FOM: non-uniqueness of the hyperreals

[Jon Barwise]

Rick Sommers writes:
>
>Friedman's justification for his view is based on the fact that "the
>reals are unique" and "all nonstandard models of arithmetic are
>pathological. There is nothing unique about any of them."  Barwise
>raises the same issue (Fri, 7 Nov 1997 08:21:16); this is a standard
>criticism.

Harvey and I draw quite different morals from the non-uniqueness.  On my
view there is, or at least was for many, many years, an intuitive concept
of an infinitesimal, a positive number that is less than 1/N for N = 1, 2,
3,...  Robinson's work showed that this intuitive concept is consistent
with the standard assumptions about the real numbers by giving us a model
of it.

Indeed, Robinson gives us a whole family of non-isomorphic models of it.
That is the rub.  What are we to make of this multiplicity of models?  Do
they correspond to different concpetions?  If so, are some conceptions of
infinitesimal better than others?  How so?

I would suggest that the answer to the first two questions is "Yes".  The
core concpet is just the one I mentioned above.  But there is a richer
concept, corresponding to countable saturation.  The richer concept is more
useful than the core concept alone, as various applications have shown.
See, e.g. Keisler's AMS Memoir.  Perhaps there is no final unique concept
here, but a family of increasingly rich notions.  That seems to me a more
fruitful response to the phenomenon than to dismiss infinitesimals because
they are not unique.  We don't reject set theory simply because there is
not a unique conception.  As Godel has taught us, we keep looking for ever
richer conceptions, conceptions that might shed light on open questions.

Why should we reject infinitesimals?  Well, my previous message did suggest
one reason, namely, it makes us uneasy if we are not sure we are talking
about the same things when we do mathematics.  But that is one (not the
only) use of the axiomatic method.  You and I may have different
conceptions of some domain but as long as we can be explicit about our
assumptions and each of us agrees that our conception satisfies the axioms,
we can proceed, each interpreting the results in our own way.  In the case
of non-standard analysis, it seems the conception that involves countable
saturation is more fruitful and so is gaining the day.  Or at least that is
my impression as an outside observer.

Jon

p.s.  Thanks for all those who have written to me or to the list pointing
out that I was right the first time, that Godel did indeed consider
Robinson's work to provide an important foundation to the concept of
infinitesimal.