: power series
Dave Marker
marker at math.uic.edu
Wed Nov 12 21:49:41 EST 1997
A couple of comments on power series and nonstandard models.
- Suppose you are just looking for nonstandard elementary extensions
of the field of real numbers.
Vaughn Pratt suggests looking at the field of Laurent series over the
reals. This won't work as not every element will have a square root.
However the field of Puisex series will work.
This is the union of the Laurent series fields R(t^{1/n}) for
n=1..infinity. In the natural (only) order t is infinitesimal. (To argue
that this is a real closed field you can
appeal to Ax-Kochen--this is a henselian valued field with real
closed residue field and divisible value group--or argue ajoining i
gives an algebraically closed field--this is done say in Walker's
book on Algebraic Curves.)
-Of course this is only appropriate for nonstandard analysis of algebraic
functions. It is impossible to define a reasonable exponential as there is
no room for exp(1/t). Van den Dries, Macintyre and I have given a
reasonably natural construction of an elementary extension of
(R,+,x,<,exp) using power series (actually a limit of limits of power
series fields). This allows you to use some algebraic nonstandard
arguments to analize asymptotics of definable functions. We used this to
answer a problem of Hardy's on rates of growth of logarithmic-exponential
functions. (a preprint is available at http://www.math.uic.edu/~marker).
-All of these models have a reasonably natural notion of "nonstandard
integers". But they are only models of quantifier free induction
(indeed the square root of two is rational) and completely unsuitable
for the type of construction you usually want to do in nonstandard
analysis.
Dave Marker
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