[FOM] Nonstandard analysis
John T. Baldwin
jbaldwin at uic.edu
Wed Jul 30 16:50:07 EDT 2003
Friedman writes;
Before replying, let me first say that we should draw a distinction between
1. Nonstandard analysis.
2. Nonstandard methods (models).
Baldwin replies:
I agree with Fridman completely in making this distinction. Perhaps he more than he I make
my living in 2); I certainly make no pretense of understanding 1). So what is the difference
between nonstardard analysis and nonstandard methods.
I think a key point is that nonstandard analysis builds in `2nd order properties' by taking not just
a model but a least a fragment of the hierarchy of sets above that model -- then forming a nonstanddard model
of the entire edifice. This leads to a notion of `internal set' - which does not arise in more ordinary
model theoretic investigations. Now
Cherlin-Hrushovski page 10 again:
"Since the extension is elemetary, all notions of set theory continue to have meaning, and (more or
less) their usual properties. In particular, for any set S occuring in the enlargement there is an
associated collection of `all' subsets of S in the sense of the enlargement; this will not actually contain all
subsets of S in general, and those which are in fact present in the enlargement are called `internal'."
Having said that I do not deny that Friedman's suggestion that this may be only a manner of speaking. My understanding
of the Henson-Keisler paper was that the omega_1-saturation principle was where one properly stepped into
the realm of nonstandard analysis. (do we have nsa and NSA :-) )
However, a large part of the argument for nonstandard analysis is precisely: manner of speaking. That infinitessimals
provide an intuitive language for discussing calculus.
As I say, in the Cherlin-Hrushovski case I don't know. I do know that the Benedikt example is simply a way of
expressing compactness and we presented it in a more standard way in our paper:
Baldwin,J.T. and Benedikt,M." ,"Stability
theory, permutations of indiscernibles and Embedded finite models, Transactions of the American
Mathematical Society, 2000}
Returning to a still earlier Friedman classification: He distinguieshed
1. Nonstandard analysis as a way of providing a model for the informal
reasoning that was prominent among mathematicians after the calculus but
before the advent of epsilon-delta (or arguably predating the calculus).
2. Nonstandard analysis as a development in standard mathematics which
purports to simplify proofs in standard mathematics, replacing them with
simpler proofs in standard mathematics.
3. Nonstandard analysis as a certain approach(es) to certain specialized
branches of mathematics, replacing certain standard theorems in certain
settings in standard mathematics with alternative standard theorems in
alternative settings in standard mathematics.
I still oontend that Cherlin-Hrushovski, Gromov-Van Den Dries-Wilkie, Benedikt fall into category 2 --although
of course except in the case of Van Den Dries-Wilkie the standard theorem is new. And certainly
a key claim of `nonstandard analyis' as a manner of speaking is that the ease of expression allows
the discovery of theorems. Robinson's celebrated work on the Bernstein problem certainly fits into that
vein.
Harvey, could you look at statement 3) again. I thought I understood what you were saying on the first
two readings but I don't see it now.
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